Is SR and GR Time Dilation the same thing?

[goodabouthood]

For instance you have time dilation in special relativity which is said to be due to something moving faster than something else relative to it.

You also have the time dilation in GR where time speeds up the further away you go from a body such as the Sun, Earth, etc.

Now could it really all be due to the same thing? Is time slowing down for an object closer to the Sun the same overall process for time slowing down for something moving fast.

Could it be that time dilation for both SR (moving faster in this case) and GR (gravity in this case) are due to the same thing?

 

[PAllen]

For instance you have time dilation in special relativity which is said to be due to something moving faster than something else relative to it.

You also have the time dilation in GR where time speeds up the further away you go from a body such as the Sun, Earth, etc.

Now could it really all be due to the same thing? Is time slowing down for an object closer to the Sun the same overall process for time slowing down for something moving fast.

Could it be that time dilation for both SR (moving faster in this case) and GR (gravity in this case) are due to the same thing?

There are similarities and differences.

Some similarities:

- clocks slow and redshift go hand in hand (really the same thing).
- clock seems locally normal - other clocks seem off

Some differences:

- time dilation between inertially moving observers is symmetric; each sees the other slow. For gravitational time dilation, it is not symmetric: surface guy thinks mountain guy is fast; mountain guy things surface guy is slow.

However, there is a way to correlate gravitational time dilation to kinematic time dilation: A free fall observer will see two static clocks accelerating such that their relative speed is not identical (distance between them is shrinking); and for this free fall observer, their relative motion explains their difference in clock speed.

 

[tom.stoer]

In a sense they are the same thing. Proper time along a curve C in spacetime is calculated according to

\tau = \int_C d\tau

Now compare two curves C and C' both connecting two points A and B in spacetime, and calculate the difference for proper times tau and tau' measured along C and C', respectively

\Delta\tau_{A\to B} = \int_{C_{A\to B}} d\tau - \int_{C^\prime_{A\to B}} d\tau

All these formulas are valid for both SR and GR and for arbitrary timelike curves. The difference arises only when looking at specific curves i.e. specific experiments
case 1) a geodesic C ('twin on earth') and a curve C' deformed by acceleration ('the twin in the spaceship')
case 2) a geodesic C ('a satellite orbiting the earth') and a curve C' with non-constant radius measuring the difference in gravitational potential

 

[pervect]

Consider an accelerating elevator or rocket.

If you use inertial coordinates, the metric tensor everywhere is Lorentzian, and you explain all observed time dilations as due to velocity.

If you use Rindler coordinates, which are natural coordinates for the accelerated observer in which the accelerating observer is always at the origin, the metric coefficients are not Lorentzian, and you explain some time dilation as being due to velocity, and other time dilation as being due to "gravitational potential".

In particular, you see clocks at the nose of the rocket ticking faster than those at the tail, even when the nose and tail are "at relative rest" in the sense of having a constant round-trip travel time for light signals.

So it's clear from the example hat the division of time dilation into "velocity" and "gravity" parts depends on your choice of coordinates.

 

[Passionflower]

A free fall observer will see two static clocks accelerating such that their relative speed is not identical (distance between them is shrinking); and for this free fall observer, their relative motion explains their difference in clock speed.

Hos does the free faller measure that the distance is shrinking?
Could you demonstrate with mathematics that that is the case or provide a reference in the literature?

case 1) a geodesic C ('twin on earth') and a curve C' deformed by acceleration ('the twin in the spaceship')

A twin on Earth does not travel on a geodesic.

 

[tom.stoer]

A twin on Earth does not travel on a geodesic.

Correct, but usually in the example for the twin paradox you neglect this tiny effect; let's assume the twin stays at the center of the Earth ;-)

 

[Fredrik]

According to GR, the reason why clocks on different floors of the same building have (slightly) different ticking rates is that the internal forces in the building are making the floors (and therefore the clocks) accelerate differently.

This effect exists in SR too. Consider a rocket that's initially at rest in some inertial coordinate system, and then accelerates gently to a speed where relativistic effects are noticeable. In the inertial coordinate system where the rocket started out at rest, the rocket is now shorter than before by a factor of gamma. This means that the rear must have had a larger acceleration than the front!

Clocks don't measure coordinate time. They measure proper time, i.e. the integral of \sqrt{dt^2-dx^2-dy^2-dz^2} along the curve that represents their motion. Constant velocity would mean dx=dy=dz=0. Any deviation from that will make the proper time smaller, because of the minus signs. If you understand that, it shouldn't be too hard to believe that if two clocks have different accelerations, the one that accelerates less (the one in the front of the rocket or the one on the higher floor of the building) will measure a larger proper time.

 

[zonde]

According to GR, the reason why clocks on different floors of the same building have (slightly) different ticking rates is that the internal forces in the building are making the floors (and therefore the clocks) accelerate differently.

This seems wrong.
http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Clock_Hypothesis" states that acceleration does not affect clock rate and it is experimentally verified.

