Postulates of SR without inertial frames?

[quasar987]

Is it ok to formulate the postulates of SR like so:

1) If two reference frames are in a constant rectilinear motion relative to one another, then the laws of physics take the same form in both of them.

2) If two reference frames are in a constant rectilinear motion relative to one another, then the speed of light in vacuum is the same for both of them (for an arbitrary emitting source).

The difference from the usual formulation is that I do not mention inertial/Galilean frames.

 

[mfb]

How do you define "constant rectilinear [relative] motion"? In which frame do you measure this motion, and how?

 

[ghwellsjr]

Is it ok to formulate the postulates of SR like so:

1) If two reference frames are in a constant rectilinear motion relative to one another, then the laws of physics take the same form in both of them.

2) If two reference frames are in a constant rectilinear motion relative to one another, then the speed of light in vacuum is the same for both of them (for an arbitrary emitting source).

The difference from the usual formulation is that I do not mention inertial/Galilean frames.

It sure looks to me like you are mentioning inertial frames, just using a definition of inertial rather than the word "inertial". What does "constant rectilinear motion" mean if not inertial?

 

[WannabeNewton]

I'm not immediately seeing the benefit of this. You are still picking out a preferred class of observers of minkowski spacetime but with the concept of a reference frame you can use a measuring apparatus to set up coordinates for the reference frame and start making measurements of observables represented by tensor fields (or more generally, in this context, spinor fields) and represent them in these coordinates.

Picking out the preferred class of observers as those whose references frames are equivalent up to a Lorentz transformation, we can conveniently relate measurements of tensor components made by one such observer to another in coordinates. So the reference frame concept has utility (tracing back to Newtonian mechanics of course) but I'm not seeing what the benefit would be in replacing it with what you said if taken at face value.

 

[quasar987]

How do you define "constant rectilinear [relative] motion"? In which frame do you measure this motion, and how?

Eh? Well, a frame of reference is made up of 3 measuring rods and a clock, so you can measure the speed of something as dr/dt, can you not?

It sure looks to me like you are mentioning inertial frames, just using a definition of inertial rather than the word "inertial". What does "constant rectilinear motion" mean if not inertial?

By "inertial/galilean frame" I mean one in which Newton's 1st law holds: a particle far enough removed from other particles moves at constant speed.

I'm not immediately seeing the benefit of this. You are still picking out a preferred class of observers of minkowski spacetime but with the concept of a reference frame you can use a measuring apparatus to set up coordinates for the reference frame and start making measurements of observables represented by tensor fields (or more generally, in this context, spinor fields) and represent them in these coordinates.

Picking out the preferred class of observers as those whose references frames are equivalent up to a Lorentz transformation, we can conveniently relate measurements of tensor components made by one such observer to another in coordinates. So the reference frame concept has utility (tracing back to Newtonian mechanics of course) but I'm not seeing what the benefit would be in replacing it with what you said if taken at face value.

I'm not claiming any objective benefits. It is simply my interpretation of the postulates, in terms I'm comfortable with, and I'm wondering if they are really correct as stated or if I'm missing some subtlety.

 

[WannabeNewton]

I'm not claiming any objective benefits. It is simply my interpretation of the postulates, in terms I'm comfortable with, and I'm wondering if they are really correct as stated or if I'm missing some subtlety.

I see. Well here's the thing regarding (1): the way you stated it, I cannot distinguish between galiliean relativity and special relativity. In Newtonian physics, $(\mathbb{R}^{4},h_{ab})$, we claim the laws of physics are the same in any inertial frame that is for any two observers $O,O'$ related by a galilean transformation, which for the simplest case is $t\rightarrow t,x^{1}\rightarrow x^{1} + vt$, the laws of physics will be the same. In SR, $(\mathbb{R}^{4},\eta _{ab})$, we redefine what an inertial frame is and instead claim the laws of physics are the same for any two observers $O,O'$ who are related by a lorentz transformation, which again for the simplest case is $t\rightarrow \gamma (t - vx^{1}),x^{1}\rightarrow \gamma (x^1 - vt)$; we say the reference frames of $O,O'$ are inertial. So you can see the difference between what inertial is in Newtonian physics and in SR.

Your (1) simply claims two observers traveling at constant velocity relative to one another will agree on the laws of physics. But how do you actually relate the coordinates set up in the frame of one observer to the coordinates set up by the other? As noted above, in both SR and Newtonian physics the preferred class of observers, for whom the laws of physics are the same, are moving with constant velocity relative to one another but the theories relate such observers differently.

