Twin Paradox with "home time"

  • #51
PeterDonis
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There is no prefered definition of the angle between the trees, but that angle is perfectly meaningful given it's conditions of measurement.
Sure, I've never said otherwise. I've never said the Einstein simultaneity convention is meaningless. I've simply said it's not required.

There is one problem with the analogy you're making here: the angle between the trees is a direct observable, while simultaneity is not. I have never said that direct observables are meaningless; indeed, as you will see me say below, they are the actual content of physics.

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.
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.

But now ask the question: is this "transit time" you speak of a direct observable? No, it isn't. (It can't possibly be, since I can make it take any value I like simply by adjusting my clock synchronization convention.) The direct observable is the proper time elapsed on the car's clock between the two markers. If you compute that using your non-standard frame with its non-standard clock synchronization, you will get the same answer as the answer you get using a standard inertial frame to do the computation. The same will be true for any other direct observable. And the actual content of physics is in the direct observables, not in the coordinate values we use to compute them.

So if you really want to claim that Einstein's simultaneity definition is preferred, you're going to have to show me a direct observable that I can't compute correctly using a non-standard frame, but which you can compute correctly using a standard inertial frame.

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 are assuming that "distance traveled divided by time", as given in the non-standard frame with its non-standard simultaneity convention, is the correct way to represent the direct physical observable "speed". It isn't. To correctly calculate the speed of the car (for example, to predict the Doppler shift a particular observer would see in light emitted by the car), you need to figure out how that observable is represented in your non-standard frame. You can't just assume it must be represented the same way as it is in a standard inertial frame. Of course you can always calculate "distance traveled divided by time"; that's not the issue. The issue is figuring out which calculation you need to do to correctly compute a particular physical observable. In general it will be a different calculation in different frames.

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.
More precisely, in which objects at rest have zero proper acceleration. If no tidal gravity is present, yes, this family of charts has a very nice property: it has a symmetry that matches the symmetry of the actual physics. That makes many calculations much easier.

However, this does not mean that nature "defines" this family of charts. Charts do not exist in nature; they exist in our minds. Nature does not define them; we do. There are no "grid lines" in nature that mark off the coordinate lines of any particular chart. (The closest approach to that is a family of inertial objects, whose worldlines can mark off the "time lines" of a local inertial frame--but we still have the freedom to choose whether or not to interpret those worldlines in that fashion.) If we choose to use a particular chart or family of charts, it isn't because nature "requires" us to; it's because the particular problem we are trying to solve has properties that make a particular chart or family of charts more suitable for solving the problem. Different problems have different properties that can make different charts more suitable; not all problems are best solved by using inertial frames.
 
  • #52
PeterDonis
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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)
This is circular; how do you pick out a "point" with respect to which you make the measurement? "Points" aren't marked out in spacetime. Only objects are. So a "point" in your measurement has to be some object that you are using as a reference, and how do you know it has zero proper acceleration?
 
  • #53
PeterDonis
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You don't "like" the non-uniqueness of coordinates, but it nevertheless exists.
Huh? Where have I said that? I think you are still mistaken about which non-uniqueness I am talking about. Consider (again) this particular paragraph in post #41:

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.
This paragraph pretty much sums up what I've been trying to get across about simultaneity. I've made some amplifying remarks in recent posts.
 
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  • #54
PeterDonis
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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)
Once again: first you have to define what "rate" means. What physical observable corresponds to "rate"? How would you measure it?

As for how you would compute it, similar remarks apply to this and the next item: you don't need to use MCIFs or Lorentz transformations. See below.

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)
You suspect incorrectly. You integrate along the worldline of the clock moving in a circle. All you need to do that is a valid coordinate chart; it doesn't have to be one that is in any way related to any of the MCIFs or that requires combining them.

(Also, the MCIF is not the same as the tangent space, and it's important not to get them confused. The MCIF is an ordinary inertial frame; the tangent space, as I said in a previous post, is an abstract space that requires some differential geometry to understand.)
 
  • #55
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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").
Thanks Peter. It's enough for today. I'm drained. o0) I don't think I'll find anything to disagree with in this post. I'll study it some more later and try to identify any missteps I've made.