On the same line. Clocks that are orbiting gravitating body should be affected by gravitational time dilation just the same.

It seems more reasonable to say that body is undergoing gravitational acceleration because clock rates are different at different heights.

 

[Fredrik]

This seems wrong.
http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Clock_Hypothesis" states that acceleration does not affect clock rate and it is experimentally verified.

I'm not saying that the properties of clocks depends on the acceleration. I'm saying that what a clock measures is a coordinate-independent property of the curve in spacetime that describes its motion. The world lines of two clocks that are glued to opposite ends of a solid rod that's accelerated gently to relativistic speeds are significantly different. The world line of the clock in the rear is more curved than the world line of the clock in the front. Perhaps I should also have mentioned that this implies that it has a higher speed in an inertial coordinate system where the rod starts out at rest.

On the same line. Clocks that are orbiting gravitating body should be affected by gravitational time dilation just the same.

The term "gravitational time dilation" usually refers to the different rates of clocks held at different heights above some fixed position on the ground of a non-rotating planet or star. I certainly don't expect a clock at the center of the Earth to remain synchronized with a clock in orbit just because they both move as described by geodesics, but I'm not sure I would like to describe the different rates as "gravitational time dilation".

Hm, didn't pervect say something about this...*scrolling*...yes he did

the division of time dilation into "velocity" and "gravity" parts depends on your choice of coordinates.

In my opinion, the term "time dilation" is only useful in very specific scenarios. In more complicated situations, it's better to drop the term and just talk about the proper time of the relevant timelike curves.

It seems more reasonable to say that body is undergoing gravitational acceleration because clock rates are different at different heights.

So instead of thinking that the geometry determines what numbers a clock displays, I should be thinking that the numbers displayed by clocks determine the geometry?

 

[A.T.]

http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Clock_Hypothesis" states that acceleration does not affect clock rate

Time dilation is not just one clock rate, which is somehow locally affected by acceleration. It is the ratio of two clock rates. Two clocks at rest in an accelerated frame (with spatial separation along the acceleration direction) will have different clock rates. This is how acceleration "generates" gravitational time dilation.

 

[zonde]

I'm not saying that the properties of clocks depends on the acceleration. I'm saying that what a clock measures is a coordinate-independent property of the curve in spacetime that describes its motion. The world lines of two clocks that are glued to opposite ends of a solid rod that's accelerated gently to relativistic speeds are significantly different. The world line of the clock in the rear is more curved than the world line of the clock in the front. Perhaps I should also have mentioned that this implies that it has a higher speed in an inertial coordinate system where the rod starts out at rest.

"Curve in spacetime" is description of your choice for physical reality. I somehow feel uncomfortable with the statement that clock measures "description" of physical reality.

The term "gravitational time dilation" usually refers to the different rates of clocks held at different heights above some fixed position on the ground of a non-rotating planet or star. I certainly don't expect a clock at the center of the Earth to remain synchronized with a clock in orbit just because they both move as described by geodesics, but I'm not sure I would like to describe the different rates as "gravitational time dilation".

We have physical fact - two clocks tick at different rates. How would you name (not describe) this physical fact?

So instead of thinking that the geometry determines what numbers a clock displays, I should be thinking that the numbers displayed by clocks determine the geometry?

Is your question like - does theory determines physical reality or physical reality determines theory?
Then of course later.

But I was trying to say something different. It was something about causal relationship between two physical facts.

Look when Einstein was trying to come up with some description for gravity he had one physical fact about gravity - objects accelerate near gravitating body. He deduced that in order for object to be free of internal stress as it accelerates it should "tick" slower in the part that is closer to gravitating body. The two things (gravitational acceleration and gravitational time dilation) are related - if we see one thing there should be the other one too.
We have experimentally verified fact that clocks tick slower when they are closer to gravitating body. So we have two physical facts.
Now we can talk about causal relationship between two things. And I am saying that gravitational time dilation causes gravitational acceleration.

 

[zonde]

Time dilation is not just one clock rate, which is somehow locally affected by acceleration. It is the ratio of two clock rates. Two clocks at rest in an accelerated frame (with spatial separation along the acceleration direction) will have different clock rates. This is how acceleration "generates" gravitational time dilation.

Object that is stationary in respect to gravitating body is not undergoing continuous length contraction that would be the case for ordinary accelerated body.
Because there is continuous length contraction in the case of ordinary accelerated body it's rear and front is moving at different speeds and have different clock rates. Just like Fredrik was saying.
There is no such thing for stationary body in gravitation field. Unless of course you want to propose that spacetime somehow "moves" at different speeds at different gravitational potentials.

 

[PAllen]

Hos does the free faller measure that the distance is shrinking?
Could you demonstrate with mathematics that that is the case or provide a reference in the literature?

The mathematics is essentially identical to the standard accelerating rocket in SR. To a free falling observer, with locally Minkowski frame, two reasonably nearby stationary clocks (say one 100 meters higher than the other) are simply accelerating clocks maintaining constant distance from the point of view of e.g. the lower clock. So, just look up the SR case and you have the math. This fact about the GR case is extremely well known.

