Linear equations and homogeneity of space and time

[khil_phys]

Einstein, in his paper "On the Electrodynamics of Moving Bodies", part 1, sec. 3, writes: "Primarily it is clear that on account of the property of homogeneity which we ascribe to time and space, the equations must be linear." What has the homogeneity of space and time to do with the degree of the equations?

 

[bcrowell]

For example, suppose that the Lorentz transformation had a form such as $x'=(x-7)^2$. Then there would be something special about the location x=7. But all that determines a Lorentz boost is the velocity v we choose, and there is no way that this choice of v would single out x=7 for special treatment. That would imply that x=7 was special not because of our choice of v but simply because x=7 had some special property of its own. This would violate the homogeneity of space.

 

[khil_phys]

I agree with you. But, after all x^'=(x-7)² is an equation in the second degree. I can single out x=7 as special through the equation x=x^'-7 too, which is linear, and it violates the homogeneity of space.

 

[fzero]

Homogeneity means that the physics at one point in spacetime is the same as that at any other point. In particular, the length of an object should be independent of its position in an inertial frame. So the length measured by the observer $O'$ must not depend on the position of the object in the inertial frame of the observer $O$ either. This implies that the infinitesimal elements $dx^{'\mu} = {\Lambda^\mu}_\nu dx^\nu,$ where the ${\Lambda^\mu}_\nu$ are independent of the $x^\mu$.

Integrating this expression leads to the Poincare transformations, which are indeed linear. This is not unusual, since in the modern viewpoint, we associate homogeneity with invariance under spacetime translations.

 

[bcrowell]

I agree with you. But, after all x^'=(x-7)² is an equation in the second degree. I can single out x=7 as special through the equation x=x^'-7 too, which is linear, and it violates the homogeneity of space.

I would put it exactly the other way around. If x=x'-7 were *not* allowed, it would violate homogeneity.

x=x'-7 doesn't do anything special at x'=7. x'=7 just happens to be where it gives x=0 -- but nothing special happens at x=0.

 

[khil_phys]

Are you saying this because on differentiating the second degree equation with respect to time, we have x=7 as a root? On the other hand, differentiating the linear equation x'=x-7 would give us a constant velocity.

 

[khil_phys]

I would put it exactly the other way around. If x=x'-7 were *not* allowed, it would violate homogeneity.

x=x'-7 doesn't do anything special at x'=7. x'=7 just happens to be where it gives x=0 -- but nothing special happens at x=0.

I got it. Thanks a lot!