# Generalizing the special principle of relativity?

I usually read the (special) principle of relativity stated along the lines of: "the laws of physics take the same form in all inertial reference frames". Here's my question: can we generalize this by saying that the laws of physics take in same form in any two reference frames-- perhaps noninertial ones -- moving with constant velocity relative to one another?

I can't think of a counterexample, but I'd like the input of other people.

Bill_K
The laws of physics are generally covariant. That says it all!

PAllen
The laws of physics are generally covariant. That says it all!

Surely you are aware that the relevance and meaning of general covariance has been disputed since 1917, at least:

The laws of physics are invariant under any continuous, differentiable remapping of positions in spacetime.

The laws of physics are also invariant under any continuous, differentiable, and local rotation of fields in spacetime.

PAllen
The laws of physics are invariant under any continuous, differentiable remapping of positions in spacetime.

The laws of physics are also invariant under any continuous, differentiable, and local rotation of fields in spacetime.

The problem is, as Kretschman showed in 1917, any laws (specifically either Newton's Gravity or Special Relativity) can be expressed in a way to meet the above propositions. See the reference I cited as well has hundreds of discussions of these points in the literature.

Bill_K
Surely you are aware that the relevance and meaning of general covariance has been disputed since 1917, at least:
No, I must admit I'm not aware of it at all. Even after a full career devoted to GR I've never heard anyone question general covariance. Perhaps that comes from talking to physicists rather than philosophers.

I didn't say that general covariance marks the distinction between special and general relativity, or anything like that. General covariance applies to special relativity just as well. "The laws of physics are generally covariant" is a universal requirement. If you want to consider coordinate systems which are not Minkowskian, you can do so in either theory, and this does not say anything at all about a property of the gravitational field.

peety
The problem is, as Kretschman showed in 1917, any laws (specifically either Newton's Gravity or Special Relativity) can be expressed in a way to meet the above propositions. See the reference I cited as well has hundreds of discussions of these points in the literature.

Indeed, I won't dispute that; in the end, this statement about, essentially, gauge invariance is indifferent to what the invariant laws of physics actually are. $G_{\mu \nu} = \kappa T_{\mu \nu}$ does not and cannot arise from that magically. Separating the basic idea of invariance under coordinate transformations from the actual physical content of the theory is something I find useful, though.

PAllen
No, I must admit I'm not aware of it at all. Even after a full career devoted to GR I've never heard anyone question general covariance. Perhaps that comes from talking to physicists rather than philosophers.

I didn't say that general covariance marks the distinction between special and general relativity, or anything like that. General covariance applies to special relativity just as well. "The laws of physics are generally covariant" is a universal requirement. If you want to consider coordinate systems which are not Minkowskian, you can do so in either theory, and this does not say anything at all about a property of the gravitational field.

MTW has a section on this, mentioning Kretschmann, and the dispute, and basically proposing a variant of Anderson's approach to an alternative principle that has meaning. Anderson's (1967 book) was my first introduction to the dispute and an approach to an alternative principle that has some real meaning.

[EDIT: MTW actually goes through a complete derivation and discussion that Newtonian physics including gravity can be formulated as a generally covariant theory.]

Last edited:
PAllen
Indeed, I won't dispute that; in the end, this statement about, essentially, gauge invariance is indifferent to what the invariant laws of physics actually are. $G_{\mu \nu} = \kappa T_{\mu \nu}$ does not and cannot arise from that magically. Separating the basic idea of invariance under coordinate transformations from the actual physical content of the theory is something I find useful, though.

I absolutely agree. Further, I think there are imperfect, but useful, alternative principles (e.g. Anderson's as discussed in the reference) that have real utility in choosing physical laws. Similarly, the Principal of Equivalence is imperfect, disputed as to its precise formulation, but I side with its conceptual utility.

Ha, well, I must admit that this discussion of GR went over my head, so I'll rephrase my question:

In special relativity, do the laws of physics take the same form in any two reference frames-- perhaps noninertial ones -- moving with constant velocity relative to one another?

PAllen
One thing I would add is that even though general covariance may not be useful as a filter of physical laws, coordinate or diffeomorphism invariance is fundamental to distinguishing observables in GR. If you compute a purported observation in GR, and it is not invariant, then it is not an observable.

Last edited:
PAllen
Ha, well, I must admit that this discussion of GR went over my head, so I'll rephrase my question:

In special relativity, do the laws of physics take the same form in any two reference frames-- perhaps noninertial ones -- moving with constant velocity relative to one another?

I'll try to phrase this in a completely non-technical way. If you express laws in terms not motivated by GR (the way special relativity was formulated early in the 20th century), then the laws do not take the same form in non-inertial frames. Further, both SR and GR predict that an inertial frame is physically distinguishable from a non-inertial frame.

I think you misunderstood my question.

Suppose you have a non-inertial reference frame R. The physics in this frame is going to be a whole lot more complicated than in an inertial frame. For example, there'll be a host of inertial forces acting on things. Now suppose we have another non-inertial frame R' which is moving relative to R at a constant velocity. We can imagine that R and R' are both spaceships with their rockets firing, both accelerating at the same rate but moving relative to one another with a constant speed. Will the physics in R' appear the same?

PAllen
I think you misunderstood my question.

Suppose you have a non-inertial reference frame R. The physics in this frame is going to be a whole lot more complicated than in an inertial frame. For example, there'll be a host of inertial forces acting on things. Now suppose we have another non-inertial frame R' which is moving relative to R at a constant velocity. We can imagine that R and R' are both spaceships with their rockets firing, both accelerating at the same rate but moving relative to one another with a constant speed. Will the physics in R' appear the same?

Yes, essentially. Some interesting details (assuming they keep accelerating the same for a long time, starting from some initial relative velocity, in the same direction):

1) In an inertial frame, if they are accelerating the same, but with some starting difference in velocity, over time, they will both be moving at essentially the same speed relative to the inertial frame - nearly c.

2) Relative to each other, their speed will not become the same [in fact, their speed relative to each other will approach c], and one will eventually 'disappear' relative to the other - it will be inside the other's Rindler Horizon, and light it emits will never catch up with the other. For example, A will see B red shift and disappear; However, B will continue to be able to see A.

However, each will have the same physics inside each rocket - the difference in starting velocity will be completely undetectable inside the rockets.

Last edited:
Ha, that's some wild stuff. Thanks!