 

[Passionflower]

The mathematics is essentially identical to the standard accelerating rocket in SR. To a free falling observer, with locally Minkowski frame, two reasonably nearby stationary clocks (say one 100 meters higher than the other) are simply accelerating clocks maintaining constant distance from the point of view of e.g. the lower clock. So, just look up the SR case and you have the math. This fact about the GR case is extremely well known.

Measuring distance in curved spacetime has complications that do not exist when measuring distance in flat spacetime.

I take it you will not demonstrate it with formulas perhaps because it is too simple and instead you leave it as an exercise for me to find out this obvious thing?

 

[Fredrik]

"Curve in spacetime" is description of your choice for physical reality.

It's only a choice of what theory to use to answer questions about motion. In SR and GR, statements about motion are statements about curves in spacetime.

I somehow feel uncomfortable with the statement that clock measures "description" of physical reality.

My statement about clocks and curves is part of the definition of both SR and GR. A theory of physics can't be defined by mathematics alone. The physics is in the statements that tell us how to interpret the mathematics as predictions about results of experiments. That's the sort of statement I made.

We have physical fact - two clocks tick at different rates.

This statement is only unambiguous at an event where the clocks are both present and have the same velocity. (If they tick at different rates at such an event, I would say that at least one of them is broken). In any other situation, we need a definition that tells us how to compare the ticking rates. I don't think the rates can be thought of as "physical facts".

How would you name (not describe) this physical fact?

I don't understand the question.

Is your question like - does theory determines physical reality or physical reality determines theory?
Then of course later.

Reality certainly determines which theories will be successful, but once you have decided what theory you're going to use to try to answer a question, reality becomes irrelevant, and all that matters is what the theory says.

But I was trying to say something different. It was something about causal relationship between two physical facts.

I think in most cases where it's possible to say that A is the reason for B, it makes just as much sense to say that B is the reason for A. For example, do we have conservation laws because of symmetries, or do we have symmetries because of conservation laws? However, I think a given theory usually makes one of the possibilities more "natural" than the other, in the sense that it will be much easier to explain. In SR and GR, there's a simple(ish) formula that tells you how to calculate the numbers displayed by a clock at different events on its world line, given a metric. I don't know if there's a way to input those numbers into a calculation that determines the metric. I wouldn't be surprised if there is, but I would still reject the suggestion that this would be a more accurate way to think about these things. As long as we're working with the standard formulation of SR and GR, it would at best be a more complicated way to think about these things.

Unless of course you want to propose that spacetime somehow "moves" at different speeds at different gravitational potentials.

Different floors in the same building accelerate by different amounts. Suppose that we pick an event A on the world line of the clock on the top floor, and draw a spacetime diagram showing what the world line looks like in a local inertial coordinate system that's comoving with the clock at A. Suppose that we do the same to the other clock, this time involving a local inertial coordinate system that's comoving with this clock at some event B. Then because the two clocks accelerate by different amounts, the two curves we draw will curve away from the time axes of these diagrams by different amounts. They will eventually have significantly different coordinate velocities in these two fixed coordinate systems.

Edit: In the special relativistic accelerating rocket scenario, we would usually draw only one spacetime diagram, but there's nothing that prevents us from drawing one for each clock. The result would be essentially the same as in the general relativistic two-clocks-on-different-floors scenario. The desynchronization of the clocks can in both cases be attributed to the coordinate velocity difference discussed above. I can't see any reason to say that we're not dealing with the same phenomenon in both cases.

 

[Passionflower]

Correct, but usually in the example for the twin paradox you neglect this tiny effect; let's assume the twin stays at the center of the Earth ;-)

Yes it is usually ignored.

It is interesting that if we toss the twin straight in the air (with adequate protection) and wait for him to come back he will be older than the twin who stayed on Earth while if we toss him harder so that he does not come back but he eventually returns by using some rockets he will be younger than the one who stayed on Earth.

 

[atyy]

SR and GR time dilation are slightly different things arising from the same underlying physics.

In SR, there are global inertial frames, and the term "time dilation" refers to differences in coordinate time assignments between frames.

In GR, there are usually no global inertial frames, and the term "time dilation" refers to an experiment result that is more like the SR doppler effect.

The underlying physics in both cases is that of a spacetime metric.

 

[A.T.]

Time dilation is not just one clock rate, which is somehow locally affected by acceleration. It is the ratio of two clock rates. Two clocks at rest in an accelerated frame (with spatial separation along the acceleration direction) will have different clock rates. This is how acceleration "generates" gravitational time dilation.

Object that is stationary in respect to gravitating body is not undergoing continuous length contraction that would be the case for ordinary accelerated body.

Length contraction is frame dependent. From the perspective of an inertial (free falling) frame an object that is stationary in respect to the gravitating body (and experiences proper acceleration) can be undergoing continuous length contraction.

