Where Does the Traveling Twin Lose Time in the Twin Paradox?

[CKH]

Some thoughts about the twin paradox (with a new question/paradox at the end):

Beginners, like myself, when confronting the twin paradox often want to know where it is that the traveling twin loses so much time. A frustrated poster trying to explain the paradox posted this:

He's been given many answers and he rejects every one because he wants an answer that fits his preconceived notion that the traveling twin's clock must lose time at a particular point in the trip.​

Still, the beginner is persistent in wanting to know where these clocks differ along the trip and by how much. There is no single answer due to the relativity of simultaneity but there are some approaches.

To answer this question, one approach is to explain what each twin sees (e.g. with a powerful telescope) on the other's clock. In this case a Doppler effect occurs in the clocks readings due to relative motion. Thus this description of the relationship between the clocks has a complicating factor: the delays in the readings by each twin of the other's clock are changing due to the the changing distance between the twins.

But there is another way of viewing the relationship between the clocks.

In the simple twin paradox, one twin stays at home and the other travels to a distant destination and back again at a substantial fraction of the speed of light. We can assume that the traveling twin accelerates to a fixed speed almost instantaneously when he leaves, when he turns around and that he stops quickly when he returns home. The traveling twin moves with constant velocity on the long outbound and inbound legs of the journey.

With this scenario in mind, here's another clock comparison method:

Prior to the trip, a number of clocks are placed at intervals along the path that the traveler will take to the destination.

All of these clocks along the path to the destination are placed so that they are stationary wrt the home clock. That is, all these clocks are at rest in the same inertial frame with the home clock.

Using Einstein's clock synchronization method, we synchronize all of these clocks with the home clock.

The clocks along the traveler's path then represent "home clock time" in the home clock's inertial frame. Since this frame is inertial throughout the trip we know that the established synchronization of these clocks with the home clock persists throughout, in the home frame.

The traveler can read these distributed "home clocks" during his trip and compare them with the readings with his own clock. Then he has an answer to how and where he is losing time relative to the home clock. The comparison is illustrated in this diagram linked from http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters_2013_Jan_1/spacetime_tachyon/index.html (sorry for the size):

Copyright John D. Norton. January 2001, September 2002; July 2006; February 3, 2007; January 23, September 24, 2008; January 21, 2010; February 1, 2012.

In the diagram the rocket twin travels at 86.6% the speed of light resulting in a time dilation factor of 1/2. The straight vertical path is the home clock's path over time (the home clock's worldline). The dogleg is the traveler's worldline. Each is marked with the proper time in years on each of the twin's clocks. The horizontal planes show the positions of the home clocks (imagine the line of clocks) and of the traveler at different times in the home frame.

The travel reads the "home time" using the clocks placed along the plane. The traveler finds the following correspondence between home time and the time on the own clock (as shown in the diagram):

Clock readings in years
Home Traveler
0 0
2 1
4 2
6 3
8 4

When the traveler returns he can show the home twin the above record, so the home twin can know in his own frame how old the traveler was when he was a certain age.

There is no mystery about where these clocks get out of sync from this perspective. They do it uniformly along the entire trip.

Acceleration does not directly affect the clocks, it is the speed of the traveler and the time over which he travels with that speed that determines the age gap. The above clock comparison exactly reflects the time dilation of the traveling twin which is constant throughout the trip.

Note that the traveling twin could, instead of traveling far away, make small circles around home at the same speed for the same amount of time and the clock results would be the same. So it appears that it has little to do with how far away he travels.

Acceleration does not affect the age difference directly but the acceleration is necessary to follow any non-straight path through flat spacetime.

The twin paradox is "why can't we just reverse the role of the home and traveler in the diagram, such that the traveler is considered stationary and the home clock is considered to move, so that the traveler gets older instead"? We cannot because the traveler's path through spacetime cannot be made a straight line in any such diagram. It is a dogleg in all inertial frames, while the path of the home clock is straight in all inertial frames.

Caveats:

1) There is no unique way to compare the clock readings, some choice of inertial frame must be made.

2) The above correspondence of clock readings is meaningful only in the home frame. When the travel reads one of the distance home clocks, he cannot conclude that this is the "current" reading on the home clock in his own frame. Remote clocks have different readings in different frames.

3) This comparison of clocks is possible because space is flat in SR and the home clock remains at rest in the same inertial frame. In more general cases such simple comparison of clocks may not be possible.

4) Einstein has analyzed this paradox using the acceleration and distance of the traveler at turn around. He claimed that, during turn around, the home clock (at home) runs much faster than the travelers clock due to the distance and the degree of the traveler's acceleration.

Question:
At least superficially, 4) is in conflict with the claim that "acceleration does not affect clock rates". Has anyone seen an analysis that demonstrates an equivalence of the two approaches? Otherwise can someone clarify why acceleration does not affect relative clock rates, while Einstein appears to claim it does?

 

[mfb]

4) Einstein has analyzed this paradox using the acceleration and distance of the traveler at turn around. He claimed that, during turn around, the home clock (at home) runs much faster than the travelers clock due to the distance and the degree of the traveler's acceleration.

That depends on your way to track time. If you apply "simultaneous" to the frame of the traveling twin, and evaluate it at each moment during the acceleration, then you will note a huge impact of acceleration on the time difference defined in that way.
In the same way, you can win or lose one day relative to the Andromeda galaxy just by starting walking towards or away from it.

 

[CKH]

That depends on your way to track time. If you apply "simultaneous" to the frame of the traveling twin, and evaluate it at each moment during the acceleration, then you will note a huge impact of acceleration on the time difference defined in that way.

Yes, as I noted a couple of times in the post: all such clock comparisons are frame dependent. It is just another view of the situation. It has the benefit that the home clock's planes of equal time are fixed. Also, such a clock comparison can be directly observed, if you have a suitable spacecraft , plenty of clocks and plenty of time.

In the same way, you can win or lose one day relative to the Andromeda galaxy just by starting walking towards or away from it.

Yes in the walkers frame, but not in Andromeda's frame (the stationary one). The home clock view is simple. The traveler's view is not simple because he accelerates.

I'm interested in a response to the question at the bottom.

 

[Dale]

4) Einstein has analyzed this paradox using the acceleration and distance of the traveler at turn around. He claimed that, during turn around, the home clock (at home) runs much faster than the travelers clock due to the distance and the degree of the traveler's acceleration

Do you have a reference for this?

 

[PAllen]

Question:
At least superficially, 4) is in conflict with the claim that "acceleration does not affect clock rates". Has anyone seen an analysis that demonstrates an equivalence of the two approaches? Otherwise can someone clarify why acceleration does not affect relative clock rates, while Einstein appears to claim it does?

This mis-states Einstein's argument. He treated the effect as due to 'gravity of a special sort' - pseudo-gravity is what most would call it. The effect on clock rates of gravity or pseudo-gravity is not due to the acceleration it is due to potential difference. It happens that potential for such a field is acceleration times distance. Thus, again, the acceleration, per se, is not the issue. For example, if a twin accelerated at 10 g in +x for a microsecond, then 10 g in --x direction for a microsecond, and continued doing this for an hour, their clock time compared to a stationary twin would be negligibly smaller. Because the distance was never large, the potential difference was never large.

As to equivalence, it is easy to formally argue the equivalence of approaches. Einstein's approach is formally equivalent to integrating proper time along the two twin world lines in Fermi-Normal coordinates for the traveling twin. But this is just a coordinate transform from inertial cartesian coordinates (producing a more complex representation of the metric). Any invariant computed with the metric remains unchanged. The proper time between two crossing of world lines, along each, is such an invariant. Thus the two methods, in every case, must come out identically.

 

[A.T.]

4) Einstein has analyzed this paradox using the acceleration and distance of the traveler at turn around. He claimed that, during turn around, the home clock (at home) runs much faster than the travelers clock due to the distance and the degree of the traveler's acceleration.

Question:
At least superficially, 4) is in conflict with the claim that "acceleration does not affect clock rates".

No it isn't. The clock rate of the clock undergoing the proper acceleration is not affected by the proper acceleration, that is what the clock hypothesis says.

What 4 refers to, are variable clock rates at different positions in non-inertial coordinates. So it's not the proper acceleration, but position and properties of a arbitrary coordinate chart that define the distant clock rates. Using a different simultaneity convention might yield a different distant clock rate.

 

[Dale]

This mis-states Einstein's argument.

Do you have a reference for this? I don't know what you guys are talking about.

 

[PAllen]

Do you have a reference for this? I don't know what you guys are talking about.

Search for paradox to find this discussion in:

http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativity

 

[CKH]

This mis-states Einstein's argument. He treated the effect as due to 'gravity of a special sort' - pseudo-gravity is what most would call it.

I deliberately misstated it because I think that, as stated by Einstein, the appeal to the equivalence principle is circular. That is, the effects of pseudo gravity that he invokes are actually derived from the analysis of an accelerated frame using SR to begin with.

The effect on clock rates of gravity or pseudo-gravity is not due to the acceleration it is due to potential difference. It happens that potential for such a field is acceleration times distance. Thus, again, the acceleration, per se, is not the issue. For example, if a twin accelerated at 10 g in +x for a microsecond, then 10 g in --x direction for a microsecond, and continued doing this for an hour, their clock time compared to a stationary twin would be negligibly smaller. Because the distance was never large, the potential difference was never large.

Yes, but then isn't the effect on clock rates dependent on the product of acceleration and distance, so acceleration does matter? Give that, how can we make the broad statement that "acceleration has no effect on clock rates"? Perhaps it is always true for two clocks at a common event, but otherwise not true in general?

Einstein's approach is formally equivalent to integrating proper time along the two twin world lines in Fermi-Normal coordinates for the traveling twin. But this is just a coordinate transform from inertial cartesian coordinates (producing a more complex representation of the metric). Any invariant computed with the metric remains unchanged. The proper time between two crossing of world lines, along each, is such an invariant. Thus the two methods, in every case, must come out identically.

Thanks. It will likely be quite some time before I'm able to understand that.

When you look at the problem using "home time" (as in the OP), acceleration appears to have nothing to do with the clock differences.

When you look at the problem from the traveler's frame, you see that it is precisely the traveler's acceleration (and distance from home) that is causing the simultaneous values on the home clock to rapidly change at turn around.

(On the other hand the traveler's rapid acceleration when he leaves and rapid deceleration when he returns has very little effect because the distance from the home clock is small.)

 

[CKH]

No it isn't. The clock rate of the clock undergoing the proper acceleration is not affected by the proper acceleration, that is what the clock hypothesis says.

