Isotropy of the speed of light

[Sagittarius A-Star]

Bradley’s derivation was based on Galilean relativity, for which simultaneity is absolute. Its real issue for light speed determination is described in my prior post.

Yes, I tried to explain Bradley’s discovery with relativity, ignoring his theoretical assumptions. I should have mentioned that and also, that then Bradley's ## \tan \alpha ## must be replaced by ## \sin \alpha ##.

 

[Sagittarius A-Star]

Einstein’s derivation showed that within SR, aberration is not a measure of speed at all, but only a measure of how null directions transform between frames.

Regarding stellar aberration measurements, you have a frame transformation between two observer frames, not between the source frame and the oberserver frame. Astronomers cannot measure the "real" angle of the star location. They look through their telescope at a star and look several months later under a changed angle at the same star. So they measure the ange difference(s) of two or more observations. The "active" aberration due to movement of the star in the sun's frame cannot be measured this way.

 

[PAllen]

Regarding stellar aberration measurements, you have a frame transformation between two observer frames, not between the source frame and the oberserver frame. Astronomers cannot measure the "real" angle of the star location. They look through their telescope at a star and look several months later under a changed angle at the same star. So they measure the ange difference(s) of two or more observations. The "active" aberration due to movement of the star in the sun's frame cannot be measured this way.

Yes, I know that. I simply didn't specify that the two frames involved were 'observer' frames rather than e.g. source/target frames.

 

[bhobba]

I have never understood discussions about the isotropy of the speed of light. It is assumed we are dealing with an inertial frame. In inertial frames the laws of physics are the same in every direction, by the very definition of an inertial frame (see Landau - Mechanics if you have never seen this definition before and it's consequences). Maxwell's equations, that govern the speed of light, are a law of nature so it must be the same in any direction. This one always has had me beat even though there are textbooks like Ohanian's on Gravitation that raise it as a serious issue. Maybe I am missing something. Now do inertial frames actually exist - that is a more interesting issue. Certainly they conceptually do.

Thanks
Bill

 

[vanhees71]

That's true, but the isotropy of space for initial observers indeed is just an assumption about the spacetime model and as any assumption it's subject to experimental tests to verify or falsify its validity. So far there's no hint at a contradiction between the assumption and the experimental tests, and that's why GR is still considered to be the most comprehensive space-time model we have.

 

[Tendex]

That's true, but the isotropy of space for initial observers indeed is just an assumption about the spacetime model and as any assumption it's subject to experimental tests to verify or falsify its validity.

Well, not all assumptions are totally subject to empirical test, for instance assumptions that are conventions in a certain mathematical framework aren't. The last of the universal mathematicians, Poincaré, wrote a lot about this in relation with geometric models of spacetime. His was the first account of mathematical conventionalism and it was used profusely in Einstein's first article to construct SR based in postulates/definitions without any possibility of mathematical contradiction since they rely on conventions inherent to geometry(flat constant curvature for SR or with variable curvature for GR).

 

[vanhees71]

You can build a lot of sound and solid mathematical models of space-time. If you claim them to describe physical space and time the only way the physicist can figure out whether you have a good model is to empirically test it.

 

[Tendex]

You can build a lot of sound and solid mathematical models of space-time. If you claim them to describe physical space and time the only way the physicist can figure out whether you have a good model is to empirically test it.

You mentioned assumptions in your previous post and I merely mentioned that certain assumptions in the physical model are conventions related to geometry or to the mathematical tools used to do the modeling, so they are outside the empirical scope(of course for the new model if it incorporates the limit of the previous already empirically valid theory like it's the case with relativity that validity, to that accuracy, is of course kept).
For instance, how do you test the one-way speed of light without a simultaneity convention that incorporates the (affine) geometry of your spacetime and light's constant velocity postulate?

 

[Vanadium 50]

In inertial frames the laws of physics are the same in every direction, by the very definition of an inertial frame

Fine, but then this becomes the question "are inertial frames realized in nature"?

 

[bhobba]

Well, not all assumptions are totally subject to empirical test

If a frame is inertial or not is one that is subject to experimental testing. It is not a convention. For simplicity we often consider a frame we know is not inertial as inertial. I think a reading or refresher of what Feynman says in the first few chapters of his Lectures on Physics would help. For many experiments and solving problems we consider a table that a spirit level shows is flat, as actually being flat in the Euclidean sense. But, as Feynman points out, when looked at closely the boundary between table and air is nebulous. It consists of molecules of the table vaporising off, and atoms/molecules of air filling the gaps. Theoretically, when we analyse problems we make simplifying assumptions to solve it. That is part of what we, as humans, do. I know the arguments against measuring the one way speed of light, it being just a convention etc eg:
https://en.wikipedia.org/wiki/One-way_speed_of_light.