 

[zonde]

My statement about clocks and curves is part of the definition of both SR and GR. A theory of physics can't be defined by mathematics alone. The physics is in the statements that tell us how to interpret the mathematics as predictions about results of experiments. That's the sort of statement I made.

Your statement was that clock measures property of the curve. It kind of implies that we are performing experiments on "curves" and clock is measurement equipment that we use in order to find some physical quantity that describes "curves". So I am wondering if you are thinking about "curves" in similar fashion as you would think about "fields".

This statement is only unambiguous at an event where the clocks are both present and have the same velocity. (If they tick at different rates at such an event, I would say that at least one of them is broken). In any other situation, we need a definition that tells us how to compare the ticking rates. I don't think the rates can be thought of as "physical facts".

We have to take something as a starting point to do some reasoning. In case of two distant clocks we can evaluate global situation that we observe and if it's fairly static and clocks have static configuration in respect to that global situation then we can claim that distance between clocks is not changing.
I think that's enough to remove ambiguity in relative ticking rates of clocks when we compare them by exchanging some signals.

Reality certainly determines which theories will be successful, but once you have decided what theory you're going to use to try to answer a question, reality becomes irrelevant, and all that matters is what the theory says.

Can not agree with that. Map is not the territory.
Established theory still does not determine physical reality. It only changes our our interpretation of physical reality.

I think in most cases where it's possible to say that A is the reason for B, it makes just as much sense to say that B is the reason for A. For example, do we have conservation laws because of symmetries, or do we have symmetries because of conservation laws? However, I think a given theory usually makes one of the possibilities more "natural" than the other, in the sense that it will be much easier to explain. In SR and GR, there's a simple(ish) formula that tells you how to calculate the numbers displayed by a clock at different events on its world line, given a metric. I don't know if there's a way to input those numbers into a calculation that determines the metric. I wouldn't be surprised if there is, but I would still reject the suggestion that this would be a more accurate way to think about these things. As long as we're working with the standard formulation of SR and GR, it would at best be a more complicated way to think about these things.

And how do you measure metric?

Different floors in the same building accelerate by different amounts. Suppose that we pick an event A on the world line of the clock on the top floor, and draw a spacetime diagram showing what the world line looks like in a local inertial coordinate system that's comoving with the clock at A. Suppose that we do the same to the other clock, this time involving a local inertial coordinate system that's comoving with this clock at some event B. Then because the two clocks accelerate by different amounts, the two curves we draw will curve away from the time axes of these diagrams by different amounts. They will eventually have significantly different coordinate velocities in these two fixed coordinate systems.

On what are these two local inertial coordinate system fixed? Physical laws for a body that is moving inertially in one of those coordinate systems are not fixed in respect to anything.

 

[zonde]

Length contraction is frame dependent. From the perspective of an inertial (free falling) frame an object that is stationary in respect to the gravitating body (and experiences proper acceleration) can be undergoing continuous length contraction.

And is your choice of coordinates physically justified? Physical context remains the same from perspective of stationary body. But from perspective of falling body physical context is undergoing continuous change. You have to apply global transformations to keep things consistent.

 

[A.T.]

From the perspective of an inertial (free falling) frame an object that is stationary in respect to the gravitating body (and experiences proper acceleration) can be undergoing continuous length contraction.

And is your choice of coordinates physically justified?

To sensibly compare SR & GR you have to compare inertial (free falling) frames for both cases. In a uniform gravitational field the inertial (free faling) observer will observe the object at constant potential in the same way as the inertial observer observes an accelerating object in SR.

 

[Fredrik]

Your statement was that clock measures property of the curve. It kind of implies that we are performing experiments on "curves" and clock is measurement equipment that we use in order to find some physical quantity that describes "curves". So I am wondering if you are thinking about "curves" in similar fashion as you would think about "fields".

I think of both as (mathematical) terms defined by theories. The statement doesn't imply that experiments are performed on mathematical objects. All it does is to specify a correspondence between something in the real world and something in the theory. There's no such thing as a theory of physics without such correspondence rules, because the mathematics alone can't tell us what the theory's predictions are.

Can not agree with that. Map is not the territory.
Established theory still does not determine physical reality. It only changes our our interpretation of physical reality.

I don't know why you think I said anything that would imply that the map is the territory or, even sillier, that theory determines reality. What I'm saying is that questions about reality can only be answered by theories that include definitions of the terms used in the question, and once you have chosen the theory that you're going to use to answer the question, then you shouldn't be thinking about anything other than what that theory says until you're done writing down the answer. Once you have an answer, reality might motivate you try another theory. But when you're trying to figure out what a specific theory says, thinking about reality will almost never help you see the answer. It's much more likely to confuse you to the point where you can't see that the question has a simple answer.

On what are these two local inertial coordinate system fixed? Physical laws for a body that is moving inertially in one of those coordinate systems are not fixed in respect to anything.

