Postulates of SR without inertial frames?

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.

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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 joke that L&L contains "not a word by Landau, not a thought by Lifgarbagez"?

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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

Staff Emeritus
Gold Member
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

Staff Emeritus
Gold Member
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.

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