As far as "the conventionality of simultaneity", I'm just not getting your point. I do not know what those word mean in your mind nor what purpose arbitrary choices can possibly serve. Particularly choices that are inconsistent in a given context (e.g. car experiment above).

Perhaps you will get it across if you continue, but I think you need to take another approach because repetition is not getting the job done.

Also, is distance not also "conventional", not to mention position? All depends on how you want to do it? But when you put something at rest in an inertial frame and you measure its length in that frame and then you claim it's length is "just conventional" I don't know what to think. I see your claim that measuring time under the same circumstances is "just conventional" the same way.:confused:
 
  • #56
PeterDonis
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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?
I think the EP was what originally started Einstein on the road to GR. In 1907, he had what he called "the happiest thought of my life", which was basically the core of the EP: that, as he put it, "if a person falls freely, he will not feel his own weight". This is what made him realize that, locally speaking, we can "emulate" the effects of gravity using acceleration: that, for example, if we are standing in an "elevator" that is accelerating at 1 g in free space, locally we will see all the same physical phenomena as if we were standing in a room at rest on the surface of the Earth--for example, if we release a rock, it will accelerate downward, relative to us, at 1 g.

However, as central as the above reasoning was in helping Einstein to work towards GR, it had no place in the actual derivation of the Einstein Field Equation, which is what the actual "derivation of GR" was. Einstein did not use the EP to derive that equation (as you can see from my previous post where I described how he did it); he only used it as a sanity check, so to speak, to verify that the equation he derived produced the EP as a consequence.

EEP speaks of "uniform acceleration". Objects with rigidity (total resistance to mechanical stress) such as a rocket ship cannot accelerate uniformly.
Yes, and this fact means that the concept of "uniform acceleration", as it was apparently originally conceived by Einstein and others, can't actually be consistently formulated. What most sources now actually mean by "uniform acceleration" is what you are describing here: an object in which all parts remain at rest relative to each other as it accelerates, which then requires that the proper acceleration of the parts varies with "height" in the object (it's larger at the bottom and smaller at the top).

Note also that, in relativity, there is no such thing as a "perfectly rigid object", i.e., there is no such thing as an object in which all parts respond instantaneously to mechanical stress. That's impossible because any change caused by a stress applied at a certain point in the object can only be propagated through the rest of the object at the speed of light. This sets a finite limit to the "rigidity" of materials in relativity. The kind of motion described above is called "rigid motion" (or sometimes "Born rigid motion", after Max Born, who did important original work in this area), but it is an idealization, only realizable if precisely timed forces are applied to all parts of the object, which is impossible in practice. It's a very useful idealization, though, which is why it's used a lot.
 
  • #57
PAllen
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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.
Heuristics leading to guess at a theory are not assumptions (in general). They could be, in a some case, but on other cases not. The equivalence principle is not contained in any exact way in GR, and it is not present in most formal derivations of it.
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.
Prior to proving a theorem, you have to get the idea that it may be true. Any heuristic you might use to do that need not have any representation in the derivation of the theorem, nor need it be a consequence of the theorm.
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.
Well, on this world class experts differ with each other. You provide a good analogy. You can get pi using polygons of ever increasing number of sides. Further, the maximum distance between a sequence of such polygons and the circle approaches zero. However, you might argue that throughout the limiting process, you have zero curvature except in a measure zero set of point. Even in the limit, with countably infinite vertices, you still have only a countably infinite - therefore measure zero - set of vertices. Therefore a circle has zero curvature except at measure zero set of points (because the polygon 'becomes' the circle in the limit). This is obviously nonsense, and shows that not everything locally true becomes true in the limit. The divergence between a chord and arc approaches zero, but a chord is always distinguishable from the arc.

In the case of GR, Synge's point was that curvature has a finite non-zero value at a point; therefore gravity is distinguishable from acceleration of a rocket even at one point. Synge's argument was mathematical (not surprising - he was primarily a mathematician, but he wrote one of the seminal textbooks on GR). However, another physicist (Ohanian) put experimental meat on Synge's point by devising an 'in principle' device that can distinguish acceleration in flat space-time from gravitation due to a mass even in the limit of zero size for the instrument. Its reading approaches a constant times one of the curvature scalars which have a fixed value even at one point. A different line of dispute with principle of equivalence is that it doesn't necessarily hold for charged bodies (though the deviation, in practice, is too small to be measured in the foreseeable future).

