Length Contraction & Mass Increase in GR | General Relativity

[Ranku]

In special relativity, an object moving at higher speed experiences time dilation, length contraction, and mass increase, compared to an object moving at slower speed. In general relativity, for an object in stronger gravitational field (i.e., with higher acceleration due to gravity), time runs slower than for an object in weaker gravitational field (i.e., with lower acceleration due to gravity). Can we expect therefore that an object in stronger gravitational field should also experience length contraction and mass increase?

 

[PeterDonis]

In special relativity, an object moving at higher speed experiences time dilation, length contraction, and mass increase, compared to an object moving at slower speed.

You left out a key qualifier: speed is relative. Which object is experiencing time dilation, length contraction, and mass increase, and to what extent, depends on your choice of frame.

In general relativity, for an object in stronger gravitational field (i.e., with higher acceleration due to gravity), time runs slower than for an object in weaker gravitational field (i.e., with lower acceleration due to gravity).

This is not correct as you state it: it is not the gravitational "field" (the "accceleration due to gravity") but the gravitational potential (how deep the object is in a gravity well) that determines an object's "rate of time flow". Also, unlike the case of relative motion in SR, the difference in "rate of time flow" between two objects at different depths in a gravity well is not frame-dependent; which one's clock is running slow is an invariant, independent of your choice of frame.

Can we expect therefore that an object in stronger gravitational field should also experience length contraction and mass increase?

No, because the two cases are not the same. See above.

 

[phinds]

In special relativity, an object moving at higher speed experiences time dilation, length contraction, and mass increase, compared to an object moving at slower speed.

Although this is not an answer to your question and is possibly just a matter of how you express yourself, it is very important to stay aware that objects do NOT "experience" time dilation or length contraction. Time and length are unchanged in the local frame (but are changed in a frame that is moving relative to the object).

EDIT: I see Peter beat me to it. Again.

 

[Ranku]

I am still looking for an explicit in-depth analysis as to why there is time dilation in SR and GR, but not length contraction and mass increase in GR.

 

[PeroK]

I am still looking for an explicit in-depth analysis as to why there is time dilation in SR and GR, but not length contraction and mass increase in GR.

Note that you do have length contraction in GR, just as in SR. GR covers the cases of relative motion (velocity-based time dilation and length contraction) and gravity (gravitational time dilation). SR covers only relative motion.

The question you really want to ask is why there is no length contraction due to gravitational potential?

Length contraction is a result of the measurement of distance between the two ends of an object that is moving relative to you:

https://en.wikipedia.org/wiki/Length_contraction

If an object is at rest relative to you, but in a different gravitational potential from you and you measure its length, then your measurement will return its proper length.

I would turn your question round: why would you expect length contraction due to gravitational field? What analysis would lead to that expectation?

Also, here is the PF insight on "relativistic mass":

https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

Here is an explicit analysis of time dilation, due to relative velocity and time dilation:

https://en.wikipedia.org/wiki/Time_dilation

 

[Ranku]

I would turn your question round: why would you expect length contraction due to gravitational field? What analysis would lead to that expectation?

I am thinking of it this way: Relative motion in SR leads to time dilation, which finds its equivalent in GR in terms of time dilation in different gravitational potentials. So what is the key difference whereby relative motion in SR which also leads to length contraction (and mass increase), however, does not find its equivalent in GR in terms of length contraction (and mass increase) in different gravitational potentials?

 

[PeroK]

I am thinking of it this way: Relative motion in SR leads to time dilation, which finds its equivalent in GR in terms of time dilation in different gravitational potentials. So what is the key difference whereby relative motion in SR which also leads to length contraction (and mass increase), however, does not find its equivalent in GR in terms of length contraction (and mass increase) in different gravitational potentials?

Let me rewrite your question:

I am thinking of it this way: Relative motion leads to time dilation, and gravitational potential leads to time dilation. So what is the key difference whereby relative motion also leads to length contraction, however, gravitational potential does not lead to length contraction?

The key difference is that one is relative motion and one is gravitational potential. There is no reason that relative motion and gravitational potential are equivalent. Why would they be?

Even the time dilations are very different in each case. Relative motion time dilation is symmetric; whereas, the time dilation associated with a gravitational potential is not symmetric.

 

[Ranku]

Even the time dilations are very different in each case. Relative motion time dilation is symmetric; whereas, the time dilation associated with a gravitational potential is not symmetric.

Could you clarify what is meant by symmetric and not symmetric?

 

[PeroK]

Could you clarify what is meant by symmetric and not symmetric?

I guess you haven't actually read anything about time dilation?

If A and B are in relative motion, then A measures B's time/clock running slow and B measures A's time/clock running slow. The situation is symmetric. Neither A nor B is "really" time dilated, just relative to each other, symmetrically.

If A and B are at a different gravitational potention (let's say A is at the lower potential), then A measures B's time/clock running fast and B measures A's time/clock running slow. This situation is asymmetric. Both agree that A's time is dilated (running slow) relative to B.