 

[bcrowell]

You might be interested in Andréka et al, http://arxiv.org/abs/1005.0960v2 . They start off by axiomatizing SR with the notation IB(b) indicating a fundamental (undefined) notion that body b's motion is inertial. The axioms refer to observers o who are inertial, IB(o). Later they axiomatize GR by throwing out the axioms distinguishing accelerated from inertial observers.

IMO it's not a particular great paper (e.g., their longwinded intro about first-order logic is dumb and irrelevant), but it might get at the issues you're interested in.

Historically, Einstein motivated GR by trying to generalize SR to noninertial frames. From the modern point of view, however, that was a mistake. SR deals just fine with noninertial frames. What modern relativists consider to be the distinction between SR and GR is curvature. So Einstein would presumably have considered your axioms doomed to failure, but he'd have been wrong to criticize them on such general grounds.

Logically, we have the same situation as in Einstein's 1905 formulation, which is that the second postulate is really a special case of the first. (Maxwell's equations are laws of physics.) The minimal set of laws of physics to which you could apply this type of axiomatization would be Maxwell's equations themselves, in which case the content of axioms 1 and 2 becomes identical. In this case, the axioms are certainly self-consistent, as well as consistent with all the experiments that established Maxwell's equations, since Maxwell's equations can be expressed in a form that is invariant under a change of coordinates, including a change to an accelerating frame: http://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism In this example, however, the postulates are needlessly weak, because they don't need the restriction to frames of reference in constant rectilinear motion relative to one another.

 

[bcrowell]

I see. Well here's the thing regarding (1): the way you stated it, I cannot distinguish between galiliean relativity and special relativity. In Newtonian physics, $(\mathbb{R}^{4},h_{ab})$, we claim the laws of physics are the same in any inertial frame that is for any two observers $O,O'$ related by a galilean transformation, which for the simplest case is $t\rightarrow t,x^{1}\rightarrow x^{1} + vt$, the laws of physics will be the same. In SR, $(\mathbb{R}^{4},\eta _{ab})$, we redefine what an inertial frame is and instead claim the laws of physics are the same for any two observers $O,O'$ who are related by a lorentz transformation, which again for the simplest case is $t\rightarrow \gamma (t - vx^{1}),x^{1}\rightarrow \gamma (x^1 - vt)$; we say the reference frames of $O,O'$ are inertial. So you can see the difference between what inertial is in Newtonian physics and in SR.

I think you've got your logic backwards up here. In Einstein's 1905 axiomatization of SR, the form of the Lorentz transformations is proved, not assumed. This also applies to other, less old-fashioned, axiomatizations of SR that I've seen. See, e.g., Laurent 1994, or Morin 2008, which uses the approach originated by Ignatowsky 1911.

Bertel Laurent, Introduction to spacetime: a first course on relativity, 1994, World Scientific
Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008, Appendix I
W.v.Ignatowsky, Phys. Zeits. 11 (1911) 972

 

[WannabeNewton]

I think you've got your logic backwards up here. In Einstein's 1905 axiomatization of SR, the form of the Lorentz transformations is proved, not assumed.

But I'm not taking the form of the lorentz transformations for granted; I wasn't trying to derive anything here. My point is that what is inertial in Newtonian physics is not what is inertial in SR even though constant velocity is involved in both cases; stating the laws of physics is the same for all constant motion observers leaves an ambiguity if the statement stands alone (say without what Quasar stated in (2)).

More generally, the isometry groups of the two theories are different.

 

[quasar987]

I see. Well here's the thing regarding (1): the way you stated it, I cannot distinguish between galiliean relativity and special relativity. In Newtonian physics, $(\mathbb{R}^{4},h_{ab})$, we claim the laws of physics are the same in any inertial frame that is for any two observers $O,O'$ related by a galilean transformation, which for the simplest case is $t\rightarrow t,x^{1}\rightarrow x^{1} + vt$, the laws of physics will be the same. In SR, $(\mathbb{R}^{4},\eta _{ab})$, we redefine what an inertial frame is and instead claim the laws of physics are the same for any two observers $O,O'$ who are related by a lorentz transformation, which again for the simplest case is $t\rightarrow \gamma (t - vx^{1}),x^{1}\rightarrow \gamma (x^1 - vt)$; we say the reference frames of $O,O'$ are inertial. So you can see the difference between what inertial is in Newtonian physics and in SR.

