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

[Dale]

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).

GR cannot be logically derived from SR, nor can it even be derived from SR + the equivalence principle. In fact, Einstein between 1905 and 1915 had several false-starts. Other theories that looked like they would be suitable generalizations of SR with the equivalence principle. That is the key problem with trying to go in that direction. There is usually more than one possible theory, as there was in this case.

The equivalence principle was what we call a "desideratum". In other words, any candidate theory should obey the equivalence principle. It allows you to eliminate any candidate theories that do not follow it, but it does not allow you to derive the theory.

What do you think? Is GR justified (derived) is some completely independent way (how?)

"Justified" and "derived" are two different things. Like all fundamental laws of physics, GR is not derived at all. The fundamental physical laws are always simply assumed. They are then justified by experimental data.

 

[PAllen]

One debate running through here (and many other threads) is whether acceleration affects (ideal) clock rates (as distinct from enables different paths between events). Within the formalism and both SR and GR, in all generality, it is trivially impossible for acceleration to have an effect on passage of proper time (readings of ideal clocks). The proper time along a world line in any coordinates at all (even ones, such as Dirac used, for which there are two lightllike and two spatial coordinates with no time coordinate at all) is the integral of the square root of the contraction of the metric with coordinate derivatives by a parameter. Derivatives, not second derivatives. Thus, there is no place in the mathematics for acceleration (second derivative by some time coordinate) to play a role. Only velocity and position can play a role, even in the most arbitrary coordinates in either SR or GR.

 

[CKH]

Regarding conventions of simultaneity that are not the same as the standard (Einstein) convention in an inertial frame, I posed this problem:

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.

I received this response:

Sure, because you've implicitly defined a "frame" that is not a standard inertial frame, and your symmetry argument only applies if measurements are taken with respect to a standard inertial frame.

It does not matter. (Measurements above that are said to be in a "local inertial frame" are the same in a "standard inertial frame"). However, to remove your objection, you may remove all references to "local" from the experiment and then agree that the argument is valid.

 

[PAllen]

Regarding conventions of simultaneity that are not the same as the standard (Einstein) convention in an inertial frame, I posed this problem:I received this response:It does not matter. (Measurements above that are said to be in a "local inertial frame" are the same in a "standard inertial frame"). However, to remove your objection, you may remove all references to "local" from the experiment and then agree that the argument is valid.

Nonstandard coordinates do not (generally) have the property of isotropy, and the laws of motion are required to have extra terms (in such coordinates). Thus, there is no contradiction. Your symmetry argument does not apply when you choose to use non-standard simultaneity. But that doesn't mean you can't do it, just that computations become more complicated.

 

[PhoebeLasa]

When I read (and re-read) Einstein's wonderful little book "Relativity", It seems clear to me that he DID believe that simultaneity at a distance (for a GIVEN inertial observer) IS meaningful in special relativity. And it seems clear that the simultaneity given by the Lorentz equations IS the simultaneity that he believed had meaning in special relativity. And that those special coordinates worked throughout all (assumed flat) spacetime, not just locally.

I'm sure Einstein realized that a given inertial observer can choose to adopt some other observer's (rest) inertial reference frame instead of his own, but I think that Einstein would say that that choice would usually be undesirable, because those alternative coordinates wouldn't be meaningful to the given observer. And I'm sure that Einstein realized that an inertial observer is even free to adopt coordinates in an almost completely arbitrary manner, but I think Einstein would have considered that to be a very stupid thing to do in special relativity ... why chose meaningless coordinates when you can have meaningful coordinates? Why choose complexity over simplicity?

Einstein didn't use, or need, differential geometry in his development of special relativity. It was only when he had finished developing special relativity, and was trying to understand how to develop general relativity, that he realized that those meaningful coordinates he used in special relativity wouldn't work in general relativity. He expressed that by saying that the "reference frame" of an observer in special relativity (which he regarded as a rigid and meaningful (mental) construction) must be replaced by a set of rather mushy, non-rigid "reference mollusks" in general relativity, with coordinates that are arbitrary and basically meaningless. And, by using the equivalence principle, applied to the rotating disk example of special relativity, he realized that Euclidean geometry doesn't work in general relativity: the ratio of the circumference of a circle to its diameter ISN'T pi in general relativity, and the sum of the three interior angles of a triangle ISN'T 180 degrees. THAT was when he realized that he needed to learn differential geometry, in order to develop general relativity.