Then I think the statement of the clock hypothesis needs to precisely state in what sense the clock rate is unaffected. From the article Proper TIme in the wiki:

In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two events than that measured by a non-accelerated (inertial) clock between the same two events. The twin paradox is an example of this effect.

Hence it seems that the clock hypothesis should stipulate that the rates are the same at the same event.

Using a different simultaneity convention might yield a different distant clock rate.

Can you suggest a different simultaneity convention for the traveler?

 

[PAllen]

Yes, but then isn't the effect on clock rates dependent on the product of acceleration and distance, so acceleration does matter? Give that, how can we make the broad statement that "acceleration has no effect on clock rates"? Perhaps it is always true for two clocks at a common event, but otherwise not true in general?

In an inertial frame it has no effect. For coordinates in which a non-inertial trajectory is at coordinate rest, it depends on the way you set up the coordinates. There is no preferred way to set such coordinates (as there is for inertial coordinates). If you set them up in a particular way, you find clock rates are affected by position as well as speed. For the traveling twin as the origin of such coordinates, the home twin's clock runs faster due to it's position. Neither clock's rate is affected by coordinate acceleration. The traveling twin is not accelerating in these coordinates. The coordinate acceleration of the home twin plays no role in its clock rate (its speed does, though).

Thus, in neither coordinates does acceleration play any role at all clock rates.

Thanks. It will likely be quite some time before I'm able to understand that.

When you look at the problem using "home time" (as in the OP), acceleration appears to have nothing to do with the clock differences.

When you look at the problem from the traveler's frame, you see that it is precisely the traveler's acceleration (and distance from home) that is causing the simultaneous values on the home clock to rapidly change at turn around.

(On the other hand the traveler's rapid acceleration when he leaves and rapid deceleration when he returns has very little effect because the distance from the home clock is small.)

Not really, see above. The traveling twin is not accelerating at all in 'his' coordinates. The home twin's acceleration in these coordinates plays no part in his clock rate. You are mixing descriptions from different coordinates.

 

[PAllen]

Can you suggest a different simultaneity convention for the traveler?

Radar simultaneity, used by the traveling twin, would produce a very different description of the home twins clock rate over their two histories. There are infinitely many choices possible.

 

[PeterDonis]

Can you suggest a different simultaneity convention for the traveler?

You have already done so. Your entire OP is a description of how the traveler will interpret his observations using the simultaneity convention of the "home" frame. Your table of corresponding values of traveler clock time vs. home clock time only makes sense if the traveler is using the simultaneity convention of the home frame.

For example, when the traveler passes the home clock reading "2" and his own clock reads "1", if he concludes that he has "lost" 1 unit of time compared to the home twin, he is implicitly assuming that the reading of "2" on the home clock he just passed is simultaneous with the reading of "2" on the home twin's clock. But if he were to use the simultaneity convention of his own rest frame at that instant, that would not be the case; the reading of "2" on the home clock he just passed would be simultaneous with an event on the home twin's worldline where the home twin's clock read much less than "1".

 

[PhoebeLasa]

Radar simultaneity, used by the traveling twin, would produce a very different description of the home twins clock rate over their two histories. There are infinitely many choices possible.

But in the link that you posted above, it seemed clear that Einstein thought that there was only one simultaneity perspective for the traveler. He didn't seem to think that some kind of choice of simultaneity "convention" had to be made.

 

[PAllen]

But in the link that you posted above, it seemed clear that Einstein thought that there was only one simultaneity perspective for the traveler. He didn't seem to think that some kind of choice of simultaneity "convention" had to be made.

Well, that was 1918 in a non-technical presentation. Einstein had general covariance as a founding principle of GR, which means any coordinates are equally valid. Also, this is the ONLY place in his writings he analyzed the twin paradox this way.

 

[Khashishi]

I get the sense that you already know this, but the "loss of time" occurs globally and not at a specific point in space-time. This is more evident in the case of gravity, where you can have a difference in time between two participants without acceleration. If closed time-like curves exist, you can go back in time over the course of a journey, but there is no one point where you can say this is where/when you go back in time.

 

[CKH]

Your entire OP is a description of how the traveler will interpret his observations using the simultaneity convention of the "home" frame. Your table of corresponding values of traveler clock time vs. home clock time only makes sense if the traveler is using the simultaneity convention of the home frame.

Yes (that was explicitly stated multiple times). When the traveler reads a "remote home clock" along his trip, what he is allowed to say is "this is the current time on the home clock in the home clock's inertial frame". I think that interpretation is OK. So what we actually see in this case, is what the traveler's clock reads at any given time in the home frame.

In this perspective on the twins, we take advantage of the constancy of the home frame to say something meaningful about the corresponding times on the traveler's clock in the home frame.

 

[PeterDonis]

Yes (that was explicitly stated multiple times).

Then do you agree that this counts as an alternative simultaneity convention for the traveler? If not, why not? If so, it seems odd that you asked for people to suggest a different simultaneity convention for the traveler, when you had already given one; that's why I took the trouble to point it out.

 

[PAllen]

I get the sense that you already know this, but the "loss of time" occurs globally and not at a specific point in space-time. This is more evident in the case of gravity, where you can have a difference in time between two participants without acceleration. If closed time-like curves exist, you can go back in time over the course of a journey, but there is no one point where you can say this is where/when you go back in time.

Yes, I know well. I never liked the Einstein 1918 analysis (pseudo-gravity) as a way to understand the twin paradox. To me, Doppler is best to understand what your measure, and metric is best for more general understanding, in particular that it doesn't make any more sense to talk about 'where the clock difference is' than it is talk about which part of a longer line is the 'extra length'.

However, since Einstein's 1918 approach was brought in, I wanted to make sure it was accurately presented rather than misconstrued to say acceleration affects proper time (clock rates).

 

[Dale]

Search for paradox to find this discussion in:

http://en.wikisource.org/wiki/Dialog_about_objections_against_the_theory_of_relativity

Thanks that is helpful. I had read that previously, but forgotten about it. Personally, I don't consider this to be an actual analysis, just a rough description of a possible analysis. It is not surprising that reading such a description someone could be left confused.

None of the analysis is actually worked out to show 1) how the reference frame of the accelerating clock is defined and 2) how the gravitational potential in that frame leads to the correct value for time dilation. If that were done, then I think it would be clear that there is no conflict involved. Especially since any mathematical analysis must show that it is not the acceleration, but something else (e.g. the potential) which determines the time dilation.

 

[CKH]

Yes, I know well. I never liked the Einstein 1918 analysis (pseudo-gravity) as a way to understand the twin paradox.

Einstein's use of pseudo gravity is circular. You should be able to make the same argument as in the 1918 paper, substituting the traveler's "acceleration" for "pseudo-gravity". I'm interested in how that description works (if it works).

To me, Doppler is best to understand what your measure, and metric is best for more general understanding, in particular that it doesn't make any more sense to talk about 'where the clock difference is' than it is talk about which part of a longer line is the 'extra length'.

Yes the doppler method is always applicable.

It makes sense to talk about time differences as well as spatial distances whenever you specify the coordinates used. There are two worldlines with fixed relationships in spacetime. What they look like at the "same time" just depends on how you choose your slice that defines "same time". It's not like clock differences and distance differences are undefinable, it's just that they are contextual.

So rather than not talk about "a time difference" it all, we can ask what it looks like in different frames. (As you can with your preferred doppler method.) We're not surprised that we get different answers, we expect that.

Rethinking the statement "acceleration does not affects proper time (clock rates)", I guess it means this:

Regardless of the acceleration of two clocks, their respective rates depend only on their current relative velocity.​

 

[CKH]

Then do you agree that this counts as an alternative simultaneity convention for the traveler? If not, why not? If so, it seems odd that you asked for people to suggest a different simultaneity convention for the traveler, when you had already given one; that's why I took the trouble to point it out.

Bad me. I was fishing but caught the only the same fish as before (radar). I'm still struggling with this "conventionality".

Here's my problem with the idea of conventions. We have a convention for individual inertial frames that is not really a convention (is there any other choice)? Based on that, we can ask about non-inertial cases. It seems legitimate to claim that at every point on a (smooth) wordline, there is an MCIF that we can choose in which the origin is the point on the worldline and constant spatial orientation . We can always assign the proper time at that point on worldline to the MCIF. The MCIF so defined is unique and has a well-defined hyperplane of simultaneity.

Thus it appears that SR already defines a convention that is perfectly justified. If we have some other convention, it likely contradicts this "natural" choice. So what's confusing me is why this freedom of choice arises and how much freedom there is?

In the case of radar, in an inertial frame, it is the same as Einstein's convention. That would appear to be a constraint on any simultaneity convention. What other rules apply to inventing simultaneity conventions?

 

[CKH]

Especially since any mathematical analysis [like Einstein's] must show that it is not the acceleration, but something else (e.g. the potential) which determines the time dilation.

Is there potential without acceleration? How is potential acquired/determined (without the circular use of pseudo-gravity please)? Perhaps the so-called potential is nothing other than relative velocity in reality?

All we have is an inertial twin and a twin far away who is moving away at high speed and has just begun to (de)accelerate.

 

[PeterDonis]

We have a convention for individual inertial frames that is not really a convention (is there any other choice)?

Sure there is. Nothing requires us to use an inertial frame. Nothing requires us to use a frame at all. It is true that, if you want to use an inertial frame, and if you are in globally flat spacetime, and if you have picked a particular inertial worldline that you want to be "at rest" in your chosen frame, then you don't have a choice about how to construct the frame; it's defined for you by the choices I just listed. But nothing requires you to make those choices. You can describe the same physics using other frames, or not even using frames at all.

It seems legitimate to claim that at every point on a (smooth) wordline, there is an MCIF that we can choose in which the origin is the point on the worldline and constant spatial orientation

Yes (with some complications in a general curved spacetime about what "constant spatial orientation" means, but I don't think those are important for this discussion). But each such MCIF is, in general, a different inertial frame. You can't combine them to make a single "non-inertial frame". You can't combine them at all. You have to either pick one, or switch between them as desired, depending on what physics you are trying to describe.

We can always assign the proper time at that point on worldline to the MCIF. The MCIF so defined is unique and has a well-defined hyperplane of simultaneity.

Sure, in the sense that any inertial frame has a well-defined hyperplane of simultaneity. But again, you have already chosen to (a) use an inertial frame, (b) pick a particular worldline (the inertial worldline that happens to be tangent to your general smooth worldline at the chosen point) to be at rest in your inertial frame. (And, if you want the frame to be globally usable, you have to be in flat spacetime.) So you've just picked a particular way of uniquely specifying an inertial frame. You're free to do that, but nothing requires you to.

it appears that SR already defines a convention that is perfectly justified.

