Time dilation plus constancy of light implies space is dilated?

[Goldbeetle]

Dear all,
I've read the following argumentation for curved spacetime:
1) From the equivalence principle one can derive that time is dilated in gravitational filed..
2) The velocity of light is constant for all local observers
3) If time dilates, then also space must dilate if one wants the velocity of light to remain constant.

Does this make any sense?

Kindest regards.

Goldbeetle

 

[bcrowell]

It doesn't make much sense to me, mainly because I don't think it means anything to talk about dilation of space.

When we say that time dilates in a gravitational field, that describes the outcome of concrete experiments. For example, you can synchronize two clocks, then move one to the top of a mountain and one to the bottom of a neighboring valley (Iijima and Fujiwara, 1978, and Briatore and Leschiutta, 1979). When you reunite the clocks, they disagree. There is no equivalent experiment for rulers.

Another good way to see this is to realize that the clock time s measured along a certain timelike world-line is given by $s=\int_A^B \sqrt{dx^cdx_c}$ (+--- metric). Time dilation is a description of the fact that the integral depends on the path from A to B. If you try to make a variation on this in which "clock time" is replaced by "ruler distance," you have to make A and B spacelike in relation to one another, and then you can't have a piece of apparatus that travels from A to B, so it's not really analogous.

A somewhat better hand-waving argument is the one about the rotating disk in the introduction of Einstein's "The Foundation of the General Theory of Relativity." But it's still just a hand-waving argument. Einstein is giving it as motivation, not as a derivation.

-Ben

 

[Goldbeetle]

Thanks, Ben! I do not think that curved spacetime can be derived. I'm just looking for strong argumentations.

Why do you regard the rotating disk as "hand-waving"? Is it because nobody ever did that experiment or because the rotating disk argumentations is fallacious?

Normally, the books I have consulted so far tend to argue this way. Since the redshift of light in a gravitational field (which incidentally can be corroborated experimentally) motivates that the special relativity metric could be modify to encompass a coefficient of dt that is space dependent. But, if this the case, the argument goes,then also the other coefficients of the metric (related to dx, dy, dx) must be dependent on space because in special relativity space and time intermingle. But does this really makes sense? Your opinion is much appreciated.

Other books say that the motivation for curved space (not time) is that the path of light is deflected. In a special relativistic flat space time this is impossible. Does this make sense? Is the geometry of special relativity based on the assumption that light travels in a straight path? If so, how? The books I have consulted so far do not go into details.
You're opinion on this is much appreciated.

Thanks,
Goldbeetle

 

[bcrowell]

Why do you regard the rotating disk as "hand-waving"? Is it because nobody ever did that experiment or because the rotating disk argumentations is fallacious?

I don't think it's fallacious, and I don't think it's necessary to do the actual experiment. I think it was an interesting was for Einstein to introduce his ideas to an audience for whom they were pretty radical. But the kind of curvature we care about in GR is intrinsic curvature, and the apparent spatial curvature of the rotating disk is not intrinsic, because it's produced by a change from a nonrotating coordinate system to a rotating one. Intrinsic curvature doesn't change when you change coordinates. Also, GR is about spacetime curvature, not just spatial curvature.

Normally, the books I have consulted so far tend to argue this way. Since the redshift of light in a gravitational field (which incidentally can be corroborated experimentally) motivates that the special relativity metric could be modify to encompass a coefficient of dt that is space dependent. But, if this the case, the argument goes,then also the other coefficients of the metric (related to dx, dy, dx) must be dependent on space because in special relativity space and time intermingle. But does this really makes sense? Your opinion is much appreciated.

Hmm...I dunno. It sounds like a reasonable plausibility argument. I don't think there is any way to derive GR rigorously from other principles, but I'm sure there are many different ways to motivate it.

Other books say that the motivation for curved space (not time) is that the path of light is deflected. In a special relativistic flat space time this is impossible. Does this make sense? Is the geometry of special relativity based on the assumption that light travels in a straight path?

IMO it's not a good thing to think of light as having any fundamental role in relativity. The moon's path is just as straight as the path of a ray of light.

 

[Goldbeetle]

"Goldbeetle: Normally, the books I have consulted so far tend to argue this way. Since the redshift of light in a gravitational field (which incidentally can be corroborated experimentally) motivates that the special relativity metric could be modify to encompass a coefficient of dt that is space dependent. But, if this the case, the argument goes,then also the other coefficients of the metric (related to dx, dy, dx) must be dependent on space because in special relativity space and time intermingle. But does this really makes sense? Your opinion is much appreciated.

"Ben: Hmm...I dunno. It sounds like a reasonable plausibility argument. I don't think there is any way to derive GR rigorously from other principles, but I'm sure there are many different ways to motivate it."

Ben,
thanks again. What I find weak in this plausibility reasoning is that proving that the metric depends on position (and time) is not sufficient to get curved spacetime. This condition is necessary but not sufficient to have a curvature not equal to zero.
Do I miss something?
Goldbeetle