 

[PeterDonis]

Nonstandard coordinates do not (generally) have the property of isotropy, and the laws of motion are required to have extra terms (in such coordinates)

Not just the laws of motion: all physical laws will, in general, have different terms in non-standard coordinates, if you expand them out from their covariant tensor formulations. (OTOH, if you write all physical laws in their covariant tensor formulations, they look exactly the same in any valid coordinate chart, whether standard or non-standard.)

There's also another key distinction here (which I know you understand but which I'm stating explicitly for the benefit of other readers of this thread). A "local inertial frame" is not just a small patch of spacetime around a chosen event: it's a small patch of spacetime around a chosen event, plus a standard inertial coordinate chart on that patch of spacetime. Using a non-standard coordinate chart (such as the non-standard simultaneity convention CKH described) on the same small patch of spacetime (such as the one in which the car scenario takes place) means you are not using a local inertial frame, even though the same small patch of spacetime can be described by a local inertial frame (by using a standard inertial coordinate chart on it). This is just a special case of the general rule that it's important to keep in mind the distinction between spacetime (or a small patch of it), the geometric object, and coordinate charts that we can use to describe it.

 

[PAllen]

When I read (and re-read) Einstein's wonderful little book "Relativity", It seems clear to me that he DID believe that simultaneity at a distance (for a GIVEN inertial observer) IS meaningful in special relativity. And it seems clear that the simultaneity given by the Lorentz equations IS the simultaneity that he believed had meaning in special relativity. And that those special coordinates worked throughout all (assumed flat) spacetime, not just locally.

I'm sure Einstein realized that a given inertial observer can choose to adopt some other observer's (rest) inertial reference frame instead of his own, but I think that Einstein would say that that choice would usually be undesirable, because those alternative coordinates wouldn't be meaningful to the given observer. And I'm sure that Einstein realized that an inertial observer is even free to adopt coordinates in an almost completely arbitrary manner, but I think Einstein would have considered that to be a very stupid thing to do in special relativity ... why chose meaningless coordinates when you can have meaningful coordinates? Why choose complexity over simplicity?

Einstein didn't use, or need, differential geometry in his development of special relativity. It was only when he had finished developing special relativity, and was trying to understand how to develop general relativity, that he realized that those meaningful coordinates he used in special relativity wouldn't work in general relativity. He expressed that by saying that the "reference frame" of an observer in special relativity (which he regarded as a rigid and meaningful (mental) construction) must be replaced by a set of rather mushy, non-rigid "reference mollusks" in general relativity, with coordinates that are arbitrary and basically meaningless. And, by using the equivalence principle, applied to the rotating disk example of special relativity, he realized that Euclidean geometry doesn't work in general relativity: the ratio of the circumference of a circle to its diameter ISN'T pi in general relativity, and the sum of the three interior angles of a triangle ISN'T 180 degrees. THAT was when he realized that he needed to learn differential geometry, in order to develop general relativity.

This is mostly true, historically (with a caveat below). However, other physicists came to disagree on interpretation of what is SR and what is GR. No physical predictions are affected by this disagreement - it is yet another philosophy debate. The disagreement began early: already Eddington in his 1922 treatise adopted the 'modern' point of view that analyzing flat spacetime with general coordinates was SR not GR. Bergmann's 1942 book introduced the whole machinery of tensor calculus in its presentation of SR.

The caveat is that, in SR, Einstein analyzed non-inertial motion only in a single inertial frame. Analyzed as a special case of GR, one notes the importance he attached to general covariance - any coordinates are good. I have never seen him use (and have looked) the concept 'planes or lines of simultaneity' for a non-inertial observer. He actually did write a paper using radar simultaneity for a non-inertial observer, but I am not able to find reference for it right now.

 

[Dale]

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