Justified once you've, once again, (a) chosen to use an inertial frame, and (b) chosen a particular inertial worldline that you want to be at rest in the frame. But nothing requires you to make those choices.

If we have some other convention, it likely contradicts this "natural" choice.

The choice is only "natural" given the prior choices I have described. Nothing requires you to make those prior choices.

what's confusing me is why this freedom of choice arises and how much freedom there is?

As much freedom as you want, as long as what you end up with is a valid simultaneity convention See below.

In the case of radar, in an inertial frame, it is the same as Einstein's convention.

Yes.

That would appear to be a constraint on any simultaneity convention.

No, it isn't. For example, the traveling twin in your scenario is perfectly free to pick the home twin's simultaneity convention, as I said in a previous post. That convention obviously does not match the Einstein convention for the traveling twin's rest frame.

What other rules apply to inventing simultaneity conventions?

The rules are simple (and they are not "other" rules, since, as above, the Einstein constraint is not a requirement):

(1) The convention must not assign two different times to the same event;

(2) The convention must not assign the same time to two events that are not spacelike separated.

(3) The convention must give a monotonic time ordering to events along any timelike or null curve, and the ordering must be oriented the same (i.e., it must pick the same direction as "future") along every such curve.

Any simultaneity convention that meets the above requirements is acceptable.

 

[A.T.]

Hence it seems that the clock hypothesis should stipulate that the rates are the same at the same event.

Or that the instantaneous rate of a clock is not affected by its instantaneous proper acceleration.

If a car goes at constant speed between two points, steering has an effect on the total distance accumulated, even if steering doesn't have any effect on the instantaneous rate the distance is accumulated at.

 

[pervect]

Bad me. I was fishing but caught the only the same fish as before (radar). I'm still struggling with this "conventionality".

Here's my problem with the idea of conventions. We have a convention for individual inertial frames that is not really a convention (is there any other choice)?

My stance on this is that you do have a choice in your definition of simultaneity if you use, for instance, Lagrangian methods, but you don't really have a choice if you want to use Newton's laws as they are taught in high school. Comments (especially how to use something less awkward and more precise than "as taught in high school) are welcome.

Based on that, we can ask about non-inertial cases. It seems legitimate to claim that at every point on a (smooth) wordline, there is an MCIF that we can choose in which the origin is the point on the worldline and constant spatial orientation . We can always assign the proper time at that point on worldline to the MCIF. The MCIF so defined is unique and has a well-defined hyperplane of simultaneity.

In the absence of gravity , i.e.in flat space time, this is true. A slightly modified version of this construct can be made to work in curved space-time, but it requires some adjustments to do so. There's even a name for this construct, it's called using Fermi Normal coordinates. I will use that term, or the abbreviation FNC, to describe this construct henceforth in this post. But there are a few important points to make about this as a coordinate system.

The first and most obvious point is that after you've stitched together different inertial frames into a unified coordinate system, Newton's laws no longer work properly, so the stitched together version (which I am again calling Fermi Normal coordinates, FNC for short) is not an inertial frame, even though you stitched these coordinates together from things that were inertial frames.

For instance, a body at rest is supposed to stay at rest, but if your reference observer is accelerating, and you use Fermi Normal coordinates a body at rest in an inertial frame of reference will not remain at rest in FNC.

So, if you use FNC, you can't use Newtons laws unmodified. Thus if one's motivation is to try to stick with physics as it's been taught in high school, this motivation has already been compromised. Also, it's not entirely clear to me how to define the correct pseudo-forces using what I've been calling "high school methods". There may be a way to do so, but for instance the simple prescription of applying a constant pseudo-force equal to the acceleration of the reference observer won't work (it may approximately work near the worldline of the observer, but it won't work in general for abitrary points). So this is also an obstacle to the application of the idea.

So one winds up having to make some modifications to physics as taught in high school to use FNC, extensions that are not obvioiusly well defined, or documented in textbooks, even in flat space-time. The problem becomes even more severe in non-flat space-time (which is unfamiliar territory to SR, but needed if one desires to move on to GR).

The second point is that FNC are well defined in a small region around the worldline of the reference observer, but may not assign unique coordinates outside that region. The problem comes in when the different hyperplanes of simultaneity uniquely defined at each point on the intersect at some point away from the worldline. At the point of intersection, one event can be assigned multiple time coordinates. Thus, the coordinates thus defined are not unique, the problem is serious enough that all the textbooks I've seen describe FNC as not covering all of space-time, but only some local region, the region where the hypersurfaces of simultaneity do not intersect.

Thus it appears that SR already defines a convention that is perfectly justified. If we have some other convention, it likely contradicts this "natural" choice. So what's confusing me is why this freedom of choice arises and how much freedom there is?

The conventions you suggest are indeed very useful in a local context, but they become very restricting as you attempt to move far away from the worldline, and ultimately become unsatisfactory. Probably the biggest issue is the second one, the fact that after the hypersurfaces intersect, the coordinates are not well defined because they are not unique.

That said, under the proper circumstances (near the worldline of the observer), you'll find quite few authors who talk about physics in terms of the FNC (Nikolic comes to mind, as do some sections of MTW), and I find them useful myself. But they aren't a panacea, or a universal solution to all problems. I hope I've explained some of the difficulties and drawbacks that they have.

If you look at papers like the radar coordinate D&G papers with the above in mind, you may see the motivation for not sticking to the FNC concept better if you keep the above difficulties in mind.

 

[DAC]

If you accept that whatever motion the space twin is subject to the Earth twin is the reciprocal of, both move equally relative to the other, therefore both age equally.

 

[Dale]

How is potential acquired/determined (without the circular use of pseudo-gravity please)?

The potential is acquired/determined from the time-time component of the metric: $g_{00}=2U-1$.

Since Einstein's description is not mathematically rigorous I am not clear about what "pseudo-gravity" is nor why you would consider it to be circular. However, there is certainly nothing circular about using the metric since it is always part of the mathematical structure of spacetime (pseudo-Riemannian manifold).

 

[Dale]

that was 1918 in a non-technical presentation.

This is a good point. This reference is essentially a pop-sci reference. So despite the fact that it is from Einstein, it is not really a reference suitable for PF.

I think we can go ahead and discuss it, while recognizing that it is on a "Brian Greene" level. When this reference is in conflict with the scientific literature then this reference is out.

 

[CKH]

Or that the instantaneous rate of a clock is not affected by its instantaneous proper acceleration.

Yes, my first statement with "at the same event" was too restrictive. I fixed it at the end of post #21 with a clearer statement.

If a car goes at constant speed between two points, steering has an effect on the total distance accumulated, even if steering doesn't have any effect on the instantaneous rate the distance is accumulated at.

Yes. But you can see how the statement "acceleration has no effect on the outcome" cannot be true without qualification, since in the absence of acceleration. the accumulated distance cannot be longer. Thus acceleration does have an effect. The effect that it has cannot be ascribed to instantaneous clock rates. The effect results not from the magnitude of acceleration but from the changes in velocities that result from acceleration. (I hope that doesn't sound like gobbledegook).

 

[PeterDonis]

If you accept that whatever motion the space twin is subject to the Earth twin is the reciprocal of, both move equally relative to the other, therefore both age equally.

Since they don't actually age equally, there is a flaw somewhere in your argument.

 

[CKH]

Nothing requires us to use an inertial frame.

If you want to apply SR (at least as I understand the theory) you cannot avoid reference to inertial frames, for those frames are the specific environments in which the laws of SR are stated (the Lorentz transformations). Any analysis that you make will ultimately depend on application of inertial frames and the Lorentz transformation.

Perhaps you know of another way of stating SR that is independent of inertial frames?

Nothing requires us to use a frame at all. It is true that, if you want to use an inertial frame, and if you are in globally flat spacetime, and if you have picked a particular inertial worldline that you want to be "at rest" in your chosen frame, then you don't have a choice about how to construct the frame; it's defined for you by the choices I just listed.

Can we keep GR out of the discussion? I understand that what we are talking about is a very special, ideal case of flat spacetime. I know you are trying to point out the limitations, but the context of the OP is SR. It would be helpful to stick with SR, while acknowledging the limits of applicability.

Yes, that's exactly the choice I must make, if I want to ask about appearances from e.g. the traveler's point of view.

Yes (with some complications in a general curved spacetime about what "constant spatial orientation" means, but I don't think those are important for this discussion).

Agreed. That's a much more difficult subject that for now I wish to avoid.

But each such MCIF is, in general, a different inertial frame. You can't combine them to make a single "non-inertial frame". You can't combine them at all. You have to either pick one, or switch between them as desired, depending on what physics you are trying to describe.

I don't understand this objection. Would you similarly reject the possibility of defining a valid continuous function whose value is the tangent vector to a curve at each point along the curve? Would you claim that the function cannot exist because the tangents "cannot be combined"? Would you claim that the normals to these tangents are invalid because they intersect? Would you claim that the path cannot be recovered correctly by integration of the tangent function?

Sure, in the sense that any inertial frame has a well-defined hyperplane of simultaneity. But again, you have already chosen to (a) use an inertial frame, (b) pick a particular worldline (the inertial worldline that happens to be tangent to your general smooth worldline at the chosen point) to be at rest in your inertial frame. (And, if you want the frame to be globally usable, you have to be in flat spacetime.) So you've just picked a particular way of uniquely specifying an inertial frame. You're free to do that, but nothing requires you to.

Justified once you've, once again, (a) chosen to use an inertial frame, and (b) chosen a particular inertial worldline that you want to be at rest in the frame. But nothing requires you to make those choices.

What requires me to do that is my goal which is to see events in spacetime from the perspective of an observer, the home twin or the traveling twin in this case. What other choice of coordinates do I have to meet that goal?

The rules are simple (and they are not "other" rules, since, as above, the Einstein constraint is not a requirement):

If the Einstein constraint is not a requirement then the convention may contradict it. When that happens, it seems to me that the original physical meaning of "simultaneous" has been abandoned.

(1) The convention must not assign two different times to the same event;

(2) The convention must not assign the same time to two events that are not spacelike separated.

(3) The convention must give a monotonic time ordering to events along any timelike or null curve, and the ordering must be oriented the same (i.e., it must pick the same direction as "future") along every such curve.

Any simultaneity convention that meets the above requirements is acceptable.

These are (to my uninitiated mind) just arbitrary rules. What is the physical motivation? Who made these rules and how are they justified?

It seems quite odd to me to have such strict rules and yet allow also allow them to be so loose that they may contradict the defined simultaneity for inertial frames. To put it another way, if I apply a convention of simultaneity that contradicts the definition given in SR when applied to an inertial frame, then that "simultaneity" is some other concept that should not be confused with the simultaneity as defined in SR.