I don't understand what you're arguing for.

 

[yoron]

Sweet definitions Fredrik. To me there is 'local time', and the time dilations I find comparing my clock to other frames of reference. Normally we use our own, local clock, as the one defining all other clocks. So finding a 'time dilation' is always a comparison between 'frames of reference' to me. And that goes for gravitational 'time dilation' too, it craves me, comparing how 'frames of reference/clocks' differs relative my local clock.

 

[A.T.]

So finding a 'time dilation' is always a comparison between 'frames of reference' to me. And that goes for gravitational 'time dilation' too,

So two clocks at rest to each other at different potentials represent two different reference frames?

 

[Passionflower]

So two clocks at rest to each other at different potentials represent two different reference frames?

While their proper distance remains the same over time both clocks have a different proper acceleration. I think it is a stretch to consider them to be in the same frame of reference.

 

[atyy]

Because there are generally no global inertial reference frames in GR, time dilation in GR is really gravitational "Doppler shift", ie. both guys use the same sorts of clocks (made of the same atoms) and send light pulses to each other. The time interval between pulses received is different from the time interval on the local clock. This is how SR Doppler shift and GR time dilation are judged. SR time dilation is not judged this way, although it is related to SR Doppler shift.

 

[Passionflower]

Because there are generally no global inertial reference frames in GR, time dilation in GR is really gravitational "Doppler shift", ie. both guys use the same sorts of clocks (made of the same atoms) and send light pulses to each other. The time interval between pulses received is different from the time interval on the local clock. This is how SR Doppler shift and GR time dilation are judged. SR time dilation is not judged this way, although it is related to SR Doppler shift.

Can you give me an example where you think that things a judged (?) differently from SR and GR.

I seriously hope you are joking because when we are going to judge things in physics we are far gone.

 

[atyy]

Can you give me an example where you think that things a judged (?) differently from SR and GR.

I seriously hope you are joking because when we are going to judge things in physics we are far gone.

GR time dilation is judged the same way as SR Doppler shift.

SR Doppler shift is not the same as SR time dilation, although they are related.

So GR time dilation is not judged the same way as SR time dilation, rather it is judged the same way as SR Doppler shift.

It's just quirky terminology, nothing deep.

 

[Passionflower]

GR time dilation is judged the same way as SR Doppler shift.

SR Doppler shift is not the same as SR time dilation, although they are related.

So GR time dilation is not judged the same way as SR time dilation, rather it is judged the same way as SR Doppler shift.

It's just quirky terminology, nothing deep.

Could you provide a working example so we can see what on Earth you are getting at, because really it does not make any sense at all to me.

And time dilation in GR is not Doppler shift, that is just nonsense.

 

[atyy]

Could you provide a working example so we can see what on Earth you are getting at, because really it does not make any sense at all to me.

And time dilation in GR is not Doppler shift, that is just nonsense.

Time dilation in GR (at least the type associated with differences in potential) is the same as gravitational red shift.

 

[Passionflower]

Let's take an example, two observers are approaching each other rapidly in some arbitrary curved spacetime, how is is GR time dilation judged the same way as SR Doppler shift?

 

[yoron]

Ahh Passion flower :) You're so right.

 

[A.T.]

So finding a 'time dilation' is always a comparison between 'frames of reference' to me. And that goes for gravitational 'time dilation' too,

So two clocks at rest to each other at different potentials represent two different reference frames?

While their proper distance remains the same over time both clocks have a different proper acceleration. I think it is a stretch to consider them to be in the same frame of reference.

Why is that stretch? They are at relative rest to each other. How is their proper acceleration relevant here? Objects at rest in a rotaing reference frame also have different proper accelerations. An yet still they have a common rest frame.

I think you and yoron are confusing "different reference frames" with "different instantaneous inertial reference frames".

 

[Passionflower]

Why is that stretch? They are at relative rest to each other. How is their proper acceleration relevant here? Objects at rest in a rotaing reference frame also have different proper accelerations. An yet still they have a common rest frame.

I think you and yoron are confusing "different reference frames" with "different instantaneous inertial reference frames".

Sure I am confused... you are simply arguing semantics.

 

[A.T.]

I think you and yoron are confusing "different reference frames" with "different instantaneous inertial reference frames".

Sure I am confused... you are simply arguing semantics.

I don't think the difference between an inertial reference frame, and a non-inertial frame is just semantics.

Objects at rest relative to the massive body at different potentials may not have a common inertial rest frame. But still do have a common rest frame.

 

[yoron]

Let us assume that time dilations are 'real', no matter if we measure them by a twin experiment, or not.

Further let us assume that they describe the same in SR as in GR. You could try to define them differently, but if we place a clock elevated (on earth) relative one standing on the ground we will, as I understands it, observe the same sort of 'time dilation' as described in the twin experiment, only its 'small scale' differing them.

This is assuming a 'arrow of time' existing naturally, locally never differing for you. And there defined by your 'clock of choice', which I find to be 'c'.