They all run up against the other evidence we have of the Earth being, to a high degree of accuracy, inertial.

Thanks
Bill

 

[bhobba]

Fine, but then this becomes the question "are inertial frames realized in nature"?

We know the answer to that - no. But yes to a high degree of accuracy. Do you doubt Euclidean Geometry because a point has position, but no size and such do not exist? Euclidean geometry can be doubted, and is indeed from GR is not true (except locally) - but that a point is an abstraction is not the reason it is doubted. An inertial frame is the same - as a conceptualisation very useful - but it's reality is up for grabs. Physical theories are models based on conceptualisations. Inertial frames are a very important part of many physical models including relativity.

We often make simplifying assumptions. However - point taken.

Thanks
Bill

 

[Vanadium 50]

Let's take it one step back. (And pretend it's before the latest unit redefinition which makes things more complicated.) The question "is the speed of light isotropic" is the same as "is the permittivity of free space the same in all directions". That is a well-defined experimental question. The answer "it must be because of the definition of an inertial frame" is something I find unsatisfying. We have to look. And if we know that it's good to ε (no pun intended) we would like to make ε as small as possible.

 

[bhobba]

Let's take it one step back. (And pretend it's before the latest unit redefinition which makes things more complicated.) The question "is the speed of light isotropic" is the same as "is the permittivity of free space the same in all directions". That is a well-defined experimental question. The answer "it must be because of the definition of an inertial frame" is something I find unsatisfying. We have to look. And if we know that it's good to ε (no pun intended) we would like to make ε as small as possible.

Yes. Everything in science is open to doubt, or as you say, without experimental confirmation, deeply unsatisfying. Indeed such would be an interesting question. I think I was jumping the gun somewhat in saying I have never understood the discussion about the one way speed of light. What I really mean is we normally assume inertial frames. The issue then is, as you pointed out, just how good an assumption is it really.

Thanks
Bill

 

[vanhees71]

You mentioned assumptions in your previous post and I merely mentioned that certain assumptions in the physical model are conventions related to geometry or to the mathematical tools used to do the modeling, so they are outside the empirical scope(of course for the new model if it incorporates the limit of the previous already empirically valid theory like it's the case with relativity that validity, to that accuracy, is of course kept).
For instance, how do you test the one-way speed of light without a simultaneity convention that incorporates the (affine) geometry of your spacetime and light's constant velocity postulate?

No, they are not outside the empirical scope. If you assume isotropy of space for (locally) inertial observers you can test this empirically. In principle nothing prevents you from finding a dependence of the speed of light on the orientation of your measurement device, and then this specific symmetry assumption would be disproven. Of course, today no such violation is known.

It's like the question of parity symmetry. Before the mid 1950ies everybody thought that "of course" nature is symmetric under spatial reflections, and famously Pauli thought Wu's experiment is superfluous, and he was clearly proven wrong by this and other experiments concerning the weak interaction. Finally it turned out that the weak interaction violates parity in a sense maximally (pure "V minus A coupling").

 

[vanhees71]

Let's take it one step back. (And pretend it's before the latest unit redefinition which makes things more complicated.) The question "is the speed of light isotropic" is the same as "is the permittivity of free space the same in all directions". That is a well-defined experimental question. The answer "it must be because of the definition of an inertial frame" is something I find unsatisfying. We have to look. And if we know that it's good to ε (no pun intended) we would like to make ε as small as possible.

Exactly, and that's why indeed isotropy is subject to experimental tests. Of course, it's true, you need a space-time model to measure time intervals and lengths. From the space-time model also the physical laws are constraint to some extent due to the symmetries implied by the assumed space-time model, and within this constraints you can build mathematical models to describe/predict real-world phenomena. Then you make observations and measurements in the real world using the underlying model to define quantitative measures for these observables. Nothing a priori ensures that everything turns out as predicted by the models, and you can check these (symmetry) assumptions by probing all kinds of consequences derived from them by building physical models.