Your (1) simply claims two observers traveling at constant velocity relative to one another will agree on the laws of physics. But how do you actually relate the coordinates set up in the frame of one observer to the coordinates set up by the other? As noted above, in both SR and Newtonian physics the preferred class of observers, for whom the laws of physics are the same, are moving with constant velocity relative to one another but the theories relate such observers differently.

Usually, in SR, the Lorentz transformations (i.e. the rules according to which coordinates in one frame relate to the coordinates in a frame in constant velocity wrt to it) are derived by combining postulate #1 with postulate #2 in a thought experiment or another. Are you saying there's something missing in my formulation that won't allow me to carry out these experiments?

 

[WannabeNewton]

Usually, in SR, the Lorentz transformations (i.e. the rules according to which coordinates in one frame relate to the coordinates in a frame in constant velocity wrt to it) are derived by combining postulate #1 with postulate #2 in a thought experiment or another. Are you saying there's something missing in my formulation that won't allow me to carry out these experiments?

Oh you are combining the two; sorry I thought you meant if (1) was standalone.

 

[Samshorn]

Is it ok to formulate the postulates of SR like so:
1) If two reference frames are in a constant rectilinear motion relative to one another, then the laws of physics take the same form in both of them.
2) If two reference frames are in a constant rectilinear motion relative to one another, then the speed of light in vacuum is the same for both of them (for an arbitrary emitting source).
The difference from the usual formulation is that I do not mention inertial/Galilean frames.

I don't think it's okay to formulate the postulates like that. When you say "reference frame" you presumably mean something like a 'rigid spatial framework', but this is already ambiguous, because if the framework is accelerating (as you would permit by not stipulating that it is inertial) then there is no unique sense of rigidity. You could specify Born rigidity, but that is defined in terms of the very space-time metrical relations that you are trying to deduce, so you can't invoke that a priori. So it's problematic to even define a "reference frame" that isn't inertial, let alone two of them that are moving uniformly relative to each other.

Even if you ignore that ambiguity, and postulate two coordinate systems that are accelerating in the inertial sense but moving uniformly relative to each other, it would be misleading to say that "the laws of physics are the same in both of them", because the laws of physics would (in general) be constantly changing in both of them (as the rate of acceleration may be varying), so you would need to stipulate some correlation between the "nows" of the two systems, and these systems would be filled with fictitious forces. Not a very promising way of trying to clearly explain the foundations of special relativity.

 

[Fredrik]

1) If two reference frames are in a constant rectilinear motion relative to one another, then the laws of physics take the same form in both of them.

2) If two reference frames are in a constant rectilinear motion relative to one another, then the speed of light in vacuum is the same for both of them (for an arbitrary emitting source).

The difference from the usual formulation is that I do not mention inertial/Galilean frames.

a frame of reference is made up of 3 measuring rods and a clock, so you can measure the speed of something as dr/dt, can you not?

If the situation isn't such that acceleration, gravity and the length and rigidity of the rods work together to ensure that the effects of acceleration and gravity can be neglected for practical purposes, then I don't know a procedure that can determine if the object we would like to think of as a measuring rod is straight. So the measurements you make this way are only going to assign coordinates as intended to events that are very close to the clock.

This makes it very difficult to assign a meaning to the statement that the frames are in constant rectilinear motion relative to one another. The first frame doesn't have a meaningful way to assign coordinates to the entire world line of the second clock, and vice versa.

 

[Fredrik]

I think I should also include my standard comments about "derivations" of the Lorentz transformation from a set of "postulates":

The Lorentz transformation isn't actually derived from the postulates, but from mathematical statements that can be thought of as making the postulates precise.

These derivations are not the reason why SR is valid as a theory. It's perfectly fine (and in my opinion much better) to just state the mathematical definitions of Minkowski spacetime, proper time, etc. without any motivation, and then just write down the correspondence rules for length and time measurements.

The point of a derivation from postulates is to show how it's possible for someone who doesn't already know SR to come up with the idea to use Minkowski spacetime in a theory of physics. If it's a joint derivation of the formulas for Galilean boosts and Lorentz boosts, it also shows what SR and pre-relativistic physics have in common. It shows the place of Galilean spacetime and Minkowski spacetime in a hierarchy of models of spacetime that have been found to be useful.