Using those definitions we find that 'c' and your 'local clock', described by splitting 'c' in arbitrarily made 'even chunks' will fit. They define your time locally as 'invariant', never changing, although you can define all other 'frames of reference' as describing a different 'time rate' than what you observe locally.

Using 'clocks', as I do to define 'frames of reference', you can also reach a theoretical definition of their (frames of references) boundary, which then to me would be 'c' propagating one Planck length in one Plank time. But then we have HUP coming into the picture, and one Plank length/time is not 'moving' at all, is it? Well, I don't see that as 'motion' at least. So where the 'macroscopic definition' of 'times arrow' should be (as in 'start') I'm not sure, although I do see it as a working definition of 'time dilations', describing them as 'one thing', the same for both SR and GR.

As for your defining it as "different instantaneous inertial reference frames". That's another way to describe it. I've seen that description and it makes sense.

Also you can think of it as me using the equivalence principle, describing mass as a 'uniform constant acceleration', a 'motion' of sorts.

 

[PAllen]

I don't think the difference between an inertial reference frame, and a non-inertial frame is just semantics.

Objects at rest relative to the massive body at different potentials may not have a common inertial rest frame. But still do have a common rest frame.

Inertial vs. non-inertial motion is certainly physics - you can locally, directly, measure the difference. Extended 'rest frames' are a matter of convention. I can define simultaneity for a rest frame via idealized rulers (at least without rotation, assuming Born rigidity), via 'no doppler', via radar ranging with 1/2 travel time definitions of simultaneity. The issue of convention is that, in general, all 3 of these will define different simultaneity for non-inertial frames, even in flat spacetime. For inertial frames in flat spacetime, they all agree, so one can consider such a global frame as preferred. In GR, they all lead to different global coordinates. So what is a non-local rest frame in GR is highly arbitrary.

 

[zonde]

To sensibly compare SR & GR you have to compare inertial (free falling) frames for both cases.

It depends. If you want to make global observations from those frames then it makes sense to compare inertial frame from SR with hovering frame from GR.

In a uniform gravitational field the inertial (free faling) observer will observe the object at constant potential in the same way as the inertial observer observes an accelerating object in SR.

Can you make your argument without involving "uniform gravitational field"?

 

[zonde]

On what are these two local inertial coordinate system fixed? Physical laws for a body that is moving inertially in one of those coordinate systems are not fixed in respect to anything.

I don't understand what you're arguing for.

Physical laws globally do not stay the same for falling observer. It has to apply continuous transformation globally to keep local laws consistent with global laws.
This is very similar to accelerated reference frame in SR. If uniformly accelerated observer takes his local physical laws as reference then he has to apply continuous transformation globally to interpret observations consistently.

 

[zonde]

Inertial vs. non-inertial motion is certainly physics - you can locally, directly, measure the difference. Extended 'rest frames' are a matter of convention. I can define simultaneity for a rest frame via idealized rulers (at least without rotation, assuming Born rigidity), via 'no doppler', via radar ranging with 1/2 travel time definitions of simultaneity. The issue of convention is that, in general, all 3 of these will define different simultaneity for non-inertial frames, even in flat spacetime. For inertial frames in flat spacetime, they all agree, so one can consider such a global frame as preferred. In GR, they all lead to different global coordinates. So what is a non-local rest frame in GR is highly arbitrary.

You can't define simultaneity using rulers. And how you define simultaneity via 'no doppler'?
As I see the third method is the only method how you can define simultaneity.

 

[Fredrik]

Physical laws globally do not stay the same for falling observer. It has to apply continuous transformation globally to keep local laws consistent with global laws.
This is very similar to accelerated reference frame in SR. If uniformly accelerated observer takes his local physical laws as reference then he has to apply continuous transformation globally to interpret observations consistently.

The common statement that the "laws of physics" are supposed to "be the same" in all inertial frames of SR, really just means that equations of motion (something like F=ma or Maxwell's equations) are tensor equations, and if you express them in terms of the components of the tensors, and replace all symbols that represent components of the metric with their values (0,1 or -1), then the result should look more or less the same, regardless of what inertial frame you used. I've never really liked such statements. I prefer to just say that when we write down theories of matter in spacetime, we need to make the equations of motion coordinate independent. We do this by using tensors (and in some cases, spinors). Note that tensor equations are coordinate independent statements. Tensors don't change when you change the coordinate system, only their components with respect to the coordinate system do. This way of looking at "covariance" works in GR too.

In this case, we're just talking about kinematics. We're talking about particles that are assumed to be constrained to move a certain way, and we don't care about what made them move that way. So equations of motion (belonging to theories of particles and/or fields in spacetime) aren't even in the picture.

And even if they were, I don't see how this sort of thing could be an argument against what I said in the quote below. That is what you're arguing against, right?