Concerning the assumption of the existence inertial reference frames it's also clear that this is also subject to experimental test (and, sorry, @Dale , when discussing this issue about experimental tests of space-time models I must use the "clocks-and-ruler definition" of a reference frame).

Of course, Newton simply assumed an absolute space and an absolute time (it's not even really a "space-time", because of the fiber-bundel structure of the Newtonian space-time model) and that was it for him. Nevertheless his view was already criticized, among other famously by his arch enemy Leibniz, who logically argued that motion cannot be absolute within Newton's own theoretical edifice, because all inertial frames are equivalent, and it's indeed necessary to define an inertial reference frame by realizing it by some reference point and three directions (realizable by rigid rods, which within Newton's physics of course exist) as well as a clock, which can be defined by a reference body assumed to move with constant velocity wrt. the (hopefully inertial) reference frame.

A naive starting point of course is, as done in any physics freshman lecture on day 1 (usually not expclitly ;-)), to just use your lab fixed at rest in Earth as an inertial reference frame, taking the ever present gravitational force of the Earth on all bodies into account as a homogeneous fource ##m \vec{g}##. As we all know, with this assumption you get very far.

It's of course clear that the Earth-fixed lab frame is for sure not an inertial reference frame. You may rather take the fixed stars as reference bodies defining an inertial frame, and then you expect that indeed the earth-fixed frame is even a rotating frame, both from the motion of the Earth around the Sun and its spin around its axis once per day. Then you develop the theory what to expect when using a non-inertial rotating reference frame and predict that the Foucault pendulum can be used to demonstrate the rotation of the Earth (wrt. the fixed-star reference frame), and as is well-known this indeed turns out to be right.

Then in the mid 19th century Maxwell developed his non-Galilei invariant electrodynamics, and many (if not all?) physicists thought that this finally fixes the reference frame for Newton's absolute space (and time). It was also theorized (including Maxwell himself) that Maxwell's electromagnetic waves are due to the vibrations of the aether, whose (global) rest frame defines Newton's absolute space. The history is known: From this it should be possible to empirically prove the existence of this absolute space and this preferred inertial aether rest frame. Then the null result of the Michelson Morley experiment, which was the first experiment being sensitive to order ##\mathcal{O}[(v/c)^2]##, showed that this idea is not correct and, even more famously, Einstein turned the argument around in 1905 and introduced a new space-time model (2 years later analyzed by Minkowski in its mathematical/geometrical structure and thus henceforth called "Minkowski space-time") valid for all of physics.

The up to now last step then was the development of GR by Einstein in his attempt to find a relativistic theory for the gravitational interaction, leading to a dynamical space-time picture. Here the important property is the equivalence of "inertial and gravitational mass", which finally in the mathematics boils down to the assumption that at any space-time point one can define a locally inertial reference frame. The extent of this local reference frame is determined by the homogeneity of the gravitational field as can be measured with test particles, and then the local inertial reference frames are defined in the neighborhood of the space-time point in question by a freely falling pointlike test body, which is then moving along a timelike geodesic of the curved Lorentzian spacetime, defining a time-like unit tangent vector (the four-velocity of the body) which then enables the construction for three space-like non-rotating unit vectors building together with the four-velocity of the body a free-falling non-rotating tetrad, defining a local inertial reference frame. It's of course only inertial to the extent defined by how accurately the gravitational field the test body is freely falling in can be regarded as homogeneous and to this accuracy the gravitational force can be regarded as equivalent to the inertial forces in a non-inertial reference frame being accerated relative to the free-falling tetrad just constructed to define the local inertial reference frame. Of course a "true gravitational field" can never be completely explained as equivalent to the inertial forces in a non-inertial local reference frame but there are always deviations from an exactly homogeneous gravitational field, leading to tidal forces measurable also in the local inertial reference frame.

 

[Vanadium 50]

t's like the question of parity symmetry.

Martin Deutsch (discoverer of positronium) told the story that in the early 1950's a grad student working on a positron experiment came up to him and said that there were more events in the left side of the detector. Marty told him that was was ridiculous. That would mean that the weak interaction was parity violating. He told the student to go fix things. Which he did.

 

[Tendex]

No, they are not outside the empirical scope. If you assume isotropy of space for (locally) inertial observers you can test this empirically. In principle nothing prevents you from finding a dependence of the speed of light on the orientation of your measurement device, and then this specific symmetry assumption would be disproven. Of course, today no such violation is known.