 

[Dale]

Perhaps you know of another way of stating SR that is independent of inertial frames?

"Spacetime is a flat pseudo-Riemannian manifold with a (-,+,+,+) metric signature."

 

[PeterDonis]

Perhaps you know of another way of stating SR that is independent of inertial frames?

Sure. SR is these two statements: (1) spacetime is globally flat; (2) the laws of physics are Lorentz invariant. Neither of those statements requires inertial frames to be defined (though of course it's much easier and more intuitive to model them mathematically with inertial frames).

that's exactly the choice I must make, if I want to ask about appearances from e.g. the traveler's point of view.

No, it isn't, not if by "appearances" you mean the actual direct observations the traveler makes. The traveler cannot directly observe what time it is on the home twin's clock "right now". He can only directly observe light signals arriving at his worldline that came from the home twin. All of those direct observations can be described without using an inertial frame, or indeed without using any frame at all. The only reason the traveler would need to define a frame is if he insists on asking questions that have no uniquely determined answers, like "what time is it on the home twin's clock right now?" But there is no need to ask any such question in order to describe or predict direct observations.

I understand that questions like "what time is it on the home twin's clock right now?" are intuitively appealing; but that doesn't mean they have uniquely determined answers. Part of understanding relativity is understanding that some of your Newtonian intuitions about what concepts are meaningful or what questions have uniquely defined answers need to be discarded.

Would you similarly reject the possibility of defining a valid continuous function whose value is the tangent vector to a curve at each point along the curve?

No, but that in itself is not sufficient to combine MCIFs. MCIFs do not just cover points on the curve; they also cover points off the curve. Pervect's post in response to this gave a good explanation of the limitations of MCIFs and Fermi Normal Coordinates.

What requires me to do that is my goal which is to see events in spacetime from the perspective of an observer, the home twin or the traveling twin in this case. What other choice of coordinates do I have to meet that goal?

The goal itself is not well-defined, because you are assuming that "the perspective of an observer" has a unique definition for events not on the observer's worldline. It doesn't. The intuitive feeling that it does is one of those intuitions that you need to discard.

If the Einstein constraint is not a requirement then the convention may contradict it. When that happens, it seems to me that the original physical meaning of "simultaneous" has been abandoned.

"Simultaneous" does not have a unique physical meaning. The intuitive feeling that it does is another of those intuitions that you need to discard.

You may be misunderstanding Einstein's reason for introducing his simultaneity convention (which basically amounts to the "radar" convention, as you have correctly noted). He was not doing it to propose a unique physical meaning for simultaneity, not even a relative one. He was doing it to show that the obvious pre-relativistic meaning of simultaneity, as applied to light signals received by an observer from two events equidistant from him, when combined with the observed fact that light propagates at the same speed in all inertial frames, requires relativity of simultaneity--i.e., it requires that the Newtonian concept of an absolute physical meaning for simultaneity must be abandoned. That doesn't mean that "simultaneity is still real, but it's relative". It means simultaneity is not "real"; it's just a convention.

Who made these rules and how are they justified?

They are the rules that are necessary to have a mathematically valid coordinate chart. An inertial frame, as it is defined in SR, is just a particular kind of valid coordinate chart, which satisfies some additional constraints.

if I apply a convention of simultaneity that contradicts the definition given in SR when applied to an inertial frame, then that "simultaneity" is some other concept that should not be confused with the simultaneity as defined in SR.

SR does not "define" simultaneity in a particular way, because SR does not require you to use inertial frames. All you are really saying is that, if a convention of simultaneity is different from the one for an inertial frame, then a frame using that other convention can't be an inertial frame. You're right; it can't. That's just a fact about non-inertial frames.

You are confusing "SR" with "SR as applied using inertial frames". The two are not the same. The fact that SR was initially introduced using inertial frames, and that it is still widely taught in introductory courses using inertial frames, does not mean that inertial frames are required for SR. It just doesn't.

 

[CKH]

The potential is acquired/determined from the time-time component of the metric: $g_{00}=2U-1$.

Try that explanation on your grandmother. ;)

Since Einstein's description is not mathematically rigorous I am not clear about what "pseudo-gravity" is nor why you would consider it to be circular.

While it's true that he does not present the detailed calculations, a careful reading makes it obvious what he's trying to do.

I'm surprised that you don't see the circularity. Einstein argues that he can view the accelerating traveler as stationary in a uniform gravitational field (by applying the equivalence principle). He then applies the physical laws of a gravitational field (that presumably come from GR) to the problem .

The circularity is that these physical laws of a gravitation field were not independently derived. They were derived directly from the physics of uniform acceleration in SR. The physics of uniform acceleration were transported to apply to a gravitational field (by invoking the equivalence principle) in the first place. That is, the gravitational law is founded on uniform acceleration in SR.

So by invoking a gravitation field, Einstein has added nothing to the analysis already done using acceleration alone in SR. In other words, the impression that he is applying something new that comes from GR is an illusion. Perhaps he had a sly smile on his face when answering his critics in this way?

 

[PAllen]

Try that explanation on your grandmother. ;)
While it's true that he does not present the detailed calculations, a careful reading makes it obvious what he's trying to do.

I'm surprised that you don't see the circularity. Einstein argues that he can view the accelerating traveler as stationary in a uniform gravitational field (by applying the equivalence principle). He then applies the physical laws of a gravitational field (that presumably come from GR) to the problem .

The circularity is that these physical laws of a gravitation field were not independently derived. They were derived directly from the physics of uniform acceleration in SR. The physics of uniform acceleration were transported to apply to a gravitational field (by invoking the equivalence principle) in the first place. That is, the gravitational law is founded on uniform acceleration in SR.

So by invoking a gravitation field, Einstein has added nothing to the analysis already done using acceleration alone in SR. In other words, the impression that he is applying something new that comes from GR is an illusion. Perhaps he had a sly smile on his face when answering his critics in this way?

You are missing a subtlety of the history of GR. In the modern view, GR is the theory of curved spacetime, SR the theory of flat spacetime that (only) is locally true in GR. However, Einstein had a different way of looking at it, which (to the best of my knowledge) he never abandoned. In his view, the derivation of the properties of non-inertial frames in SR was part of GR, which also extended this to cover significant mass with curved spacetime. Thus, he is using the features of accelerated coordinates in SR displaying a position dependent potential (which you can see in the Rindler metric - to which I believe I referred you earlier), all derived for flat spacetime (no Einstein field equations of GR involved). Einstein just viewed this physics as special case of GR rather than part of SR.

Thus, with Einstein's packaging, there is no circularity.

It seems you have still not fully grappled with my posts #5 and #11. Especially the point that in the non-inertial coordinates in which the metric shows a potential, the traveling twin is not accelerating. It is the home twin that is accelerating in these coordinate, and that acceleration plays no role in the clock rate of the home twin.

 

[Dale]

Try that explanation on your grandmother. ;)

My grandmother is dead, but were she alive I am sure that I would be able to explain to her how to add 1 and divide by 2 to go from the metric to the potential.

While it's true that he does not present the detailed calculations, a careful reading makes it obvious what he's trying to do.

Then please present the detailed calculations that are so obvious to you. They are far from obvious to me.

I'm surprised that you don't see the circularity. Einstein argues that he can view the accelerating traveler as stationary in a uniform gravitational field (by applying the equivalence principle). He then applies the physical laws of a gravitational field (that presumably come from GR) to the problem .

The circularity is that these physical laws of a gravitation field were not independently derived. They were derived directly from the physics of uniform acceleration in SR. The physics of uniform acceleration were transported to apply to a gravitational field (by invoking the equivalence principle) in the first place. That is, the gravitational law is founded on uniform acceleration in SR.

That is a matter of historical happenstance, not logical circularity. There is no reason that you cannot logically start with GR and use it to derive results that apply to the special case of flat spacetime. In fact, logically it is more sound to start with a general principle and derive special cases than to start with a special case and then generalize (despite the fact that historically it rarely happens that way).

So by invoking a gravitation field, Einstein has added nothing to the analysis already done using acceleration alone in SR.

I agree completely on this.

 

[CKH]

Sure. SR is these two statements: (1) spacetime is globally flat; (2) the laws of physics are Lorentz invariant. Neither of those statements requires inertial frames to be defined (though of course it's much easier and more intuitive to model them mathematically with inertial frames).

1) Spacetime is globally flat.

That's a geometric statement about a mathematical abstraction called spacetime (see "block spacetime" for a debate about its physical reality). I'm guessing it would be described mathematically as a four dimensional manifold with 0 curvature throughout. How does this statement acquire a meaning in the physical world? Physics in SR is about the behavior of objects (while ignoring their gravitational properties).

Allow me to attempt to relate these mathematical abstractions to the physical world and replace that abstract description with a physical one.

Under the conditions that no forces act on objects, objects at mutual rest remain at mutual rest. Objects in motion move at uniform speed in straight lines. These statements appear to be equivalent to "each object remains at rest in an inertial frame". This is the physical equivalent to the abstract mathematical statement "spacetime is flat".

2) The laws of physics are Lorentz invariant.

Let's translate that statement into a more detailed one as suggested in the Lorentz Covariance wiki article which states:

An equation [a physical law in this case] is said to be Lorentz covariant if it can be written in terms of Lorentz covariant quantities (confusingly, some use the term "invariant" here). The key property of such equations is that if they hold in one inertial frame, then they hold in any inertial frame; this follows from the result that if all the components of a tensor vanish in one frame, they vanish in every frame. This condition is a requirement according to the principle of relativity, i.e., all non-gravitational laws must make the same predictions for identical experiments taking place at the same spacetime event in two different inertial frames of reference.

(I believe that this second statement is what distinguishes SR from Newton's theory.)

This is like a game of hide and seek. We can attempt to hide the inertial frames in some abstract mathematical statement. But when we dig into the physics behind the abstraction, we find that the inertial frames are still there.

I had the same difficulty in the thread Einstein says objects do not fall to the Earth? In that thread, a 4-vector equation for force, mass and acceleration was presented which includes a term which compensates for the fictitious forces that appear in any specific non-inertial or non-cartesian coordinate system. This is done in order to make the law "coordinate independent". I claim that the equation is dependent on the concept of inertial motion. No one wanted to admit that this "fictitious force" term is constructed relative to inertial motion, even though they stated that the term vanishes in all inertial frames. It was claimed that inertial frames or inertial motion or whatever you want to call the "absence of proper acceleration" was nowhere to be found in that equation. I do not understand that claim.