Different floors in the same building accelerate by different amounts. Suppose that we pick an event A on the world line of the clock on the top floor, and draw a spacetime diagram showing what the world line looks like in a local inertial coordinate system that's comoving with the clock at A. Suppose that we do the same to the other clock, this time involving a local inertial coordinate system that's comoving with this clock at some event B. Then because the two clocks accelerate by different amounts, the two curves we draw will curve away from the time axes of these diagrams by different amounts. They will eventually have significantly different coordinate velocities in these two fixed coordinate systems.

Edit: In the special relativistic accelerating rocket scenario, we would usually draw only one spacetime diagram, but there's nothing that prevents us from drawing one for each clock. The result would be essentially the same as in the general relativistic two-clocks-on-different-floors scenario. The desynchronization of the clocks can in both cases be attributed to the coordinate velocity difference discussed above. I can't see any reason to say that we're not dealing with the same phenomenon in both cases.

 

[harrylin]

It's only a choice of what theory to use to answer questions about motion. In SR and GR, statements about motion are statements about curves in spacetime.[..]

As Zonde already suggested, that depends on your choice of how you prefer to describe physical reality by means of those theories. In fact, I did not find such a formulation at all in early SR nor in early GR. Those theories do not depend on such descriptions.

 

[A.T.]

It depends. If you want to make global observations from those frames then it makes sense to compare inertial frame from SR with hovering frame from GR.

If you compare inertial with non-inertial frames, you will obviously get different effects. So this comparison seems pointless to answer the question if certain effects from SR & GR are equivalent. A sensible comparison between SR & GR effects should be based on the equivalence principle, and the correspondence of frames stated there.

Can you make your argument without involving "uniform gravitational field"?

The equivalence principle assumes a uniform gravitational field and works only for that special case, because special relativity is just a special case of general relativity. It makes no sense to compare SR & GR effects for other cases, which SR cannot even model.

 

[Fredrik]

As Zonde already suggested, that depends on your choice of how you prefer to describe physical reality by means of those theories. In fact, I did not find such a formulation at all in early SR nor in early GR. Those theories do not depend on such descriptions.

I don't see how to make sense of this. Are you saying that there's a version of GR that's not a theory of space, time and motion, or that there's a version of GR that is a theory of space, time and motion, in which curves in spacetime don't have anything to do with motion? I don't see what else you could mean.

I don't think early publications are of much use in these discussions. In the early days, physicists probably weren't at all concerned about the exact definition of the theory. This is something that physicists in general aren't very concerned with.

 

[PAllen]

You can't define simultaneity using rulers. And how you define simultaneity via 'no doppler'?
As I see the third method is the only method how you can define simultaneity.

I was admittedly cryptic in my descriptions of alternate ways of setting up coordinates. I thought they might be familiar to you. Here are each of 3 methods (many others are possible) described in more detail:

1) Rulers. Here, I really mean a purely mathematical construction which may or may not overlap will with realizable physical rulers. Given a chosen origin world line (not necessarily a geodesic, since we aim to cover non-inertial observers; but we assume no rotation), at each event on it, extend the family of spacelike geodesics 4-orthogonal to the world line. These define a hypersurface of simultaneity, along which proper distance defines your position coordinates.

2) Doppler. The idea here is actually related to 'at rest' for a 'rest frame'. This indirectly defines simultaneity. Procedure: start with an origin world line as in (1), and an initial surface of simultaneity (either by convention in (1) or (3), below). Then define the congruence of world lines through this initial surface such that redshift/blueshift is zero between nearby world lines (maintaining this condition at all times). Declare t=0 at the intersection of this congruence with the initial surface. Then, each later surface of simultaneity is defined by the set of events a fixed proper time from zero along the 'at rest' congruence of world lines. Having thus defined a foliation of simultaneity surfaces, distances are again proper distance per such surfaces.

3) Radar. What I actually had in mind was radar used to define both simultaneity and distance. Again, pick an origin world line, again not necessarily geodesic. Time is simply proper time on this world line. Simultaneity is defined by radar convention: the time of distant event is halfway along the interval from sending and receiving a signal, measured from the origin world line. For distance, one can use proper distance, or define distance as local c times 1/2 round trip time (this conventions gives constant c, globally, from the origin, but converts Shapiro time delay to Shapiro orbital bump).

So, here we have 3 general methods, with a couple of detail choices for each, for establishing large scale coordinates [None of these methods will give you a single global chart in the general case. For (1) and (2), the issue is that coordinate lines or surfaces may intersect at some point, so you can't specify a 1-1 mapping; for 3, any instance of an Einstein ring or similar severe gravitational optical distortion defeats 1-1 mapping.]

Then, my main point remains: uniquely in the case of inertial frames in flat spacetime, all of these are identical. For non-inertial observers in flat spacetime, and any observers in GR, these are generally all different - each abstracting a different feature of inertial coordinates to emphasize.

Thus, I strongly re-iterate: "So what is a non-local rest frame in GR is highly arbitrary. "

 

[harrylin]

I don't see how to make sense of this. Are you saying that there's a version of GR that's not a theory of space, time and motion, or that there's a version of GR that is a theory of space, time and motion, in which curves in spacetime don't have anything to do with motion? I don't see what else you could mean.