 

[bcrowell]

I don't think it's okay to formulate the postulates like that. When you say "reference frame" you presumably mean something like a 'rigid spatial framework', but this is already ambiguous, because if the framework is accelerating (as you would permit by not stipulating that it is inertial) then there is no unique sense of rigidity. You could specify Born rigidity, but that is defined in terms of the very space-time metrical relations that you are trying to deduce, so you can't invoke that a priori. So it's problematic to even define a "reference frame" that isn't inertial, let alone two of them that are moving uniformly relative to each other.

You raise an interesting point, but I don't think it quite works.

First off, there is nothing wrong with introducing rigid rulers as things that are simply assumed to exist. In fact, this is what Einstein did in his 1905 paper on SR: "The theory to be developed is based—like all electrodynamics—on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of coordinates), clocks, and electromagnetic processes. [...] If a material point is at rest relative to this system of coordinates, its position can be defined relative thereto by the employment of rigid standards of measurement and the methods of Euclidean geometry, and can be expressed in Cartesian coordinates." Even in formal mathematical theories, we always have certain primitive objects that we just assume to exist, e.g., the empty set or the geometrical objects referred to in Euclid's postulates.

But I think we can make everything more conceptually clear by only assuming that clocks exist. Once you have a clock and the ability to send signals using light (which is implicit in the OP's 2nd postulate), you have the ability to measure the spacetime line element between any two nearby events, i.e., you have a metric. Once you have a metric, you can define Born-rigidity as the vanishing of the expansion tensor (see https://www.physicsforums.com/showthread.php?t=430381 ). Born-rigidity imposes some restrictions (e.g., you can't apply an angular acceleration to an object with nonvanishing internal area), but I don't see how that would be a problem for the OP, who just needs rigid one-dimensional rulers. It would be interesting to hear from the OP whether the intention was to allow rotating coordinate systems.

Even if you ignore that ambiguity, and postulate two coordinate systems that are accelerating in the inertial sense but moving uniformly relative to each other, it would be misleading to say that "the laws of physics are the same in both of them", because the laws of physics would (in general) be constantly changing in both of them (as the rate of acceleration may be varying), so you would need to stipulate some correlation between the "nows" of the two systems, and these systems would be filled with fictitious forces. Not a very promising way of trying to clearly explain the foundations of special relativity.

I don't think the laws of physics have to be constantly changing just because there is a varying rate of acceleration for the coordinate system. As a counterexample, let's say that the laws of physics are Maxwell's equations in a vacuum, expressed covariantly as F^{bc}_{;a;a}=0, where F is the electromagnetic tensor and the semicolons are covariant derivatives. Suppose they have this form in some inertial frame, and suppose we then transform to a new frame according to x\rightarrow x'=x+a\sin(\omega t), where |a\omega|<1 so that the mapping is one-to-one. Then Maxwell's equations have exactly the same form in the x' frame as in the original frame.

One thing that I think *is* going to be a little awkward in the OP's system is that if we have clocks, then certain world-lines will result in minimal time on a clock. Gosh, what's so special about those (curved) world-lines?

 

[quasar987]

I don't think it's okay to formulate the postulates like that. When you say "reference frame" you presumably mean something like a 'rigid spatial framework', but this is already ambiguous, because if the framework is accelerating (as you would permit by not stipulating that it is inertial) then there is no unique sense of rigidity. You could specify Born rigidity, but that is defined in terms of the very space-time metrical relations that you are trying to deduce, so you can't invoke that a priori. So it's problematic to even define a "reference frame" that isn't inertial, let alone two of them that are moving uniformly relative to each other.

No unique sense of rigidity? You lost me.

Even if you ignore that ambiguity, and postulate two coordinate systems that are accelerating in the inertial sense but moving uniformly relative to each other, it would be misleading to say that "the laws of physics are the same in both of them", because the laws of physics would (in general) be constantly changing in both of them (as the rate of acceleration may be varying), so you would need to stipulate some correlation between the "nows" of the two systems[...]

What do you mean? I have a clock. You have a clock. You move at constant speed relative to me and we know that when you were at the same place as me, both our clocks read 0:00.

 

[Fredrik]

No unique sense of rigidity? You lost me.

In SR, if an accelerating object is rigid in inertial one coordinate system, in the sense that the coordinate distance between any two component parts is constant, then it's not rigid in another.