Thus, quite objectively, one can say that Einstein's formulation of the principle of equivalence is false in GR. Most physicists (myself included) feel this is still (very clever) nitpicking. Almost all measurements, to any desired precision, are consistent with the principle of equivalence, intelligently applied, so it remains very useful. But, it clearly can't be either an axiom or a consequence of GR if it is technically false in GR.
 
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  • #58
PeterDonis
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As far as "the conventionality of simultaneity", I'm just not getting your point. I do not know what those word mean in your mind nor what purpose arbitrary choices can possibly serve. Particularly choices that are inconsistent in a given context (e.g. car experiment above).
Some of my latest posts might help with the car experiment. But let me give another illustration, one which is given early on in MTW, to illustrate the sense in which coordinates themselves are "conventional".

MTW describe a thought experiment which they call "the centrifuge and the photon". Suppose we have a "centrifuge", which we idealize as a ring rotating in its own plane (i.e., about an axis perpendicular to its own plane and running through the geometric center of the ring) at a constant angular velocity. At some instant, we fire a photon across the ring, so it is emitted at some point on the ring and received (some time later) at some other point on the ring. The photon has a known fixed frequency when it is emitted. What is its frequency when it is detected? I.e., is it redshifted, blueshifted, or neither?

You might try to solve this using frames--perhaps an inertial frame in which the geometric center of the ring is at rest, or even using MCIFs at the events of emission and reception. But MTW point out that you can solve this problem without using coordinates at all. All you need is the following:

(1) The photon is described by a 4-momentum vector which does not change along its trajectory. By "does not change" we mean that it behaves the same as the 4-velocity of an observer moving inertially--i.e., the "proper acceleration" of the photon (I put the term in quotes because it's not quite the same as an ordinary object's proper acceleration) is zero, so it moves along a geodesic, and its 4-momentum vector at a given point is just the tangent vector to that geodesic at that point. (Physically, this means that the photon is moving freely through vacuum, with no interactions with anything between emission and absorption, and that there is no tidal gravity or anything else that might affect its energy or momentum.)

(2) The frequency of the photon, as measured by a given observer, is just the inner product of the observer's 4-velocity with the photon's 4-momentum. (Strictly speaking, this gives the photon's energy, but quantum theory gives a direct relationship between energy and frequency--the former is just the latter times Planck's constant. For this problem that is sufficient.)

Therefore, if we write ##\mathbf{e}## for the 4-velocity of the point on the ring where the photon is emitted, at the instant it is emitted, ##\mathbf{p}## for the 4-momentum of the photon, and ##\mathbf{r}## for the 4-velocity of the point on the ring where the photon is received, at the instant it is received, we have (using ##E## for the frequency since the difference is just Planck's constant, as above) ##E_e = \mathbf{e} \cdot \mathbf{p}## and ##E_r = \mathbf{r} \cdot \mathbf{p}##. Note that ##\mathbf{p}## is the same in both formulas.

Now we just need to evaluate the two inner products. The inner product, as you can see from the way I wrote it, is just the generalization of the ordinary "dot product" in vector analysis to the case of 4-d spacetime; in other words, it is just the (cosine of the) "angle" in spacetime between two vectors. The only complication in spacetime is that "angle" contains the time dimension as well as the space dimensions; but this turns out to just be relative velocity (or at least that's a good enough way of looking at it for this problem--see below for how it works).

So what are the respective angles here? First, since the ring is rotating at a constant angular velocity, and since there is no tidal gravity (so we can directly compare velocities at different points in space), the relative velocity of the point of emission and the point of absorption is the same at the instant of emission as it is at the instant of absorption. This means that the "time" portion of the "angle in spacetime" between the photon and the ring's 4-velocity is the same at both emission and reception (since the photon itself always moves at ##c##).

