Is there a proof that all observable predictions of special relativity follow from the Lorentz transformations?

I have edited my question with concrete examples, so that it can be understood more easily.

The formulation of special relativity I have in mind is that exposed by Einstein in 1905, before it was reformulated in term of the Minkowski spacetime. The original formulation rests on the following assumptions:

  • The principle of relativity
  • The constancy of the one-way speed of light
  • Homogeneity of space and time

From these assumptions can be derived the Lorentz transformations. Then from the Lorentz transformations can be derived experimental consequences, i.e. predictions.

Consider the following Lorentz transformation between two relatively moving observers:

$$ x' = \gamma(x - v t) \\ t' = \gamma(t - \frac{v x}{c^2}) $$

By setting $$x = vt$$ one gets $$t' = \frac{t}{\gamma}$$

If the two observers were initially at the same location and synchronized ($x=x'=0$ when $t=t'=0$) then the Lorentz transformations predict that when the observer's clock at the origin shows the time $t$, the moving observer's clock shows the time $t' = \frac{t}{\gamma}$ . This is an example of a non-observable (or non-testable, non-verifiable, non-falsifiable, ...) prediction, for we have no instantaneous signals to test if this is indeed the case.

However the relation $t' = \frac{t}{\gamma}$ leads to testable predictions. For instance it predicts that when the moving observer is at a distance $d$ from the origin, his clock will show the time $$t' = \frac{\frac{d}{v}}{\gamma}$$

It also predicts that if at time $t'$ the moving observer instantly turns around and move back towards the origin at velocity $-v$, then when the two observers meet again the moving observer's clock will show the time $2t'$, while the clock of the observer who stayed at the origin will show the time $2t'\gamma$. This is an observable prediction, that can be checked experimentally.

Similarly the Lorentz transformations allow to derive many other observable predictions, such as that a signal of frequency $f$ emitted by the moving observer towards the origin will be measured by the observer at the origin to have the frequency $f\sqrt{1+\beta\over 1-\beta}$. In other set-ups it allows to derive the angle of aberration of distant light sources.

All these observable predictions are a consequence of the assumptions of special relativity, since the Lorentz transformations are derived from them. But these predictions can also be arrived at without using the Lorentz transformations, by starting from the assumptions of the theory. So how do we know that the Lorentz transformations encompass all the possible observable predictions that can be made, starting from the assumptions of special relativity?

In other words, if we call A the assumptions of special relativity, B the Lorentz transformations, C the observable predictions derived from the Lorentz transformations, and D the observable predictions derived the assumptions of special relativity, how do we know that C is exactly the same as D and not just a subset of D? Is it possible to prove that C and D are the same?

As to the motivation behind this question, I am wondering whether two theories that start from different assumptions but from which Lorentz transformations can be derived are necessarily indistinguishable experimentally. My point of view is that if the Lorentz transformations do not encompass all the predictions that can be derived from the assumptions of each theory, then in principle the two theories may be distinguishable.


closed as off-topic by CuriousOne, Daniel Griscom, Gert, user36790, yuggib Jan 7 '16 at 14:09

