Special Relativity & General Relativity: Length, Time, and Motion

[yourdadonapogostick]

special relativity has to do with constant velocity motion, spacetime, and observers. it is a special case of general relativity (hence the name 'special'). it says that constant velocity motion contracts lenth, dilates time, and increases massenergy.

general relativity says the the effects of acceleration and gravity are indistinguishable. i know that gravity dilates time, but does it contract length?

 

[pervect]

Yes, this follows directly from the metric for the Schwarzschild geometry (the standard metric for a spherical body).

See for instance http://vishnu.mth.uct.ac.za/omei/gr/chap8/node8.html

 

[yourdadonapogostick]

so, acceleration contracts length as well...

 

[pervect]

so, acceleration contracts length as well...

Hmmm - no. Just as the Schwarzschild metric is the standard metric for an observer on a massive spherical body, the Rindler metric is the standard metric for an accelerating obsever. This metric for someone accelerating with an acceleration 'g' in the z direction is

dx^2 + dy^2 + dz^2 - (1+gz)dt^2

Looking at the metric coefficients, this means that an accelerating observer sees time slow down or speed up depending on whether an object is "above" him or "below" him (the value of 'z') - an effect quite comparable to gravitational time dilation.

But the spatial metric is not affected - in this case there is no gravitational length contraction.

That's why the url I linked to noted that "you can't predict this from the equivalence principle", I would guess.

 

[yourdadonapogostick]

damn...there goes my other question...

 

[yourdadonapogostick]

i'm confused...the equivelence principle doesn't always work?

 

[εllipse]

The reason gravity contracts length is because of the equivalence principle. An actual acceleration itself does not cause length contraction, but it does imply motion relative to an instantaneous SR reference frame (terminology I've never heard used, but just made up, so there's probably something faulty in it). It may help you to read this: http://www.bartleby.com/173/23.html

 

[Chronos]

GR is the four dimensional equivalent of Pythagoras' theorem. The sum of all vectors result in the same product. Does that help?

 

[yourdadonapogostick]

i thought that was sr. it shows time dilation by moving in other dimensions and such.

 

[pervect]

i'm confused...the equivelence principle doesn't always work?

The equivalence principle always works. However, length contraction due to planetary or black-hole gravity cannot be derived from the equivalence princple alone. Length contraction comes from inspecting the metric resulting from the full solution of Einstein's field equations. The example of the Rindler metric shows that length contraction does not follow directly from the equivalence principle in isolation from the rest of the full theory.