Born rigid motion is the kind of motion that a solid does when it's accelerated gently (or not at all): For each point p on the object, the coordinate distance to nearby points, in the inertial coordinate system that's comoving with p, is approximately constant. The closer those other points are, the better the approximation.

This is one of several reasons why it seems like a bad idea to use a grid of rulers and synchronized clocks to define a coordinate system or a frame (in the mathematical sense) on a yet-to-be-determined spacetime, under less than ideal circumstances. How do you keep the clocks synchronized? (If a given method is suitable or not depends on the spacetime and the type of motion the grid is doing). How do you make sure that the rulers are aligned in the right direction? How do you even know if they're straight enough? Is the idea to build a grid in a laboratory where gravity is negligible and accelerometers display 0, and then accelerate it? Then how do you accelerate it? Do you push or pull? How do you account for the stretching/compression? Do you just assume that the grid is "rigid"? Rigid in what coordinate system?

 

[quasar987]

Let me ask a related question.

According to SR, if we take two inertial frames, then the coordinate transform between them is a Lorentz transformation. In the language of GR, taking an inertial frame corresponds to taking a chart of the space-time manifold M. Conversely, if we take two charts on M related by a Lorentz transformation, do they correspond to inertial frames?

Or maybe this makes no sense either? :) Thanks.

 

[Fredrik]

When we're dealing with Minkowski spacetime, if x and y are global inertial coordinate systems such that $x^{-1}(0)=y^{-1}(0)$, then $x\circ y^{-1}$ is a Lorentz transformation.

Most other spacetimes do not have global coordinate systems at all, so global inertial coordinate systems are out. There is a sort of local inertial coordinate system, but the problem is, there's more than one kind. Let p be a point in spacetime. Take any four geodesics through p such that one of them is timelike, the others spacelike, and the four are mutually orthogonal. Now we can partially define a coordinate system by taking these four geodesics as the "axes" and label points on them by their proper time/distance from p along the geodesic they're on. There are now multiple ways to assign coordinates to points that are not on these four geodesics. We could e.g. use all the other geodesics through p, or we could for each point q on the timelike geodesic, use all the geodesics through q that are orthogonal to the timelike geodesic.

I think both of these constructions and several others give us coordinate systems such that $g_{\mu\nu}(p)=\eta_{\mu\nu}$ and $\partial_\sigma g_{\mu\nu}(p)=0$. I think any of those could be considered a local inertial coordinate system. Even if the definition can be extended to all of spacetime, it's not likely to have a lot in common with the inertial coordinate systems of SR, except very close to the point the coordinate system sends to 0. In particular, I expect that if x and y are two local inertial coordinate systems such that $x^{-1}(0)=y^{-1}(0)$, then $x\circ y^{-1}$ is not going to be a Lorentz transformation, but its effect on points close to 0 will be very similar to the effect of a Lorentz transformation.

In other words, if $\Lambda=x\circ y^{-1}$, then I expect that $u^T\Lambda^T\eta\Lambda u\approx u^T\eta u$ will be a good approximation for $u\in\mathbb R^4$ with small Euclidean norm.

 

[Samshorn]

Let me ask a related question. According to SR, if we take two inertial frames, then the coordinate transform between them is a Lorentz transformation... Conversely, if we take two charts on M related by a Lorentz transformation, do they correspond to inertial frames?

Clearly not. In a certain region, define an arbitrary system of coordinates (which typically will not be inertial), and then apply a Lorentz transformation to those coordinates to give another set of coordinates over that region. Neither of these will be inertial coordinate systems. That was the point of my earlier post: The fact that two coordinate systems have some linear relationship between each other does not imply that either of them is inertial. Also, in general a Born-accelerating coordinate system can only be extended over a finite region (without going singular), so even if you invoke this construction (which would be circular in any case) it clearly doesn't have the properties of a well-behaved global "frame".

By the way, Born rigidity doesn't require gentle accelerations, and it isn't an approximation, but it requires that each point of the object move on a coordinated hyperbolic trajectory, "centered" on a particular event. Essentially Born rigid motion is just a rotation of a locus in space-time about a given pivot point. This is not the kind of motion that a solid ever makes naturally. It requires that each particle be given its own precise hyperbolic profile of accelerations, e.g., attaching little rockets to each particle, with different instructions for each one about how to accelerate in a coordinated dance to maintain the proper length of each interval in terms of its instantaneously co-moving inertial frame. Also, even if you contrive to create Born rigid motion of an object (like a ruler, which could only be done over a finite extent), the phase relations between the different parts of the object would be constantly changing, so even this is inherently dis-similar to an inertial ruler.