No, it isn't, not if by "appearances" you mean the actual direct observations the traveler makes. The traveler cannot directly observe what time it is on the home twin's clock "right now". He can only directly observe light signals arriving at his worldline that came from the home twin. All of those direct observations can be described without using an inertial frame, or indeed without using any frame at all. The only reason the traveler would need to define a frame is if he insists on asking questions that have no uniquely determined answers, like "what time is it on the home twin's clock right now?" But there is no need to ask any such question in order to describe or predict direct observations.

I can't help it if you don't like such questions. You seem to feel that such questions should be prohibited because the answers are coordinate dependent or because they only work in flat spacetime?

I agree with you that observations made at a single point in spacetime are very limited. In that case you can only speak of about signals or objects that intersect your worldline. You can use "radar" as means of measurement. It is a constrained method of observation but by no means the only way to make observations. So, must we dispense with a concept like an inertial frame in which time is defined at more than one position? When I look at my watch, would you argue that it says nothing about what time it is a block away? I think that's taking it a bit too far.

I understand that questions like "what time is it on the home twin's clock right now?" are intuitively appealing; but that doesn't mean they have uniquely determined answers. Part of understanding relativity is understanding that some of your Newtonian intuitions about what concepts are meaningful or what questions have uniquely defined answers need to be discarded.

If you really think I don't understand that, then I suspect you only skimmed the OP and just jumped to the conclusion that I was deluded and that I claimed uniqueness. All measurements are relative to something, right? There are no absolutes. There are invariants, but they too have conditions of measurement that are relative to something.

No, but that in itself is not sufficient to combine MCIFs. MCIFs do not just cover points on the curve; they also cover points off the curve.

So do tangents; do you have something against them? They answer questions like "which way am I going now"? "What points in space are normal to my current direction and how far away are they?" "If I had some clocks available, synchronized with my own in my momentary state of motion, what would they read someplace else?"

I suspect you don't like these questions because you believe that they mislead novices and you want to avoid that (which is of course a noble cause). Well, perhaps that is true, but the questions are not without meaning when properly stated. If you claim meaninglessness (for example with the reply "mu") to such carefully worded statements then you may also create confusion in the minds of novices who think they now require a zen master to convey to them some ineffable truth.

Pervect's post in response to this gave a good explanation of the limitations of MCIFs and Fermi Normal Coordinates.

I will get to it. Responding to all these objections is time consuming. However, I also must admit that I was a bit put off by pervect's repeated references to "high school" and "Newtonian physics". Along the way in these discussion, I am learning from experts like yourself, but it is not helpful to ask me to respond to specious complaints about carefully worded statements that apparently have not been carefully read.

The goal itself is not well-defined, because you are assuming that "the perspective of an observer" has a unique definition for events not on the observer's worldline. It doesn't. The intuitive feeling that it does is one of those intuitions that you need to discard.

"Simultaneous" does not have a unique physical meaning. The intuitive feeling that it does is another of those intuitions that you need to discard.

See above. This is not news to me. I did not speak of intuitions in my OP. I spoke of conditions and results in the context of those conditions. If you and other posters continue to deny that then it's very hard to communicate usefully.

In fairness to you, you wish to exclude my statements because you believe they may be misinterpreted (oddly even by myself). It may well be that other skimmers might draw incorrect conclusions, but there is nothing unusual about that problem.

You may be misunderstanding Einstein's reason for introducing his simultaneity convention (which basically amounts to the "radar" convention, as you have correctly noted). He was not doing it to propose a unique physical meaning for simultaneity, not even a relative one. He was doing it to show that the obvious pre-relativistic meaning of simultaneity, as applied to light signals received by an observer from two events equidistant from him, when combined with the observed fact that light propagates at the same speed in all inertial frames, requires relativity of simultaneity--i.e., it requires that the Newtonian concept of an absolute physical meaning for simultaneity must be abandoned. That doesn't mean that "simultaneity is still real, but it's relative". It means simultaneity is not "real"; it's just a convention.

Perhaps, but I think you are conjecturing what Einstein intended in the context of mathematical approaches developed long after 1906 (that is an anachronism). My own interpretation is he established a non-local definition of time (as well as distance, which he does not explicitly define) for an inertial frame. You believe he did so only to appease those stuck on Newtonian notions? Did he not use these very constructs (inertial frames with their coordinates of time and space) to derive the Lorentz transformations from the PoR and the constancy of the speed of light?

Peter, before you lecture me about these fact of non-uniqueness could you do me the courtesy of actually reading my OP? I think you will find that every time I mentioned "home time" it was qualified with the frame that defines it.

They are the rules that are necessary to have a mathematically valid coordinate chart. An inertial frame, as it is defined in SR, is just a particular kind of valid coordinate chart, which satisfies some additional constraints.

Perhaps you mean "mathematically tractable"? The fact that different inertial frames assign different coordinates to the same events is not "un-mathematical".

SR does not "define" simultaneity in a particular way, because SR does not require you to use inertial frames. All you are really saying is that, if a convention of simultaneity is different from the one for an inertial frame, then a frame using that other convention can't be an inertial frame. You're right; it can't. That's just a fact about non-inertial frames.

But you appear to insist that we cannot find instantaneous inertial frames in arbitrary motion and work with them mathematically. I have no idea why you want to insist on that. In SR, how do you determine that rate of a clock moving in a circle relative to a stationary clock at the center? How do you add up those differences around the circle?

You are confusing "SR" with "SR as applied using inertial frames". The two are not the same. The fact that SR was initially introduced using inertial frames, and that it is still widely taught in introductory courses using inertial frames, does not mean that inertial frames are required for SR. It just doesn't.

Here again, I feel that you attempt to hide the centrality of inertial frames to SR. You claim that this concept is somehow transcended in a more lofty mathematical formulation, but you haven't convinced me, yet. You need only look at the foundation of the mathematics to see that the concept is still there (see above).

Your resistance to acknowledging carefully worded conclusions doesn't help my understanding and it may confuse others as well.

I am indeed appreciative of the time and work it requires of you and others to respond. But please don't put statements in my mouth that I did not make and then declare me wrong or misguided.

 

[CKH]

I agree completely on this.

Well, then you understand why it is circular. GR does not derive by some special magic what happens in a uniform gravitational field. It comes directly from SR and requires no knowledge about gravity itself aside from the belief in the equivalence principle.

 

[Dale]

Well, then you understand why it is circular.

Not at all. It is completely unnecessary to bring in GR and I agree that it "adds nothing" to the SR analysis to talk about, but it is not circular.

GR does not derive by some special magic what happens in a uniform gravitational field. It comes directly from SR

I think that you have the logic here exactly backwards. From a logical standpoint it is actually impossible to derive GR from SR. GR is not a derivation of SR, it is a generalization of SR. Do you understand the difference? You can derive a special-case from a general-case (GR logically implies SR), but you simply cannot logically derive a general-case from a special-case (SR does not logically imply GR).

It is clear that historically SR came first. However, it seems that you are being distracted by the historical order of the development and mistaking that for a logical dependency. A good special-case (e.g. SR) and some reasonable intuition (e.g. the equivalence principle) can allow a good guess at a useful generalization (e.g. GR), and this is historically what happened. But intuition is not derivation and historical order is not logical inference.

GR is not logically derivable from SR, so it is not circular logic for Einstein to analyze a scenario in flat spacetime using GR and his gravitational fields. It is just unnecessary (it "adds nothing" as you said and to which I agreed).

 

[PeterDonis]

How does this statement acquire a meaning in the physical world?

Spacetime curvature is the same thing as tidal gravity. If spacetime is flat, with zero curvature, that means no tidal gravity is present anywhere. Of course the real universe we live in does not have that property; SR, as a theory, does not describe our actual world in terms of global properties.

Physics in SR is about the behavior of objects (while ignoring their gravitational properties).

And, as I just noted, SR does not actually apply to the actual universe, because there is tidal gravity present in our actual universe, so the behavior of objects does not exactly match the predictions of SR. (Note that we can detect the presence of tidal gravity by making measurements on very small objects whose gravity is negligible; attributing the presence of tidal gravity to the presence of gravitating masses is separate, conceptually speaking, from detecting the presence of tidal gravity itself.)

Under the conditions that no forces act on objects, objects at mutual rest remain at mutual rest.

This is ok except for the term "forces"; a better way of stating it would be "under the conditions that objects have zero proper acceleration". With that stipulation, yes, this is equivalent to stating that there is zero tidal gravity, and therefore zero spacetime curvature.

We can attempt to hide the inertial frames in some abstract mathematical statement. But when we dig into the physics behind the abstraction, we find that the inertial frames are still there.

But "inertial frame" in the quote you gave has a subtly different meaning than the one we've been using up to now. Note that the quote specifies measurements made at a single spacetime event. So the two different "inertial frames" being used to describe measurements at that spacetime event don't have to cover all of spacetime; they only have to cover an infinitesimal region of spacetime around the chosen event (enough to define derivatives of quantities at that event).

In other words, the "inertial frame" here is really what is called in GR a "local inertial frame"--by definition it only covers a small patch of spacetime. So the claim that the laws of physics look the same in all inertial frames is a much weaker statement here than it would be if "inertial frame" had its usual meaning (which we have been using up to now) of a global inertial frame, covering all of spacetime. And a better way of saying what I have been saying is that SR does not require global inertial frames. It does require local inertial frames, or something equivalent, in order to make sense of the concept of Lorentz invariance. But, as I just noted, that is a much weaker requirement.

No one wanted to admit that this "fictitious force" term is constructed relative to inertial motion

That's because it isn't. It arises out of perfectly general terms in tensor equations that are written without making any assumptions about the state of motion. The fact that those terms happen to vanish for a coordinate chart constructed in a particular way in a particular spacetime--the kind of chart that defines a global inertial frame in flat spacetime--does not mean those terms require the concept of inertial motion for their definition.

I suggest taking some time to get familiar with differential geometry. I learned it from the section in MTW on differential geometry, which may not be the best reference; Carroll's online lecture notes also cover it.

You seem to feel that such questions should be prohibited because the answers are coordinate dependent or because they only work in flat spacetime?

I have never said such questions should be prohibited. I have only said you should not expect the answers to mean something they don't mean. If you're okay with that, ask away. But when you talk as though there is some preferred definition of simultaneity, for example, based on inertial frames, you are attributing a meaning to the answers to those questions that is simply not there. There is no preferred definition of simultaneity; there just isn't. If asking those questions and getting answers to them makes you think there is, then you need to either stop asking the questions, or stop attributing a meaning to the answers that they don't have.

must we dispense with a concept like an inertial frame in which time is defined at more than one position?

I have never said we must dispense with the concept of an inertial frame. I have only said you should not attribute a meaning to it that it doesn't have. See above.

