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If an object was sliding on an infinitely long friction-less floor on Earth with relativistic speeds (ignoring air resistance), would it exert more vertical weight force on the floor than when it's at rest?

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The question has to make explicit how the answer is supposed to treat the effect of the curvature of earth's surface. –  Johannes May 9 '13 at 15:34
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@Johannes Or we could just assume a hypothetical universe in order to answer the spirit of the question –  Jim May 9 '13 at 15:35
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I am not a relativist and I hesitate to write an "answer", but my understanding is something like The "relativistic mass" is energy and all energy couples to gravity, so the short answer is "yes", but there are some caveats because it also shows up in other elements of the stress-energy tensor generating contributions with the opposite sign so that you can not, say, accelerate a body until it forms a black hole. Not sure what the final effect is. –  dmckee May 9 '13 at 15:41
    
Interesting question –  Andrew Palfreyman May 14 '13 at 19:28
    
The answer is just yes. –  Ron Maimon Aug 22 '13 at 22:52
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3 Answers 3

up vote 11 down vote accepted

First off, your question is phrased in terms of relativistic mass, which is an obsolete concept. But anyway, that's a side issue.

The question can be posed in terms of either the earth's force on the puck or the puck's force on the earth. We expect these to be equal because of conservation of momentum.

In general relativity, the source of gravitational fields is not the mass or the mass-energy but the stress-energy tensor, which includes pieces representing pressure, for example. The puck has some stress-energy tensor, and this stress-energy tensor is changed a lot by the puck's highly relativistic motion. Therefore the puck's own gravitational field is definitely changed by the fact of its motion. However, the change is not simply a scaling up of its normal gravitational field. The field will also be distorted rather than spherically symmetric. Yes, the effect is probably to increase its force on the earth. The earth therefore makes an increased force on the puck.

Here is a similar example that shows that you can't just naively use $E=mc^2$ to calculate gravitational forces. Two beams of light moving parallel to each other experience no gravitational interaction, while antiparallel beams do. See Tolman, R.C., Ehrenfest, P., and Podolsky, B. Phys. Rev. (1931) 37, 602, http://authors.library.caltech.edu/1544/

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"Two beams of light moving parallel to each other experience no gravitational interaction, while antiparallel beams do." Ahhhh... This general relativity thing is still too difficult for me... –  Calmarius May 9 '13 at 18:38
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I wonder how close the anti-parallel beams have to be in order to orbit each other. –  Thomas May 9 '13 at 20:58
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My previous answer proved to be wrong. Energy density ("relativistic mass") does contribute to gravity - and the fact that the object is moving at relativistic speeds does affect the space-time around it.
There is an interesting document that explains the problem in further context.

Besides, when we think about it, if energy density didn't contribute to the force of gravity, photons, mass of which consists of their energy density, would not be affected by gravity. But we have provided many times, that the lightwaves - thus photons - are affected by gravity.

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It's okay to say that the object's rest mass remains the same and classically gravity depends on rest mass. But, don't we consider mass-energy? (which increases exponentially with velocity). I think it still affects the stress-energy tensor. –  Waffle's Crazy Peanut May 9 '13 at 15:33
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I'm just reading some works that actually prove that my answer is wrong. I'll edit it. –  Tomáš Zato May 9 '13 at 15:35
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Gravity does not couple to the rest mass. –  Johannes May 9 '13 at 15:36
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Gravity does not couple to the rest mass. - That would mean that still objects are not affected by the gravity - wouldn't it? After reading a few documents, I'm quite concerned that gravity is affected by the total $m$ which equals to: $\frac{m_0}{\sqrt{1-v^2/c^2}}$ so the rest (or invariant) mass affects gravity too. I might be confused by the terminology though. –  Tomáš Zato May 9 '13 at 15:46
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What I mean is that gravity couples to energy and momentum (and their fluxes) and not to rest mass (which your earlier answer stated). –  Johannes May 9 '13 at 18:49
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This is only a mere approximation to General Relativity.

Level 0 Approximation Aristotlean Physics. Ughhh......

Level 1 Approximation Newtonian Gravity

Here, the gravitational force, or the "weight", is given by:

$$F=G\frac{m_1m_2}{r^2 }$$.

Level 2 Approximation Newtonian Gravity with Special Relativistic corrections.

Ok, this is what your question's all about. The gravitational force (weight) here, is given by:

$$F=G \frac{\gamma\left(v_1\right)\gamma\left(v_2\right) m_1 m_2}{ r^2 }$$

Note, here, that even the definition of $r$ has changed. It has also undergoe Lorentz transformations.

Here, this $\gamma\left(v_1\right)m_1$ and $\gamma\left(v_2\right) m_2$, are the so-called "relativistic mass"es (an obselote term) of the objects.

Level $\bf{\infty}$ Approximation General relativity itself.

For the Schwarzschild metric in the case where $ \mbox dr=0 $, for example, What is the weight equation through general relativity?

In general, it is just,

$$W=W_{\operatorname{Newton}} c_0 \frac{ 1 }{\left(\frac{\operatorname ds}{ \operatorname{d} t }\right)}$$ \mbox

So, to answer your question, : .

Yes, at a Level 2 Approximation, but the real answer is given by GR.

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Minor quibble: Any reason to avoid refer to GR as level 3 approximation? If level infinity is taken, where does that leave a future QG theory? –  Johannes Jul 20 '13 at 7:50
    
@Johannes: Note that I said "Approximations to GR", not ST or QG . . –  Dimensio1n0 Jul 20 '13 at 7:59
    
The actual level 2 approximation does not change r in a straightforward way, it has an attraction given by the product of the energies (relativistic masses) but with a peculiar radial dependence given by the gravitomagnetic contribution, which is best to work out by boosting to the rest frame of one of the objects, where the gravitational field is known. –  Ron Maimon Aug 22 '13 at 22:53
    
@RonMaimon: Oh, yes, I forgot about the $r$. –  Dimensio1n0 Aug 23 '13 at 5:01
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protected by Qmechanic Aug 9 '13 at 4:45

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