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    $\begingroup$ But this is missing the point of my question which you have misinterpreted. By "verifiable predictions" I mean "testable predictions", in contrast to predictions that cannot be tested (such as that the one-way speed of light is c). And I am not asking whether special relativity is consistent with all experiments (although I would argue that in its domain of application it is unless I missed something), what I am asking is whether the postulates of special relativity and the Lorentz transformations make all the same testable predictions. $\endgroup$ – Peter Jan 6 '16 at 9:55
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    $\begingroup$ I'm voting to close this question as off-topic because it seems to belong into philosophy. $\endgroup$ – CuriousOne Jan 6 '16 at 10:14
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    $\begingroup$ It does not belong to philosophy, you are the one introducing philosophy by trying to lecture me on the difference between verification and falsification because you misinterpreted my question. Then once I refer to a philosophical work to show why one of the claims you make in the comment section is wrong, claim totally unrelated to my question, you conclude then that my question must belong to philosophy, but that's a non sequitur, logically. $\endgroup$ – Peter Jan 6 '16 at 10:26
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    $\begingroup$ My question is about physics and logic. The postulates of special relativity have experimental consequences. The Lorentz transformations have experimental consequences. The Lorentz transformations are one consequence of the postulates of special relativity. This does not imply that all experimental consequences of special relativity are consequence of the Lorentz transformations. That you do not comprehend my question does not transport it to the realm of philosophy. $\endgroup$ – Peter Jan 6 '16 at 10:28
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    $\begingroup$ You are the one who is mistaken, I explained why. That I am a new user on this website does not mean that I am new to science, physics or its philosophy, i.e. what it does. Have a nice day too. $\endgroup$ – Peter Jan 6 '16 at 10:38

Special relativity is the spacetime geometry described by the Minkowski metric. It is the vacuum solution to Einstein's equation with the lowest ADM energy. All the properties of the geometry, time dilation, lorentz contraction, etc, are described by this metric.

Starting with the Minkowski metric it is possible to derive the Lorentz transformations. They are a special case where all motion is unaccelerated. If you wish to describe accelerated motion you need to go back to using the metric directly.

So there is no:

proof that all experimentally verifiable predictions of special relativity follow from the Lorentz transformations

because they do not - they are a special case. As for a proof that all verifiable predictions of SR are described by the Minkowski metric, that's a tautology because the Minkowski metric is special relativity.

  • $\begingroup$ Thank you for your answer. But isn't it possible to describe accelerated motion by treating it as successive uniform motions on infinitesimal time intervals, thus using Lorentz transformations with a specific velocity during each such time interval? $\endgroup$ – Peter Jan 6 '16 at 11:50
  • $\begingroup$ Also to clarify further discussion and so that you know where my question is coming from, the formulation of special relativity I have in mind is that of Einstein in 1905, prior to the introduction of Minkowski's metric and Einstein's field equations. The way I see it all the consequences of special relativity should be derivable from the postulates in this 1905 formulation (principle of relativity, constancy of the speed of light, homogeneity of space and time), but some consequences of these postulates may not be consequences of the Lorentz transformations. $\endgroup$ – Peter Jan 6 '16 at 11:52
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    $\begingroup$ I don't think that "they are a special case" is correct. If you start from the Lorentz group and the assumption that it preserves the spacetime metric (which is just another phrasing of it being a symmetry group, or the group that relates inertial frames), it follows that the spacetime metric is Minkowski, and then the whole of SR follows. $\endgroup$ – ACuriousMind Jan 6 '16 at 17:28
  • $\begingroup$ The last sentence of this answer --- and ACuriousMind's comment --- seem to me to be clearly correct and responsive to the OP's question. It also seems clear that questions like "What are the implications of Lorentz symmetry", "What are the implications of SR", and "Are those two sets of implications identical?" are unambiguously questions about physics. Although this particular question has a very easy answer, I'm glad that the comments intended to derail this question did not deter John from answering. $\endgroup$ – WillO Jan 6 '16 at 19:49
  • $\begingroup$ @WillO Neither the answer nor the comments are correct. The OP clearly and specifically brings up homogeneity and the answer ignores that by throwing away every other manifold with the Minkowski metric without any mention that the others aren't homogeneous. And the comments repeat it by using the Lorentz group instead of the Poincaré group. $\endgroup$ – Timaeus Jan 7 '16 at 3:24

Historically, Lorentz transformations were discovered before Einstein's relativity. But only special relativity with its 2 postulates formed today's theoretical base for Lorentz transforms. So, today there is no doubt that Lorentz transforms may be derived from SR.