I don't think early publications are of much use in these discussions. In the early days, physicists probably weren't at all concerned about the exact definition of the theory. This is something that physicists in general aren't very concerned with.

No. For this particular case (GR) I had in mind the publications of Einstein who was very much concerned with exact definitions - he was even the one who labeled "GR" and "SR". GR was already rather well defined in 1916, here:
http://www.Alberteinstein.info/gallery/gtext3.html

Now, getting back to Zonde's comment on what you were saying:

"[what a clock measures is a coordinate-independent property of the] curve in spacetime [that describes its motion]" is description of your choice for physical reality.

So, it sounded as if you were not merely referring to the application of the mathematical toolbox of GR to the concepts of "space" and "time" (as Einstein did), but as if you were identifying an invisible, metaphysical item as physical cause. Right? Your next reply in #22 sounded like a denial, but then I don't understand what you could have meant with the sentence that zonde commented on; surely you did not mean that a clock measures a mathematical curve.

The choice of portraying a curve in spacetime as an invisible physical thing is not inherent to GR, but merely reflects a certain view of physical reality by those who use such expressions.

Harald

 

[harrylin]

[clarifying "what is a non-local rest frame in GR is highly arbitrary":]

I was admittedly cryptic in my descriptions of alternate ways of setting up coordinates. I thought they might be familiar to you. Here are each of 3 methods (many others are possible) described in more detail:

1) Rulers. Here, I really mean a purely mathematical construction which may or may not overlap will with realizable physical rulers. [..]

2) Doppler. The idea here is actually related to 'at rest' for a 'rest frame'. [..]

3) Radar. What I actually had in mind was radar used to define both simultaneity and distance. [..]

Then, my main point remains: uniquely in the case of inertial frames in flat spacetime, all of these are identical. For non-inertial observers in flat spacetime, and any observers in GR, these are generally all different - each abstracting a different feature of inertial coordinates to emphasize.

Thus, I strongly re-iterate: "So what is a non-local rest frame in GR is highly arbitrary. "

I wonder if that matters - are the concepts under discussion here not independent of the choice of simultaneity? Einstein predicted that "a clock would go more slowly in the neigbourhood of ponderable masses" - a clock "at rest in a gravitational field".
It doesn't matter for the observed Doppler shifts what times we attribute to those distances.

 

[Fredrik]

So, it sounded as if you were not merely referring to the application of the mathematical toolbox of GR to the concepts of "space" and "time" (as Einstein did), but as if you were identifying an invisible, metaphysical item as physical cause. Right? Your next reply in #22 sounded like a denial, but then I don't understand what you could have meant with the sentence that zonde commented on; surely you did not mean that a clock measures a mathematical curve.

I thought I explained that part. The only thing that can answer a question about reality is a theory. A theory is defined by a piece of mathematics and a bunch of additional assumptions that tell us how to interpret the mathematics as predictions about results of experiments. My statement about clocks is such a statement.

I have no idea what it would mean to "identify an invisible, metaphysical item as physical cause".

 

[harrylin]

I thought I explained that part. The only thing that can answer a question about reality is a theory. A theory is defined by a piece of mathematics and a bunch of additional assumptions that tell us how to interpret the mathematics as predictions about results of experiments. My statement about clocks is such a statement.

I have no idea what it would mean to "identify an invisible, metaphysical item as physical cause".

Thanks for your clarification! However, what you meant remains a bit foggy to me, for if I plug in that purely mathematical meaning (with which I fully agree), then I obtain: "what a clock measures is a coordinate-independent property of the of the [mathematical description] of its motion".

How can a clock measure a description of its motion? What does that mean?

 

[Fredrik]

Thanks for your clarification! However, what you meant remains a bit foggy to me, for if I plug in that purely mathematical meaning (with which I fully agree), then I obtain: "what a clock measures is a coordinate-independent property of the of the [mathematical description] of its motion".

How can a clock measure a description of its motion? What does that mean?

Thanks for letting me know that you found my choice of words confusing. I would like to be able to explain these things in a way that won't be misunderstood by anyone.

The purely mathematical parts of both SR and GR define a function \tau that takes piecewise smooth timelike curves to positive real numbers. The number \tau(C) is called the "proper time" of the curve C.

A real-world physical clock that moves in a way that's represented by a piecewise smooth timelike curve C in the purely mathematical part of the theory, will display a number at the end of its real-world physical journey and another at the start of it. The difference between those numbers is \tau(C)[/color].

Now, the purely mathematical parts of SR and GR don't say that. They just associate the term "proper time" with the function \tau. So we need to consider the preceding paragraph a part of the definition of each of these two theories.

Let me know if this is still unclear.

Edit: The statement I colored brown is the more precise version of what I've been saying as "A clock measures the proper time of the curve in spacetime that represents its motion".