That only leaves the "space" portion of the angle, but that is just the ordinary angle between the spatial vectors tangent to the ring and to the photon's motion in space. If you draw a diagram, you will see that these two angles are the same. So both the "space" and "time" portions of the angle in spacetime are the same at emission and reception. That means there is zero frequency shift!

The fact that we can obtain this answer without ever using coordinates at all (let alone any particular kind of coordinates, such as inertial ones) shows the sense in which coordinates are "conventional": you can do physics without them. And since a definition of simultaneity is just part of a definition of coordinates, since you can do physics without coordinates, you can do physics without simultaneity, so simultaneity is "conventional" in the same sense as coordinates are.

Furthermore, even if you choose to use coordinates, nothing requires you to use a particular kind of coordinates. For example, we could choose to solve the above problem using ordinary inertial coordinates (the obvious ones to use are ones in which the geometric center of the ring is at rest). But we could also choose to solve it using non-inertial coordinates in which the ring is at rest (these are called "Langevin coordinates" after Paul Langevin, who introduced them). These non-inertial coordinates use a different simultaneity definition from that of any of the MCIFs of points on the ring--in fact they use the same definition (i.e., the same sets of simultaneous events) as the inertial frame I just described. (But they are not the same coordinates, because, as I said, in these coordinates the ring is at rest, whereas it isn't in the inertial frame.) Or we could solve it using two MCIFs, those of the emission and absorption events, and use a Lorentz transformation to convert quantities from one to the other. Or we could use any other valid coordinate chart.

So none of these coordinates can be said to be necessary for solving the problem, nor can any particular definition of simultaneity. (In this particular case, all of the definitions turned out to be based on the Einstein convention somehow; but in other problems that would not be the case.) That is the sense in which these things are "conventional".

is distance not also "conventional", not to mention position?
If "conventional" means "depends on your choice of coordinates", then sure. Or if it means "depends on your method of measuring them", then again, sure. I'm not sure this is quite the same sense of "conventional" as the one I was using; see above.

Let me expand on this a bit more by using a simpler example. I can measure the distance between New York and Los Angeles along the great circle that connects the two, by laying a ruler end to end repeatedly starting at New York and ending at Los Angeles, taking care to make sure I lay the ruler "straight" along itself. (Or I can do it by other methods that give equivalent results.) The result of such a measurement is a geometric fact about the shape of the Earth's surface; it is only "conventional" in the sense that I picked which geometric fact I wanted to measure.

However, I can also compute this geometric fact using different coordinate charts. For example, I could use a standard Mercator chart, or I could use a stereographic projection centered on the North Pole. These charts are "conventional" in a sense in which the measurement itself, the geometric fact, is not.

In the case of spacetime, direct observables, like the elapsed time on a particular clock between two events on its worldline, or the length of a particular spacelike geodesic between two events (which could be interpreted as a "distance" between two points of interest), are geometric facts like the distance from New York to Los Angeles. They are only "conventional" in the sense that I can pick which ones I'm interested in. But coordinate charts are "conventional" in a different, stronger sense, the sense in which the charts used to describe the Earth's surface are "conventional". This is true even of "natural" charts like inertial coordinates in flat spacetime or latitude and longitude on the Earth; they happen to match particular symmetries of the objects being described, but that doesn't stop them from being conventional; it just means they're more suitable for certain problems.
 
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  • #59
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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.
Only for the particular point of view that the traveller is always in absolute rest, as you clarified yourself in your summary! That implies simultaneity conventions. But as I mentioned earlier*, it doesn't really work anyway.
* https://www.physicsforums.com/threa...paradox-as-paradox.780185/page-7#post-4921049
 
  • #60
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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.
That's quite correct except on one point: Einstein derived the properties of accelerated frames in SR as part of SR. From that he next derived, by means of the EEP (which was in his view part of GR), the properties for an equivalent non-accelerated reference frame in a gravitational field.

And with that correction, your explanation gains clarity:
[..] 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.
Yes, exactly :)
 
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  • #61
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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.
 
  • #62
PAllen
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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.
 
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  • #63
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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.
 
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  • #64
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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.
 
  • #65
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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.
 
  • #66
PeterDonis
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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.
 
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  • #67
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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.
 
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