So do tangents

No, they don't. Tangent vectors only "cover" a single point. This is another area where you need to learn some differential geometry; learning it will show you why the concept of "vector" you may be used to, where a vector is an arrow going from one point in space (or spacetime) to another, doesn't work, and needs to be replaced with the concept of "tangent vector", which is only "attached" to a single point in spacetime. (More precisely, at each point in spacetime, there is something called the "tangent space", and tangent vectors--and all other vectors, tensors, and geometric objects used in the math of differential geometry--are defined in the tangent space.)

They answer questions like "which way am I going now"? "What points in space are normal to my current direction and how far away are they?" "If I had some clocks available, synchronized with my own in my momentary state of motion, what would they read someplace else?"

The first question is answered by the tangent vector to your worldline, yes.

The second is not answered by a tangent vector by itself. There are a couple of different ways to answer it using differential geometry, but a tangent vector alone is not enough.

The third is also not answered by a tangent vector by itself. You need a synchronization convention. The Einstein convention is one possible one, but not the only one.

Once again, I strongly recommend taking some time to learn differential geometry.

I did not speak of intuitions in my OP. I spoke of conditions and results in the context of those conditions.

But you keep on talking as if you think those conditions are somehow privileged or preferred. You keep on talking as if the Einstein synchronization convention, and the other machinery that defines an inertial frame, are somehow privileged or preferred. They aren't.

before you lecture me about these fact of non-uniqueness could you do me the courtesy of actually reading my OP? I think you will find that every time I mentioned "home time" it was qualified with the frame that defines it.

That's not the kind of non-uniqueness I'm talking about. I know that you know that different inertial frames have different simultaneity conventions, based on the Einstein simultaneity definition. What I'm not sure you understand is that there is nothing that requires an observer, even if he is moving inertially, to use the Einstein simultaneity definition. Using that definition is a choice--a convention. There is nothing in physics that requires it. Two observers both moving inertially, and both at rest relative to each other, and both in flat spacetime, could perfectly well choose different simultaneity definitions; and as long as they both constructed valid coordinate charts based on their respective definitions, they could both make correct physical predictions.

Perhaps you mean "mathematically tractable"?

No, I meant exactly what I said.

The fact that different inertial frames assign different coordinates to the same events is not "un-mathematical".

Different inertial frames are different coordinate charts. The rules I gave apply to a single coordinate chart. Different coordinate charts give different descriptions of spacetime and what happens in it; the rules I gave are what is required for a single description to be valid.

you appear to insist that we cannot find instantaneous inertial frames in arbitrary motion and work with them mathematically.

I have never said that. I have only said that you can't combine multiple "instantaneous inertial frames" along a non-inertial worldline into a single consistent "frame".

In SR, how do you determine that rate of a clock moving in a circle relative to a stationary clock at the center?

First you need to decide what "rate" means. If it means "rate in the inertial frame in which the clock at the center is at rest", then it's easy. If it means something else, you need to decide what. For example, the two clocks could exchange light signals and use the round-trip travel times to determine their rates.

How do you add up those differences around the circle?

You integrate the rate of the clock moving in a circle (determined based on how you define "rate", as above) along its worldline.

You need only look at the foundation of the mathematics to see that the concept is still there (see above).

See my response above.

please don't put statements in my mouth that I did not make and then declare me wrong or misguided.

I'm not sure what statements you think I've misattributed to you. Are you saying that inertial frames are central to SR (which you just said in your post), but are not "required" for SR (which is how I worded your claim in the quote you gave)? If that's your position, it seems odd.

 

[CKH]

However, it seems that you are being distracted by the historical order of the development and mistaking that for a logical dependency. A good special-case (e.g. SR) and some reasonable intuition (e.g. the equivalence principle) can allow a good guess at a useful generalization (e.g. GR), and this is historically what happened. But intuition is not derivation and historical order is not logical inference.

If indeed GR is an independent theory (from SR), then you should not require the equivalence principle to arrive at GR. Is it true the the equivalence principle is unnecessary for GR? I'm asking because I haven't actually followed any derivation of GR (the math is still difficult for me). So, I was under the perhaps false impression that the equivalence principle is the central justification (the bridge that allows us beginning with SR to discover how gravity behaves in general).

It's possible however that the equivalence principle is actually just "window dressing" while GR has an origin in which that principle is entirely absent and unnecessary, but some other principles are used instead.

What do you think? Is GR justified (derived) is some completely independent way (how?) but after the fact, we notice that the equivalence principle happens to be consistent with this independently derived theory?

 

[PAllen]

If indeed GR is an independent theory (from SR), then you should not require the equivalence principle to arrive at GR. Is it true the the equivalence principle is unnecessary for GR? I'm asking because I haven't actually followed any derivation of GR (the math is still difficult for me). So, I was under the perhaps false impression that the equivalence principle is the central justification (the bridge that allows us beginning with SR to discover how gravity behaves in general).

It's possible however that the equivalence principle is actually just "window dressing" while GR has an origin in which that principle is entirely absent and unnecessary, but some other principles are used instead.

What do you think? Is GR justified (derived) is some completely independent way (how?) but after the fact, we notice that the equivalence principle happens to be consistent with this independently derived theory?

One arrives at a theory via heuristics, guesses, etc. (hopefully good ones!). Once you have a theory, you derive consequences, and the initial heuristics become irrelevant - unless they are also formally derived consequences.

The mathematical statement of GR nowhere has the principle of equivalence. However, it is a derivable approximate, local, consequence. Given the caveats, some highly esteemed GR experts (e.g. J. L. Synge), argued it should be abandoned, because as formal, mathematical statement it is simply false everywhere in GR (due the the approximate, local nature). Most physicists find the principle of equivalence remains a highly useful guide to intuition in GR, but one that always must be used with care, and no derivation based on it can be relied on without some additional formal validation.

 

[PeterDonis]

Is it true the the equivalence principle is unnecessary for GR?

Yes. There are actually multiple ways of arriving at GR from a simple set of starting assumptions; MTW lists six of them. The first two, which are the methods that Einstein and Hilbert, respectively, used in 1915 to obtain the Einstein Field Equation, are:

(1) Use automatic conservation of the "source" of gravity as the key requirement. This involves finding a tensor to describe gravity that is constructed from the metric and its derivatives and whose covariant divergence is identically zero; this tensor turns out to be the Einstein tensor, $G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R$. Then the field equation for gravity is obtained by simply setting this tensor equal to the tensor describing the source (the stress-energy tensor), times a proportionality constant whose value can be fixed by looking at the weak-field, slow-motion Newtonian limit.

(2) Use the principle of least action. This involves constructing a Lagrangian for gravity, and then using standard variational methods to find the field equation corresponding to that Lagrangian, which turns out to be the Einstein Field Equation.

Notice that the EP does not appear anywhere in the above. Notice also that neither of the above methods look anything like "start with SR and add gravity". Neither do three of the other four methods MTW lists.

The fifth method MTW lists is the only one that involves anything like the EP or starting with SR and adding gravity; it is to start with the field theory of a massless, spin-2 field in flat spacetime, and notice that such a theory makes correct predictions for gravity in the weak-field, slow-motion Newtonian limit. However, this theory is not self-consistent, and there turns out to be no way to make it self-consistent without making the "background" flat spacetime that we started with completely unobservable--all objects move as though they were in the curved spacetime obtained from the full metric tensor $g_{\mu \nu}$ that includes all the contributions from the spin-2 field (which is nonlinear because it interacts with itself, so there are an infinite series of contributions, and it took some very smart physicists about a decade to figure out how to extract a self-contained answer from all this). The fact that all objects move in this curved spacetime can be viewed as a manifestation of the EP--all objects see the same (curved) spacetime geometry, regardless of their mass or composition.

The problem with viewing the above as showing that the EP is "required" for GR, or that GR can be derived by taking SR and adding gravity, is twofold. First, as noted, there are multiple ways of deriving the Einstein Field Equation, and the others do not require the EP or taking SR and adding gravity. Second, if you interpret the above spin-2 field method as "SR plus gravity", then you are implicitly limiting yourself to curved spacetimes that have the same topology as flat Minkowski spacetime. Unfortunately, that excludes most of the key solutions to the EFE that are extensively used in GR, including all the black hole solutions and the FRW solutions used in cosmology. So the best that the spin-2 field approach can give us is a way of understanding, at least heuristically, how gravity might arise locally from some underlying spin-2 field; it can't give us the full range of predictive power that GR can.

 

[CKH]

Spacetime curvature is the same thing as tidal gravity. If spacetime is flat, with zero curvature, that means no tidal gravity is present anywhere. Of course the real universe we live in does not have that property; SR, as a theory, does not describe our actual world in terms of global properties.

Which is of course the same as "objects at rest remain at rest" etc. in ordinary language (of physics). It is nothing more. The statement "spacetime is flat" is couched in a more sophisticated geometric language which has been found useful as a mathematical tool in GR. Manifolds, curvature, tensors etc. are concepts are not needed in SR. No one needs to speak of gravity in SR.

Agreed, SR does not describe "our actual world in terms of global properties". We think GR does, but not with absolute certainty. However, SR is a pretty good approximation for some purposes.

I conjecture that your emphasis on GR in a discussion about SR arises from your concern that novices may misapply it to GR. That's OK, but why keep repeating it and why apparently contradict statements in the OP on that basis? Why not just accept them for what they are by acknowledging correctness under the conditions repeatedly specified? Or , show these conclusions to be actually wrong under the conditions specified.

In the OP, we have something more than an abstract mathematical argument. We have one in which physical clocks have been placed and synchronized in a well-defined manner. We can then make conclusions about the coincident (same event) readings on both physical clocks. If you are saying that those conclusion are wrong, then it is almost certainly myself who is wrong because you are an expert. In that case I need your help to fix my error.

And, as I just noted, SR does not actually apply to the actual universe, because there is tidal gravity present in our actual universe, so the behavior of objects does not exactly match the predictions of SR.

Right, objects at rest do not remain at rest. You don't even need to know what gravity is or a tide is.

(Note that we can detect the presence of tidal gravity by making measurements on very small objects whose gravity is negligible; attributing the presence of tidal gravity to the presence of gravitating masses is separate, conceptually speaking, from detecting the presence of tidal gravity itself.)

Well yes, but if we are interested in a physical cause then we need to explain this through the presence of gravitating masses.

This is ok except for the term "forces"; a better way of stating it would be "under the conditions that objects have zero proper acceleration". With that stipulation, yes, this is equivalent to stating that there is zero tidal gravity, and therefore zero spacetime curvature.

I'm OK with that, but I hope you also see that "objects at rest stay at rest..." is equivalent to the geometric expression "zero spacetime curvature". The former is a physical statement, the latter is an abstract mathematical statement that needs some "translation" to become physically meaningful.