Now you are asking if it would be possible to derive conversely the two SR postulates from Lorentz transforms. This is not the case, because SR postulates imply certain constellations which are not considered by Lorentz transforms. In particular, certain parts of Minkowski spacetime such as the light cone, zero spacetime intervals and lightlike movements are not taken into consideration by Lorentz transforms, the transformations being limited to reference frames.

Edit/ Observable predictions: The movement at speed of light cannot be predicted because Lorentz transforms are referring only to reference frames, and massless particles do not have any reference frame.

  • $\begingroup$ Thank you, so this would answer the second part of my question, that B does not imply A. In the end the core of my question is: is there any observable prediction of special relativity, which can be derived from its postulates but not from the Lorentz transformations? By observable I mean something which could be in principle detected in an experiment. $\endgroup$ – Peter Jan 6 '16 at 14:00
  • $\begingroup$ See my edit in my answer. $\endgroup$ – Moonraker Jan 6 '16 at 14:47
  • $\begingroup$ That makes sense, since the Lorentz transformations are not defined for v = c. But can the motion at the speed of light really be construed as a prediction of special relativity but not of Lorentz transformations? For in order to apply the Lorentz transformations to experimental situations one has to measure the velocity of another observer, and in order to do that one has to already have a set of synchronized clocks, and in order to have that one has to synchronize clocks by exchanging light signals, which assumes that light travels at c in straight lines in the first place... $\endgroup$ – Peter Jan 6 '16 at 16:05
  • $\begingroup$ That means, in order to put Lorentz transforms into experimental practice you need SR. $\endgroup$ – Moonraker Jan 6 '16 at 16:55
  • $\begingroup$ Well it means you need the postulate of the constancy of the speed of light, but do we really need the principle of relativity to apply the Lorentz transforms to experimental situations? $\endgroup$ – Peter Jan 6 '16 at 17:00

The Lorentz transformations by themselves have no physical content and make no predictions. They basically take things called frames, each of which contains some ordered four tuples of numbers, and describes a particular bijection between the set of four tuple in one fixed frame and the set of four tuples in another fixed frame.

If you wanted to, you could do all of SR without ever picking or using a frame. You could have a flat homogeneous isotropic manifold with a metric of Lorentzian signature.

This is an example of a non-observable (or non-testable, non-verifiable, non-falsifiable, ...) prediction, for we have no instantaneous signals to test if this is indeed the case.

It's an oxymoron to say you have a non-falsifiable prediction. A prediction is a precise statement of something that could be observed to happen and could also be not observed to happen.

In this case you'd need to use SR to make a model. The model could be that for a each inertial frame, clocks can exist at rest in that frame and the inertially moving clocks measure time durations between two events where the clock is situated according to the time coordinate difference for the four tuples corresponding to the two events.

The model required that you correspond events in reality to events in the mathematical model. The mathematical model could be a 4d manifold. It could also be a 4d vector space. Each real event could also be identified with an equivalence class of four tuples of coordinates, one four tuples from each frame, specifically chosen so the transformation maps the corresponding four tuples to each other.

You could even start with the manifold and then select frames as different assignments of four coordinates to each 4d point on the 4d manifold. Which you start with, and even whether you choose to use Lorentz transformations is up to you.

Then you can make a synchronization convention so that a bunch of inertially moving clocks all with zero relative velocity (which can also be defined in the framework of SR) all basically record the time coordinate of that frame in which they are at rest.

So now you simply take a clock at rest in each frame that record 0 when both are the first event. And take another pair of clocks at rest in each frame that record $t$ and $t'$ when both are at the second event.

Since clocks record times and since the second pair are both at one and the same event, the comparison is directly achievable and the record can be sent to the future for analysis.

SR has a common future for any events, so its actually one of the features that any fact that can be recorded and transmitted in two regions can be analyzed jointly and together in some region. So if you think otherwise you've failed to grasp which models the theory allows.

