Is SR and GR Time Dilation the same thing?

[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".

 

[tom.stoer]

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.

That's exactly what I was saying in my first post

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_{C,C^\prime} = \Delta\tau_{C_{A\to B}, C^\prime_{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

The difference is that in SR we are asking special questions, whereas in GR we are allowed to ask more general questions. The mathemical difference is that in SR the underlying manifold on which the curve is defined is restricted to a flat manifold [which allows for a metric which is globally diag(+1, -1, -1, -1)] whereas in GR the manifold can be any Riemannian manifold.

In that sense the time dilation in SR is nothing else but the effect of an arbitrary curve on a fixed, flat manifold, whereas in GR time dilation is due to arbitrary curves on arbitrary manifolds - for which disentangling effects due to the curve itself and due to the manifold is no longer possible.

Remark: I guess one source of confusion is that quite often time dilation in SR is explained w/o restricting the two curves to intersect at a common end point. I think that in general cases in GR this is no longer allowed, the two curves C and C' must connect two points A and B in spacetime.