 

[Fredrik]

This is not the kind of motion that a solid ever makes naturally.

Not exactly no, but if you accelerate a solid gently, its motion will approximate Born rigid motion. This is why you can push a rod gently at one end and still have an inertial observer see it contract according to the Lorentz contraction formula.

 

[atyy]

By "inertial/galilean frame" I mean one in which Newton's 1st law holds: a particle far enough removed from other particles moves at constant speed.

I think it's fine. I have usually considered an inertial frame in Newtonian mechanics as one in which eg. the laws have the form F=dp/dt, F=GMm/r², ie. by specifying what I mean by "the same form". So your (1) looks like specifying an inertial frame as WannabeNewton says, and (2) specifies that it's a Lorentz inertial frame.

However, thinking about it, I don't quite know how to specify "the same form" if (1) is to include Galilean and Lorentz inertial frames. Would it be sufficient to specify F=dp/dt (which isn't that different from saying Newton's first law holds, except that it allows you to define an inertial frame even if all particles always interact)?

I suppose it would be cleaner if we specified F=dp/dt and p=γmv, with the Galilean case being the c→∞ limit. But then you wouldn't be able to "derive" the corrected second law, since it is now assumed.

 

[bloby]

If I assume the standard postulates of SR, consider an inertial frame and a frame slowly rotating about some point O. Then I can singled out this frame among all it's friends: an object at O will stay there. There is no such point in the frames in constant rectilinear motion relative to it. The new postulate 1 seems false.

 

[WannabeNewton]

I think any of those could be considered a local inertial coordinate system

Easiest way is to use Riemann normal coordinates. They are the arbitrary Riemannian manifold analogue of the usual result for regular surfaces where all you need is the inverse function theorem and taylor series to find such coordinates. Here, the normal coordinates make use of $\text{exp}$.

 

[Fredrik]

Easiest way is to use Riemann normal coordinates. They are the arbitrary Riemannian manifold analogue of the usual result for regular surfaces where all you need is the inverse function theorem and taylor series to find such coordinates. Here, the normal coordinates make use of $\text{exp}$.

I haven't actually done this, so I don't know what's easier. I think Riemann normal coordinates are the ones that use all the geodesics through 0, and Fermi normal coordinates are the ones that use all geodesics through the 0 axis that are orthogonal to the 0 axis.

 

[atyy]

Logically, we have the same situation as in Einstein's 1905 formulation, which is that the second postulate is really a special case of the first. (Maxwell's equations are laws of physics.) The minimal set of laws of physics to which you could apply this type of axiomatization would be Maxwell's equations themselves, in which case the content of axioms 1 and 2 becomes identical. In this case, the axioms are certainly self-consistent, as well as consistent with all the experiments that established Maxwell's equations, since Maxwell's equations can be expressed in a form that is invariant under a change of coordinates, including a change to an accelerating frame: http://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism In this example, however, the postulates are needlessly weak, because they don't need the restriction to frames of reference in constant rectilinear motion relative to one another.

In order not have (1) and (2) be identical, so that (1) can be consistent with Lorentzian and Galilean inertial frames, would it work to specify the "laws of physics having the same form" as being derivable from a Lagrangian that is covariant under space and time translations, and in which the spatial metric has the form diag(1,1,1) and derivatives of the metric do not appear?

 

[WannabeNewton]

Well I know for sure that Landau and Lifgarbagez derive Galilean invariance that way. An inertial frame is defined as one in which space is both homogenous and isotropic and time is homogenous. In such a frame, the lagrangian must be independent of position and time due to homogeneity so $L$ should only be a function of the velocity but the isotropy of space tell us that there is no preferred direction for velocity which implies that the lagrangian must be a function of the magnitude of velocity alone $L = L(v^2)$ hence by lagrange's equations $\frac{\partial L}{\partial \mathbf{v}} = \text{const.}$ so $\mathbf{v} = \text{const.}$; this is of course the law of inerta. In particular if we consider another frame moving uniformly with respect to this inertial frame, the law of inertia is preserved i.e. the motion.

On the other hand, I cannot find a similar thing for minkowski space - time in L&L's classical theory of fields.