Maybe I'm not getting my point across, I think Vanadium 50 got It right. It's the inertial frame what cannot be empirically discarded without mathematical contradiction. So inertial frames are postulated together with the 2 postulates of SR, and then, as Einstein underlined several times no contradiction will be found in the theory. So inertial motion existence is a primitive definition/postulate of the theory just in order to not enter into contradiction and it is not subject to empirical test as an assumption. In other words its presence guarantees the theory is not contradictory, therefore its absence leads to contradiction. This is known as the Law of the excluded middle. If a physical model that claims to be based on mathematics failed it it wouldn't be taken too seriously.

 

[Sagittarius A-Star]

No, they are not outside the empirical scope. If you assume isotropy of space for (locally) inertial observers you can test this empirically. In principle nothing prevents you from finding a dependence of the speed of light on the orientation of your measurement device, and then this specific symmetry assumption would be disproven. Of course, today no such violation is known.

But that is only valid for the 2-way-speed of light. The isotropy of the one-way speed of light in an inertial (= not accelerated) frame is not an assumption. It is a definition. If you synchonize clocks at the light source and at the light detector differently from the Einstein-synchonization, then you will measure a non-isotropic one-way-speed of light.

 

[hutchphd]

The isotropy of the one-way speed of light in an inertial (= not accelerated) frame is not an assumption. It is a definition.

It is an assumption of convenience. As is the standard method of synchronization. It is not a definition because the theory is consistent without it being specified.

 

[Sagittarius A-Star]

It is an assumption of convenience. As is the standard method of synchronization. It is not a definition because the theory is consistent without it being specified.

No, it is a definition:

you declare: "I maintain my previous definition nevertheless, because in reality it assumes absolutely nothing about light. There is only one demand to be made of the definition of simultaneity, ...

Source

 

[Dale]

I think that there is a good point that @Tendex is making. There are assumptions that we make that are purely conventional and are not subject to empirical testing. That includes all definitions of terms and many mathematical conventions, like using the right hand rule, and other conventions like positive charge for protons or units. None of those assumptions are subject to testing.

There are other assumptions that we make which are empirically testable, like isotropy of the two-way speed of light, or conservation of momentum. So we need to test these testable assumptions and we need to distinguish when an assumption is testable or not so that we don't waste time trying to test untestable assumptions and so that we don't mentally elevate our convention assumptions to the status of facts about nature.

 

[Sagittarius A-Star]

like isotropy of the two-way speed of light, or conservation of momentum.

I think, the conservation of "one-way momentum" also depends on the definition of simultaneity, so it should be: "like isotropy of the two-way speed of light, or conservation of two-way momentum."

 

[PeterDonis]

I think, the conservation of "one-way momentum" also depends on the definition of simultaneity, so it should be: "like isotropy of the two-way speed of light, or conservation of two-way momentum."

There is no such thing as "one-way momentum" vs. "two-way momentum". There is just momentum.

 

[Sagittarius A-Star]

There is no such thing as "one-way momentum" vs. "two-way momentum". There is just momentum.

Yes. But consider a coordinate chart, that creates an anisotropy of the one-way light speed in (+/-) x-direction and describe in this coordinate chart an explosion. The momentum will not be conserved.

 

[PeterDonis]

Yes. But consider a coordinate chart, that creates an anisotropy of the one-way light speed in (+/-) x-direction and describe in this coordinate chart an explosion. The momentum will not be conserved.

This is not correct. Conservation of momentum is not a coordinate-dependent law. In the coordinate chart you describe, the one-way light speed won't be the only thing that is different from a standard inertial chart. Other things will be different as well, in just the right way to preserve momentum conservation.

 

[Sagittarius A-Star]

Conservation of momentum is not a coordinate-dependent law.

I think that is only the case, as long you also use the Einstein definition of simultaneity:

Salmon (1977, 273) argues, however, that the standard formulation of the law of conservation of momentum makes use of the concept of one-way velocities, which cannot be measured without the use of (something equivalent to) synchronized clocks at the two ends of the spatial interval that is traversed; thus, it is a circular argument to use conservation of momentum to define simultaneity.

Source:
https://plato.stanford.edu/entries/spacetime-convensimul/

 

[vanhees71]

I think that there is a good point that @Tendex is making. There are assumptions that we make that are purely conventional and are not subject to empirical testing. That includes all definitions of terms and many mathematical conventions, like using the right hand rule, and other conventions like positive charge for protons or units. None of those assumptions are subject to testing.