Also keep in mind that the Lorentz Transformations are basically maps of vector spaces. General Relativity can use them for the set of tangent vectors at a point. And Special Relativity could use them on event themselves. But those are totally different uses of the same mathematics and the same symmetry.

what if I characterize Lorentz transformations by: "If by the assumptions of SR A infers an event to happen at (x,y,z,t) and B infers an event to happen at (x',y',z',t'), then the relations between (x,y,z,t) and (x',y',z',t') are called the Lorentz transformations", would you agree then that these relations allows to make predictions?

Firstly, no those still make no predictions. All you are doing is identifying a set of labels for one person with a set of labels for another person. And secondly you should use Poincaré Transformations for SR. Lorentz transformations always identify (0,0,0,0) for one person and (0,0,0,0) for another person. But even then, still you have no predictions.

Someone could identify points from different people's vector spaces and then not use the vector space to make predictions. Even if they used them, they could have equations like $F=ma$ or equations like $E=mv$ or $G=mj$ ($j$ for jerk) there is no rule that they have mass, inertial motion, momentum, second order differential equations for dynamics. And in fact Maxwell doesn't have mass for electromagnetic fields and does have first order dynamics. And relativistic fluid mechanics and particle mechanics does have mass and has second order dynamics. So there is at least two types, and they are coupled.

Really you should think of SR as either a meta theory telling what symmetries your actual theory should respect. Or else you need way more to it than just giving a bunch of people some vector spaces and identifying points in them.

Without either, you could ignore the vector space and instead use something else to make all your predictions and your predictions could be arbitrarily different than physics as we know it. Predictions come from models. Models

in order to synchronize the clocks one has to assume that light travels at c in all directions.

That isn't actually true. Clock synchronization is basically a convention. If light "really did" move at different speeds in two directions, we could still use the radar time convention to synchronize clocks and postulate that a clock moving inertially between two events that have the same spatial coordinates in some frame measures and records the time coordinate difference in that frame where the spatial coordinates don't change. Then you've connected clocks to you coordinates. And the synchronization allows clocks that are mutually at rest to 1) be identified as mutually at rest 2) to be identified as synchronized if synchronized according to the convention. This making of a model required way way way more than just having lots of vector spaces and identifying one vector from each with an event. We have to connect aspects of the mathematics to thibgs that can be made, done, seen, and observed in the real world.

For instance we can connect clock readings to time coordinates. We could also connect ruler readings to spatial coordinates. We can say that objects track out differential (or at least connected) parameterized curves in the vector spaces, or in the spacetime and call those parameterized curves worldlines. This tells us that particles can move in the spacetime (we didn't have that before, the LT by itself didn't tell us that). We could assert that some move so that only a time coordinate changes, and we could call those worldlines inertial motion. And we could say those particles have mass and we can assert that every frame agrees it has the same mass and that it never changes. We can assert that some particles have their x and t coordinates change an equal amount. We can assert that those particles have an energy-momentum four-vector that points in the same direction as the tangent to the worldline. We could say that the massive particles don't have to move inertially (though the inertially moving ones do have to have mass).

Obviously the LT by themselves do not tell us that because they don't mention mass or momentum.

Even in introductory physics you probably remember the section on kinematics and then dybamkcs came later. Kinematics didn't make predictions. The LT don't make any predictions either. They merely connect some vector spaces for different people. I'm flabbergasted that you can even imagine the LT making any predictions. It's like if some people named their cats and someone else named the cats their own names. The Lorentz Transformations could tell you which cat names the first person uses relate to which cat names the second person uses. But it doesn't matter. Cats don't respond to their names (and if you think they do use fish names or something else instead). Identifying labels doesn't make predictions unless those labels are used to make predictions.