But "inertial frame" in the quote you gave has a subtly different meaning than the one we've been using up to now. Note that the quote specifies measurements made at a single spacetime event. So the two different "inertial frames" being used to describe measurements at that spacetime event don't have to cover all of spacetime; they only have to cover an infinitesimal region of spacetime around the chosen event (enough to define derivatives of quantities at that event).

In other words, the "inertial frame" here is really what is called in GR a "local inertial frame"--by definition it only covers a small patch of spacetime. So the claim that the laws of physics look the same in all inertial frames is a much weaker statement here than it would be if "inertial frame" had its usual meaning (which we have been using up to now) of a global inertial frame, covering all of spacetime. And a better way of saying what I have been saying is that SR does not require global inertial frames. It does require local inertial frames, or something equivalent, in order to make sense of the concept of Lorentz invariance. But, as I just noted, that is a much weaker requirement.

Agreed. In the terminology used in physics, an "inertial frame" has global extent in time and space. The concept (if there is one so-called) of "inertial motion" is not global. If "local inertial frame" is the correct terminology for "a local cartesian coordinate system in which proper acceleration is zero", then that term is usually preferable for the sake of generality and applicability in GR as well as SR. However, in SR we can always speak of the global concept "inertial frame" which is otherwise a term of very limited applicability.

That's because it isn't. It arises out of perfectly general terms in tensor equations that are written without making any assumptions about the state of motion. The fact that those terms happen to vanish for a coordinate chart constructed in a particular way in a particular spacetime--the kind of chart that defines a global inertial frame in flat spacetime--does not mean those terms require the concept of inertial motion for their definition.

You still disagree that the "fictitious force" term has anything to do with "inertial motion"? How do you calculate or measure the coefficients in that term without reference to some local cartesian coordinates with "0 proper acceleration" (i.e. local inertial motion)? If you allow the coefficients to be computed or measured in a local coordinate system which is non-inertial or non-cartesian, the law will be wrong.

So if you want to insist, that no concept of inertial motion exists in that equation, then tell me how you physically measure the coefficients with no such reference.

I suggest taking some time to get familiar with differential geometry. I learned it from the section in MTW on differential geometry, which may not be the best reference; Carroll's online lecture notes also cover it.

I will, just that it takes time and perhaps I waste far too much time arguing against misinterpretations of my own statements.

I have never said such questions should be prohibited. I have only said you should not expect the answers to mean something they don't mean. If you're okay with that, ask away. But when you talk as though there is some preferred definition of simultaneity, for example, based on inertial frames, you are attributing a meaning to the answers to those questions that is simply not there. There is no preferred definition of simultaneity; there just isn't. If asking those questions and getting answers to them makes you think there is, then you need to either stop asking the questions, or stop attributing a meaning to the answers that they don't have.

This is not so mysterious as you make it sound. You make it sound as if "simultaneity" (more generally an interval of time) is some especially slippery undefinable measurement with the implication that the concept is best abandoned.

So let me make an analogy. Suppose there are two trees in the distance and I measure the angle between them. Would you so vigorously complain that that the angle is meaningless because if you measure it from another point it is different? There is no prefered definition of the angle between the trees, but that angle is perfectly meaningful given it's conditions of measurement. We could not survey land without the belief that these angles have meaning. Of course, in a (small) survey we take for granted the "flatness of space".

In the OP, the conditions have not been left to implication. The condition of SR is explicitly stated and the use of inertial frames to make measurements is explicitly stated.

I have never said we must dispense with the concept of an inertial frame. I have only said you should not attribute a meaning to it that it doesn't have. See above.

Do you mean by this that "inertial frame" implies "global inertial frame" which is different from "local inertial frame"? What meaning have I attributed to "inertial frame" that it doesn't have, globalness? If so, perhaps I misused the word. and should have said "local inertial frame". Does that completely throw you off the rails of comprehension?

No, they don't. Tangent vectors only "cover" a single point. This is another area where you need to learn some differential geometry; learning it will show you why the concept of "vector" you may be used to, where a vector is an arrow going from one point in space (or spacetime) to another, doesn't work, and needs to be replaced with the concept of "tangent vector", which is only "attached" to a single point in spacetime. (More precisely, at each point in spacetime, there is something called the "tangent space", and tangent vectors--and all other vectors, tensors, and geometric objects used in the math of differential geometry--are defined in the tangent space.)

OK. No real problem here.

We know that a tangent can only intersect a curved line at a single point, fine. But as soon as you introduce the notion of "tangent space" you are not sticking with your rule that "tangent vectors only 'cover' a single point". You are using them as a basis in this tangent space which extends from that point to points off of the curve.

The first question is answered by the tangent vector to your worldline, yes.

The second is not answered by a tangent vector by itself. There are a couple of different ways to answer it using differential geometry, but a tangent vector alone is not enough.

The third is also not answered by a tangent vector by itself. You need a synchronization convention. The Einstein convention is one possible one, but not the only one.

Once again, I strongly recommend taking some time to learn differential geometry.

You mean "not enough" because we need orientation, rotation etc.? What an MCIF is, is more than a vector. Is that what you mean?

Consider the consequences is using a different convention than Einstein's in a local inertial frame. Let's imagine two mile markers along the x-axis within the boundaries within this local inertial frame. We will have one clock at each marker, but they will be de-synchronized wrt to Einstein's convention by 1 hour to satisfy some other "convention of simultaneity". We perform two experiments in which a car (starting from rest in the frame) undergoes a certain proper acceleration for a certain amount of proper time and then stops accelerating (all within the local inertial frame). After accelerating, in each experiment the car cruises past the two milestones (but in the opposite direction).

The measurement system (the milestones/clocks) have no proper acceleration. We are forced to conclude (prior to experiment) by symmetry that the transits times are equal (whether in proper time or coordinate time in that local frame). But using these clocks they are not. So as a "convention for simultaneity for these clocks" one might ask "what good is that one"? It's "meaningless nonsense" or if you prefer "arbitrary and inconsistent with the physical symmetry". In what sense can you claim that they are synchronized?

But you keep on talking as if you think those conditions are somehow privileged or preferred. You keep on talking as if the Einstein synchronization convention, and the other machinery that defines an inertial frame, are somehow privileged or preferred. They aren't.

I just explained why I think the Einstein synchronization convention is "preferred" in a local inertial frame, so you can address the weakness in that explanation. Actually I'd go so far as to say it is the required convention for consistency of physical measurement. Without consistency of measurement, what physics can you apply?

I also explained that the angle between two trees is not a privileged measurement. What is it that you think I'm missing? You project onto my mind by implication that I "think those conditions are somehow privileged". You must be talking to someone else for I certainly never claimed any such thing.

What I'm not sure you understand is that there is nothing that requires an observer, even if he is moving inertially, to use the Einstein simultaneity definition. Using that definition is a choice--a convention.

Then justify your other convention in the above example of a moving car.

Nobody can force you to do anything (in a free world), but if your results are nonsense it places your choice in doubt.

There is nothing in physics that requires it.

Not even consistency? Not symmetry? Not anything? That makes "simultaneity" a word devoid of all meaning, does it not?

Take my tree analogy. If you say there is nothing forcing me to measure an angle between trees in a certain way from a certain point in space, then there is no conclusion that can be reached about angles and the concept of angle is useless physically. It cannot provide any consistent result.

Two observers both moving inertially, and both at rest relative to each other, and both in flat spacetime, could perfectly well choose different simultaneity definitions; and as long as they both constructed valid coordinate charts based on their respective definitions, they could both make correct physical predictions.

Would you call it a "correct physical prediction" that the car in the above example moves at different speeds depending upon the direction it is going? You could, I suppose but I see no purpose in so degrading the meaning of "physical prediction" as to make the prediction arbitrary and claim that all such predictions are equally correct. When you do that you say bye bye to physics, for you can no longer measure anything in a consistent manner.

Different inertial frames are different coordinate charts. The rules I gave apply to a single coordinate chart. Different coordinate charts give different descriptions of spacetime and what happens in it; the rules I gave are what is required for a single description to be valid.

The rules you impose are not required for nature to be valid, they are imposed by your difficulties in mathematics with multi-value relations.

Sure you can define any coordinate chart you want, but there is a convenient family of charts defined by nature herself, without need of reference to some other chart, a chart in which all proper acceleration vanishes. Any chart is related to this chart by it's non-zero proper accelerations. Proper acceleration is an absolute because it is measured in a specified manner that does not allow for different conditions of measurement. Zero proper acceleration is particularly easy to measure. We need only free an object at a point and see if it moves from that point (to eliminate rotations we actually need more than one point). We care not how by much it moves or in what direction, nor how fast.

In the "coordinate independent" equation of inertial motion (you know, f=ma in modern form), there is an implicit local chart which is any of the family of charts with 0 proper acceleration and cartesian coordinates. The coefficients in the "fictitious force" term are calculated with reference to anyone of those special charts. Or to put it another way those coefficients are calculated wrt to 0 proper acceleration and cartesian coordinates. All measurements are relative to something. This is precisely why the coefficients are 0 in any local inertial frame. It's no coincidence and that fact the they are 0 is not meaningless. That fact allows me to claim that the equation is defined wrt to 0 proper acceleration (i.e. any local inertial frame). Why do you deny that?

I have never said that. I have only said that you can't combine multiple "instantaneous inertial frames" along a non-inertial worldline into a single consistent "frame".

By your definition of "consistent", no you cannot. You don't "like" the non-uniqueness of coordinates, but it nevertheless exists. There is no mathematically flaw that I see, simply a mathematical inconvenience.

First you need to decide what "rate" means. If it means "rate in the inertial frame in which the clock at the center is at rest", then it's easy. If it means something else, you need to decide what. For example, the two clocks could exchange light signals and use the round-trip travel times to determine their rates.

If we get desperate we can do that experimentally. How would you calculate the result without doing that experiment?

You integrate the rate of the clock moving in a circle (determined based on how you define "rate", as above) along its worldline.

My point is this:
1) How do you compute the rate of the clock moving in a circle (relative to the "stationary" clock at the center)? (I suspect you apply a MCIF and the Lorentz transformation)
2) How can you integrate those rates? (I suspect you integrate over these tangent MCIFs (or, in case you complain, "tangent local MCIFs"), and yet you claim that is not valid because they cannot be combined)

These are just suspicions, not claims of course.

I'm not sure what statements you think I've misattributed to you.

Here's one:

But you keep on talking as if you think those conditions are somehow privileged or preferred. ​

You keep implying that I believe there is something unique about simultaneity when I never said any such thing. And, if you read my OP you would see that I qualified the meaning of clock readings every time I referred to them, specifically to avoid such a misinterpretation of my words. Put yourself in my shoes and see if you find it annoying.