There are other assumptions that we make which are empirically testable, like isotropy of the two-way speed of light, or conservation of momentum. So we need to test these testable assumptions and we need to distinguish when an assumption is testable or not so that we don't waste time trying to test untestable assumptions and so that we don't mentally elevate our convention assumptions to the status of facts about nature.

That's true. You always need both theory and operational definitions that relate the theoretical (mathematical) elements to the phenomena you measure, i.e., make quantifiable and describable by abstract mathematical models (e.g., to use the real numbers to measure the distance between points in Euclidean geometry, which is a pretty modern finding by Hilbert).

Nevertheless even conventions like the definition of units are in principle subject to experimental test, i.e., when there accumulates evidence of inconsistencies between measurement results of a certain phenomenon using theory to define these units, it may be that the theory is not (precisely) correct. E.g., if the fine structure constant is not really a constant our definition of the Ampere (or Coulomb) in the SI is not consistent, and indeed a possible time dependence of the fine structure constant is indeed considered (with no hint so far that this may really be the case).

 

[vanhees71]

I think that is only the case, as long you also use the Einstein definition of simultaneity:

Source:
https://plato.stanford.edu/entries/spacetime-convensimul/

Momentum conservation is a direct consequence of the assumption of homogeneity of space for any inertial observer. As such it does not depend on any choice of coordinates, because everything observable is independent of the choice of coordinates by construction.

 

[Sagittarius A-Star]

Momentum conservation is a direct consequence of the assumption of homogeneity of space for any inertial observer. As such it does not depend on any choice of coordinates, because everything observable is independent of the choice of coordinates by construction.

Is this also true in an anisotropic inertial frame?

Salmon argued that momentum conservation in its standard form assumes isotropic one-way speed of moving bodies from the outset.
...
In addition, Iyer and Prabhu distinguished between "isotropic inertial frames" with standard synchrony and "anisotropic inertial frames" with non-standard synchrony.[25]

Source:
https://en.wikipedia.org/wiki/One-way_speed_of_light#Inertial_frames_and_dynamics

 

[vanhees71]

What do you mean by "anisotropic inertial frame"? If the space of an inertial observer is not isotropic you change the standard space-time model itself. This cannot be done by simply choosing some coordinates in Minkowski space. All geometrical properties are independent of the choice of coordinates.

It's as in Euclidean space: Only because you use spherical coordinates, implying to choose an arbitrary point as the origin and a direction as the polar axis, you don't destroy isotropy and homogeneity of Euclidean (affine) space.

 

[Sagittarius A-Star]

What do you mean by "anisotropic inertial frame"?

I mean the primed frame in:
$$x' = x \ \ \ \ \ y' = y \ \ \ \ \ z' = z \ \ \ \ \ t' = t + \frac{kx}{c}$$
Source:
https://www.mathpages.com/home/kmath229/kmath229.htm

 

[Tendex]

I mean the primed frame in:
$$x' = x \ \ \ \ \ y' = y \ \ \ \ \ z' = z \ \ \ \ \ t' = t + \frac{kx}{c}$$
Source:
https://www.mathpages.com/home/kmath229/kmath229.htm

That's using a synchronization convention different than Einstein's that makes you use non-inertial coordinates that are more contrived. Doesn't affect what vanhees said.

 

[Sagittarius A-Star]

That's using a synchronization convention different than Einstein's that makes you use non-inertial coordinates that are more contrived. Doesn't affect what vanhees said.

That are inertial coordinates. Reason: This frame is not accelerated. Therefore, it does not contain fictitious forces.

 

[PeterDonis]

I mean the primed frame in:
$$x' = x \ \ \ \ \ y' = y \ \ \ \ \ z' = z \ \ \ \ \ t' = t + \frac{kx}{c}$$
Source:
https://www.mathpages.com/home/kmath229/kmath229.htm

This is not an inertial frame. The article you reference does not say that it is.

 

[PeterDonis]

That are inertial coordinates. Reason: This frame is not accelerated. Therefore, it does not contain fictitious forces.

"Not accelerated" is a necessary condition for "no fictitious forces", but not a sufficient one. Try describing the motion of a free body (i.e., one that would have zero coordinate acceleration in a standard isotropic inertial frame) in your anisotropic frame. What is its coordinate acceleration?