 

[atyy]

Well I know for sure that Landau and Lifgarbagez derive Galilean invariance that way. An inertial frame is defined as one in which space is both homogenous and isotropic and time is homogenous. In such a frame, the lagrangian must be independent of position and time due to homogeneity so $L$ should only be a function of the velocity but the isotropy of space tell us that there is no preferred direction for velocity which implies that the lagrangian must be a function of the magnitude of velocity alone $L = L(v^2)$ hence by lagrange's equations $\frac{\partial L}{\partial v} = \text{const.}$ so $v = \text{const.}$; this is of course the law of inerta. In particular if we consider another frame moving uniformly with respect to this inertial frame, the law of inertia is preserved i.e. the motion.

On the other hand, I cannot find a similar thing for minkowski space - time in L&L's classical theory of fields.

Hmmm, I had imagined the difficulty would be on the Galilean side, but I guess that's ok after all. I wonder if it's ok for Minkowski space even though they don't mention it.

BTW, have you heard this terrible http://www.aip.org/history/ohilist/4915_2.html that L&L contains "not a word by Landau, not a thought by Lifgarbagez"?

 

[WannabeNewton]

Can't say I have. I'll have to read that once I finish my statistical mechanics homework. It's 11:40 PM here and I still have like three HWs left unfinished that are due tomorrow haha. I'll also try to take a look again later in classical theory of fields to see if I missed something.

By the way to quasar, L&L also has a neat proof regarding the rigidity issue talked about before by Fredrik.

 

[Fredrik]

An inertial frame is defined as one in which space is both homogenous and isotropic and time is homogenous.

I don't think I understand this definition.

I prefer the approach that focuses on the functions that change coordinates from one global inertial coordinate system to another, instead of the global inertial coordinate systems themselves. The principle of relativity suggests that the set of such transformations should form a group, and that each of them should have a well-defined velocity (corresponding to the velocity of one inertial observer in an inertial coordinate system that's comoving with another). Then we make a few technical assumptions that can be intepreted as making the principle of translation invariance and the principle of rotation invariance mathematically precise, and prove that these assumptions imply that the group is a subgroup of the Poincaré group or the Galilean group. The only thing left undetermined is what types of reflections are included in the group.

As long as we intend to define spacetime with underlying set $\mathbb R^4$, the global inertial coordinate systems can be identified with the members of this group. In other words, we can define a global inertial coordinate system as a member of the group.

 

[WannabeNewton]

As a member of what? The Poincare group in the case of SR? I don't see how that would work. A global coordinate chart, for this space - time is simply the usual pair $(\mathbb{R}^{4},\phi )$ and the coordinate system is the 4 - tuple of functions given by $\phi(p) = (x^{0}(p),...,x^{3}(p)),p\in \mathbb{R}^{4}$. If this coordinate system happened to be set up by an inertial observer and if we have another global coordinate chart $(\mathbb{R}^{4},\phi')$ then whether or not the associated coordinate system was set up by another inertial observer can be determined by the transition map.

On the other hand, each $g\in G$, with $G$ being the Poincare group, has associated a right translation $\chi _g$. The killing fields of $(\mathbb{R}^{4},\eta _{ab})$, that is the vector fields $\xi ^{a}$ such that $\mathcal{L}_{\xi }\eta _{ab} = 0$, then correspond to the resulting right invariant vector fields on $G$. So the elements of the Poincare group are intimately related to the geometrical symmetries of Minkowski space - time. I'm not seeing how we can make coordinate systems be members of this.

 

[Fredrik]

Yes, the Poincaré group in the case of SR. You don't have to define the Poincaré group in terms of killing fields or anything else that involves differential geometry. It's perfectly adequate to define it as the set of maps $x\mapsto \Lambda x+a:\mathbb R^4\to\mathbb R^4$ such that $\Lambda$ is linear and $\Lambda^T\eta\Lambda=\eta$. With this definition, it's just a subgroup of the permutation group of $\mathbb R^4$. Since the underlying set of spacetime is $\mathbb R^4$, any smooth permutation of $\mathbb R^4$ can be thought of as a coordinate system.

The Poincaré group can also be defined as the group of isometries of the Minkowski metric, i.e. as the set of all diffeomorphisms $\phi:\mathbb R^4\to\mathbb R^4$ such that $\phi^*g=g$, where $\phi^*$ is the pullback function associated with $\phi$. This definition is equivalent to the simple one above. I prefer it over that stuff involving killing fields, but maybe that's just because I understand this approach much better.