You need dynamics to make predictions. Something like Maxwell. Or something like $F=ma$ and some force laws. Or something about how clock readings are related to time coordinates. Or something about how ruler readings are connected to x,y, and Z coordinates. Or lots of those.

along with the assumption of the constancy of the speed of light which is implied in order to apply the relations experimentally

If you have the LT and you relate the time coordinates to clocks and the spatial coordinates to rulers, then you really think you need to add an extra assumption of constancy of speed c? That doesn't even make sense. You can't just add an assumption to an existing mathematical framework and not change the framework and expect that you are using it. The closest you can get to that is like Newton's third law where you decide to restrict your changes in momentum so that when something loses some momentum it gives an amount equal momentum what it lost to something else. Before you had that law you were allowed to do otherwise, and you placed a restriction on your allowed models to not do that. Or with Maxwell you could asd the equation $\vec \nabla \cdot \vec B=0$ and you've now restricted your initial magnetic fields to have zero divergence, so you don't allow just any initial field. In your case if you wanted to try to add this assumption about c, the closest you could get is if you are trying to restrict how you assign clock readings and ruler readings to coordinates to make it so that c will be constant. That's unnecessary if you assign them the regular way, and its too vague if you haven't specified how to connect coordinates to measurements. For instance you allow that people could also have 5d vector spaces in addition to their 4d vector spaces and they could have 5d LT between them and have rulers and clocks and such related to the 5d vector space in the usual way and now c will be constant but this is a different theory with four independent spatial directions. The old 4d vector spaces simply aren't being used to make predictions becasue you never required that have to be.

In case you can't tell, the issue with the LT is that coordinates and LT by themselves don't tell you anything at all. You have to have models that have mathematics and connections between the mathematics and real world stuff. You need to say how coordinates are related to things that can be measured.

The LT don't tell you whether there are things. Or whether they can move inertially. Or whether they can move at c. Or whether they can move at faster than c. The LT by itself is consistent with all of those and with forbidding all of those.

When we do mechanics we'll say things like that the tangent to a worldline of a massive particle always has the same sign for the square. Or we will make force laws and say $F=dp/dt$ and say that just like the massless particles the energy-momentum four vector is tangent to the worldline.

Lots of stuff and connections to the real world have to be made before you get predictions.

  • $\begingroup$ Thank you for your answer. So you are saying the Lorentz transformations make no prediction, thus that my question is meaningless. However, what if I characterize Lorentz transformations by: "If by the assumptions of SR A infers an event to happen at (x,y,z,t) and B infers an event to happen at (x',y',z',t'), then the relations between (x,y,z,t) and (x',y',z',t') are called the Lorentz transformations", would you agree then that these relations allows to make predictions? And then are all the predictions derivable from the assumptions of SR also predictions of these relations? $\endgroup$ – Peter Jan 7 '16 at 8:10
  • $\begingroup$ Where A and B are each in constant uniform motion in an inertial frame $\endgroup$ – Peter Jan 7 '16 at 8:12
  • $\begingroup$ And where in order to apply the relations to experimental situations one has to have a set of synchronized clocks to measure velocities, and in order to synchronize the clocks one has to assume that light travels at c in all directions. So basically by "Lorentz transformations" I mean the mathematical relations between what A and B infer to happen (derived from the assumptions of SR), along with the assumption of the constancy of the speed of light which is implied in order to apply the relations experimentally $\endgroup$ – Peter Jan 7 '16 at 8:20
  • $\begingroup$ @Peter Firstly, no those still make no predictions. All you are doing is identifying a set of labels for one person with a set of labels for another person. And secondly you should use Poincaré Transformations for SR. Lorentz transformations always identify (0,0,0,0) for one person and (0,0,0,0) for another person. But even then, still you have no predictions. $\endgroup$ – Timaeus Jan 7 '16 at 15:51
  • $\begingroup$ @Peter I edited my answer. And you might be able to see why this isn't physics. You have some math and you didn't connect it to any observations and so its non science which is fine. But if you act like its science when it isn't that is a huge problem. And something that some people do nonstop and never learn to do otherwise, so you set off every warning sign. $\endgroup$ – Timaeus Jan 7 '16 at 16:44

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