 

[Herman Trivilino]

I am thinking of it this way: Relative motion in SR leads to time dilation, which finds its equivalent in GR in terms of time dilation in different gravitational potentials.

I do not think they're equivalent. Time dilation is indeed due to relative motion and it is symmetrical. It's a comparison of a proper time to coordinate time.

Clocks at different gravitational potentials run at different rates, but this is not symmetrical. It's a comparison of two different proper times.

 

[PeterDonis]

I am still looking for an explicit in-depth analysis as to why there is time dilation in SR and GR, but not length contraction and mass increase in GR.

You have already been given the answer: the two cases you described are not equivalent. I described why in post #2. Do you have questions about what I said there?

 

[Ranku]

Do you have questions about what I said there?

What I am trying to get at is what elements of SR survived and what did not in GR, and how and why. After all, the reason it is called GR is because the principle of relativity (the laws of physics are the same for observers in all inertial reference frames) was extended from the inertial reference frame of SR to the locally inertial reference frame in the global non-inertial reference frame of GR. Through further mathematical development, involving tensor mathematics, this allows, for example, what is observed as time dilation in relative inertial reference frames in SR to emerge as time dilation in different gravitational potentials in GR.

Since I do not have the mathematical expertise of tensor mathematics of GR, I am wondering if there is a way to visually and conceptually demontrate this transformation. If so, can it therefore be also conceptually demonstrated as to why the other aspects of relative inertial reference frames in SR such as length contraction and mass increase also does not get tranformed as length contraction and mass increase in different gravitational potentials in GR?

 

[Ibix]

The difference between SR and GR is the same as the difference between Euclidean geometry and spherical geometry (or other curved surface geometry) - almost literally so. SR applies in flat spacetime, as Euclidean geometry applies on a flat plane. General relativity applies on any curved background, including flat spacetime since "flat" is just zero curvature. Like Euclidean geometry, which is a decent approximation for small areas of a curved surface (hence you can pretend your floor is Euclidean), SR applies as a good approximation in small patches of spacetime even when curvature is present.

I think that your thinking on this is completely wrong. I don't think you can have some sort of a tick list of "things that happen in SR that do/do not happen in GR". The reason for time dilation happening in the two cases you are talking about are similar in some senses (it boils down to: the path length between two geometric planes depends on the path you take), and completely different in others (the reason for picking the planes to travel between is radically different). And I'm not sure there's even a way to define length contraction between objects at different heights in a gravitational field.

One last thing - please forget about mass increasing with velocity. This is using an archaic definition of mass called "relativistic mass". Modern textbooks do not use it because it's unhelpfully confusing. Reputable sources stopped using it decades ago - it persists in pop-sci sources that are more concerned with sounding cool than helping you learn. The modern definition of mass is what used to be called "rest mass" or "invariant mass".

 

[Herman Trivilino]

What I am trying to get at is what elements of SR survived and what did not in GR, and how and why.

They all did. Understanding how and why means, I would think, that you would have to understand GR itself.

After all, the reason it is called GR is because the principle of relativity (the laws of physics are the same for observers in all inertial reference frames) was extended from the inertial reference frame of SR to the locally inertial reference frame in the global non-inertial reference frame of GR.

Reasons for naming schemes are not necessarily helpful and may in fact be misleading. They are created during the infancy of ideas, before they are refined and an understanding is complete.

Through further mathematical development, involving tensor mathematics, this allows, for example, what is observed as time dilation in relative inertial reference frames in SR to emerge as time dilation in different gravitational potentials in GR.

As has been stated repeatedly in this thread, those two things are different.

Since I do not have the mathematical expertise of tensor mathematics of GR, I am wondering if there is a way to visually and conceptually demontrate this transformation. If so, can it therefore be also conceptually demonstrated as to why the other aspects of relative inertial reference frames in SR such as length contraction and mass increase also does not get tranformed as length contraction and mass increase in different gravitational potentials in GR?

If it could that would be a disaster! You'd have a theory predicting something that contradicts what we observe.

And by the way, the increase in mass you mention is not a consequence of the Principle of Relativity. It is instead a consequence of a choice to re-define what we mean by mass. Such a re-definition is not necessary, advisable, or even common.

 

[PeterDonis]

Through further mathematical development, involving tensor mathematics, this allows, for example, what is observed as time dilation in relative inertial reference frames in SR to emerge as time dilation in different gravitational potentials in GR.

This is not correct, and I don't know where you are getting this idea from. Relative motion is not the same thing as different gravitational potentials. You should not expect them to work the same, and GR does not say they do.

Since I do not have the mathematical expertise of tensor mathematics of GR, I am wondering if there is a way to visually and conceptually demontrate this transformation.

The short answer is no, at least not the way you are apparently thinking. Reasoning by analogy the way you are trying to does not work in physics.