What you are attributing to me in that statement exist in your head. I don't know why, but I'll take a guess: 1) your failure to carefully read my words, 2) you projection upon me of the confusion of others with whom you have discussed this in the past.

I'd appreciate it if you would not attribute the mistakes of others in other threads to me. It's not helpful to respond with complaints that result from you own failure to read carefully (if that is actually the case). And it's especially annoying to attribute thoughts to me that are manifestly not mine.

Are you saying that inertial frames are central to SR (which you just said in your post), but are not "required" for SR (which is how I worded your claim in the quote you gave)? If that's your position, it seems odd.

It seems (from my perspective of course) that it is your business to find something odd about about things I say, almost as a matter policy.

From my perspective, if I make a mistake in terminology, it is rather like talking to a computer that says "that does not compute". If and when I make a mistake in terminology, perhaps you could use your intellect to see through the mistake and correct the terminology without denying all meaning to my statement with "I find that odd". I understand that can be difficult. Nevertheless, I am truly puzzled by the difficulties you express (in spite of your apparent intellectual abilities) in understanding me. Why do you keep misinterpreting my OP and ascribing misperceptions to me?

If I did contradict myself, could you show me where so I can try to straighten out what I may have misstated?

 

[CKH]

Yes. There are actually multiple ways of arriving at GR from a simple set of starting assumptions; MTW lists six of them. The first two, which are the methods that Einstein and Hilbert, respectively, used in 1915 to obtain the Einstein Field Equation, are:

I can read your words but they are mostly over my head. So I will take your word for it that EP is unnecessary without being able to understand what the other principles of derivation actually are. My problem of course is that when you raise the discussion to levels of abstraction not familiar to myself, I cannot tell what they conceal. ;)

But seriously, I do stand corrected. It was my belief that EP is what opened the door to understanding gravity. Since there are yet other ways, that is interesting. It seems as if Einstein built GR on a foundation of EP. But I doubt we can claim that it has since been discovered that no foundation is required. Rather, I would conjecture this, these alternative foundations contain assumptions about gravity they can be shown to be equivalent to EP.

Concerning the historical view of the foundation of Einstein's GR, do you believe that Einstein did not develop his theory around EP and therefore his use of pseudo-gravitational fields was indeed non-circular in the context of his own derivation?

As an aside, I read (actually sort of read because the math was over my head) a paper not too long ago, in which the author presented a different theory of relativity (actually slightly different in physical prediction) and remarked that in his theory, the EP is a consequence, not an assumption. Given what you have said, I suppose that is also true for GR when derived in some other way. In each of these derivations there of course must be some assumptions about gravity, for we cannot go on to describe gravity without assumptions (however well-founded in experiment).

I have a question about EP that is bothering me. EEP speaks of "uniform acceleration". Objects with rigidity (total resistance to mechanical stress) such as a rocket ship cannot accelerate uniformly. So, in a cartoon version of EEP we have an observer in a spaceship who cannot distinguish his accelerated motion from a uniform gravitational field. What coordinate system is it in which there is no distinction? The coordinates of the rigid rocket or exactly uniformly accelerated coordinates?

 

[PeterDonis]

If you are saying that those conclusion are wrong

Remember that I came into this thread because you asked about simultaneity conventions other than the Einstein one. I responded by pointing out that the very scenario you described in the OP, where the traveling twin uses clocks synchronized with the home twin's clock using the Einstein simultaneity definition, to see how much time his clock has "lost" compared to the home twin's clock, amounted to the traveling twin adopting a different simultaneity convention from the Einstein one for his rest frame. (It is still the Einstein convention for a different inertial frame, the home twin's, in which the traveling twin is not at rest--more on that below.)

None of what I said contradicted what you said in your OP. I was simply trying to make an additional point: that simultaneity is a convention. You have repeatedly responded by (apparently, to me) maintaining that the Einstein definition of simultaneity is somehow privileged--that there is something in physics that requires, at the very least, that someone who is moving inertially must adopt the Einstein simultaneity convention for the inertial frame in which he is at rest (or at any rate that he is somehow missing out on some physics if he does not do so). I have pointed out two ways in which that view is mistaken: first, that there is nothing stopping an inertial observer from adopting the Einstein simultaneity convention for a different inertial frame (such as the traveling twin adopting the home twin's convention, as described in your OP); and second, that there is nothing even requiring any observer to use the Einstein simultaneity convention in the first place, whether he is moving inertially or not--he could just as well use some other definition of simultaneity, as long as it meets the basic requirements I gave.

Most of the time, nobody does this for inertial observers; they use standard inertial frames in which those observers (at least one of them) are at rest. And most of the time, people using non-inertial frames use one of a few well-known simultaneity conventions that work, at least over a limited range of space (and possibly time), for non-inertial frames (such as the radar convention). But there's nothing in physics that requires any of this. You can make up any wacky simultaneity convention you like, and as long as it meets the basic requirements I gave, you can describe all of the same physics that you can describe using a standard inertial frame with the standard Einstein simultaneity convention.

If you agree with what I said in the above paragraphs, then my purpose in entering this thread is accomplished (though I'm certainly willing to answer other questions, and I'll respond to your question about the EP in a separate post). If you don't, then I think that's where our discussion needs to focus. That will make it clear that I'm not disputing what you said in the OP, and I'm not disputing any of what you say about how inertial frames work, given that you've already chosen to use an inertial frame and its associated simultaneity convention.

I'll respond to other things in your post separately, because I want to keep the issue I just described separate from the rest of the discussion.

 

[PeterDonis]

I hope you also see that "objects at rest stay at rest..." is equivalent to the geometric expression "zero spacetime curvature". The former is a physical statement, the latter is an abstract mathematical statement that needs some "translation" to become physically meaningful.

Yes (with the added clarification that "at rest" means "at rest relative to each other" and that "objects" means "objects that are close together and are moving inertially").

In the terminology used in physics, an "inertial frame" has global extent in time and space.

More precisely, in the terminology used in SR and in Newtonian physics. In GR, there is no such thing as a global inertial frame (except in the idealized case of perfectly flat spacetime), so nobody uses the term to refer to it since it doesn't exist.

The concept (if there is one so-called) of "inertial motion" is not global.

No, but it's not quite "local" either. It applies to a worldline, or a segment of a worldline. Mathematically, an inertial worldline, or segment of one, has zero path curvature ("path curvature" is the mathematical representation of proper acceleration in differential geometry). Physically, an inertial worldline is one such that an object whose motion is described by that worldline feels zero acceleration (an accelerometer attached to the object reads zero). The terms "free fall" and "weightless" are often used to describe this state of motion.

If "local inertial frame" is the correct terminology for "a local cartesian coordinate system in which proper acceleration is zero"

Not quite, because your phrasing invites the question "proper acceleration of what"? Also, it leaves out the main reason for using local inertial frames in GR: the presence of spacetime curvature/tidal gravity. Since you prefer physical descriptions, consider: suppose we have two free-falling objects (I prefer that terminology) that are close together and, at some instant, are at rest relative to each other. We pick one object (call it A) and set up an inertial frame with its worldline as the "time axis", and the origin (the point t = 0, x = 0, y = 0, z = 0) as the event on its worldline that occurs at the instant at which the other object (call it B) is at rest relative to it.

Now, if there is no tidal gravity, both objects will remain at rest forever in this inertial frame--it's a perfectly normal global inertial frame such as we use all the time in SR. But suppose tidal gravity is present. Then we have a problem. The two objects do not remain at rest relative to each other. That means that, if object A is at rest in the frame (which it is, because we've constructed the frame that way), object B is not, except at the instant t = 0. This fact immediately forces us to admit that, whatever this thing is that we've constructed, it can't work the same as an ordinary inertial frame in SR.

However, even though the thing we've constructed can't be exactly the same as an SR inertial frame, it can still work approximately the same, for a small region of spacetime around the origin. How small a region depends on two things: how strong tidal gravity is (i.e., how much object B's motion deviates, over a given interval of time, from what it would be if no tidal gravity were present), and how accurate we need our measurements to be (i.e., how big the deviation of object B's motion needs to be before it affects whatever we are trying to calculate). Within that small region of spacetime, we can work with our local inertial frame just as if it were a corresponding small patch of an ordinary SR inertial frame.

 

[CKH]

One arrives at a theory via heuristics, guesses, etc. (hopefully good ones!). Once you have a theory, you derive consequences, and the initial heuristics become irrelevant - unless they are also formally derived consequences.

These "heuristics made formal statements of fact" are what I referred to as "assumptions" (about gravity) in my previous post. Surely you don't mean that you can abandon your assumptions once you formulate a theory? That sounds like building a castle on the ground, then claiming the ground is no longer needed to support the castle.

To make another analogy, in mathematics we prove a theorem from predicates. When we are done, we cannot say that those predicates can now be abandoned and that we have a statement of truth independent from them.

The mathematical statement of GR nowhere has the principle of equivalence. However, it is a derivable approximate, local, consequence. Given the caveats, some highly esteemed GR experts (e.g. J. L. Synge), argued it should be abandoned, because as formal, mathematical statement it is simply false everywhere in GR (due the the approximate, local nature).

I'll take that claim with a grain of salt since calculus itself based on derivation from local approximations. I don't see how we can abandon that method, but if we can, I'm interested.

Of course it depends on exactly what his complaint is about the statement of EP. If it needs to be stated as a local (infinitesimal relationship) to make sense, then it ought to be stated that way rather than complaining that it is always false. It is always false that a segment of a circle is straight, but that has not prevented us from finding pi that way.

Most physicists find the principle of equivalence remains a highly useful guide to intuition in GR, but one that always must be used with care, and no derivation based on it can be relied on without some additional formal validation.

Sure. But I doubt anyone seriously finds the derivation from EP (and other assumptions) wrong when properly applied. EP is certainly not sufficient since we do not have uniform gravitational fields. Other assumptions must be brought into play in addition to EP to get the job done.

BTW, I think we can agree that there is no assumption that can be properly used without care. Such a warning is always applicable and goes without saying. We all learn that sooner or later.

 

[PeterDonis]

as soon as you introduce the notion of "tangent space" you are not sticking with your rule that "tangent vectors only 'cover' a single point". You are using them as a basis in this tangent space which extends from that point to points off of the curve.

No, it doesn't. The tangent space is not a space of points that correspond to points in spacetime. It's an abstract space in which vectors, tensors, and other objects "live" that are attached to a particular single point in spacetime. Please learn differential geometry before making further statements in this area, or at least ask questions instead of making statements (but at this point it's going to be hard to answer further questions in this area without writing a book about differential geometry--it's better for you just to take some time out to study it).