Speed & gravity

LeonhardEuler

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In SR, when an object starts moving quickly, it seems to have more mass in the sense that it requires more force to accelerate it at a given rate. Dose the object also behave as if it has more mass in the sense that other objects are more attracted to it gravitationaly?
 

JesseM

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LeonhardEuler said:
In SR, when an object starts moving quickly, it seems to have more mass in the sense that it requires more force to accelerate it at a given rate.
As jtbell points out in post #180 of this thread, it's not as simple as saying the object just behaves like it's more massive, because the force required to accelerate a moving object will depend on the direction of the force relative to the direction of motion--it's hardest to accelerate the object parallel to the direction of motion, but easier in other directions.
LeonhardEuler said:
Dose the object also behave as if it has more mass in the sense that other objects are more attracted to it gravitationaly?
This is discussed in this question from the Usenet Physics FAQ:
If you go too fast do you become a black hole?

According to relativity the following are true facts:

1. As an object approaches the speed of light, its kinetic energy increases without limit.
2. Energy is related to mass by the formula E = mc^2.
3. As an object approaches the speed of light, its length contracts towards zero.
4. If enough mass is squeezed into a sufficiently small space it will form a black hole

Put these facts together and it looks like we should be able to conclude that an object which moves at a speed sufficiently close to the speed of light should collapse to form a black hole. We could even argue that if you move fast enough relative to a star then that star must appear as a black hole to you because of its increased energy as observed by you. This would be paradoxical since we would expect things to appear very differently to an observer who is stationary relative to the star. So what has gone wrong?

In fact objects do not have any increased tendency to form black holes due to their extra energy of motion. In a frame of reference stationary with respect to the object, it has only rest mass energy and will not form a black hole unless its rest mass is sufficient. If it is not a black hole in one reference frame, then it cannot be a black hole in any other reference frame.

In part the misunderstanding arises because of the use of the concept of relativistic mass in the equation E = mc^2. Relativistic mass, which increases with the velocity and kinetic energy of an object, cannot be blindly substituted into formulae such as the one that gives the radius for a black hole in terms of its mass. One way to avoid this is to not speak about relativistic mass and think only in terms of invariant rest mass (see Relativity FAQ Does mass change with velocity?).

The statement that "If enough mass is squeezed into a sufficiently small space it will form a black hole" is rather vague. Crudely speaking we would say that if an amount of mass, M is contained within a sphere of radius 2GM/c^2 (the Schwarzschild radius) then it must be a black hole. But this is based on a particular static solution to the Einstein field equations of general relativity, and ignores momentum and angular momentum as well as the dynamics of space-time itself. In general relativity, gravity does not simply couple to mass as it does in the Newtonian theory of gravity. It also couples to momentum and momentum flow; the gravitational field is even coupled to itself. It is actually quite difficult to define the correct conditions for a black hole to form. Hawking and Penrose proved a number of useful singularities theorems about the formation of black holes, and from astrophysics we know that the theorems should apply to sufficiently massive stars when they reach the end of their life and collapse into a small volume.
This stuff was also discussed at length on this thread.
 

LeonhardEuler

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Thanks!oiasd
 

pervect

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LeonhardEuler said:
In SR, when an object starts moving quickly, it seems to have more mass in the sense that it requires more force to accelerate it at a given rate. Dose the object also behave as if it has more mass in the sense that other objects are more attracted to it gravitationaly?
You might also want to take a look at this thread where the issues has been discussed previously.
 

Aer

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LeonhardEuler said:
Dose the object also behave as if it has more mass in the sense that other objects are more attracted to it gravitationaly?
This is a very good question for JesseM and Pervect to answer directly
:biggrin:

Go ahead.
 

pervect

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I already have, though I've rambled on a bit more than is necessary. One of these days I'll probably try to write a clearer summary with nice diagrams. Don't hold your breath waiting, though, you might turn blue :-).

One of the things I really need to do before I write the summary is to sit down and investigate energy pseudo-tensors more, because I think they should provide the clearest answer to the quesiton (via an anology to Gauss's law).

I think it's unfortunate that this question doesn't seem to be addressed more fully (with worked examples) in the literature or in the textbooks (at least the couple of textbooks I've looked at). MTW seems to treat everything else in detail, for instance (hence the size of the book :-)), but their treatment of this issue is downright skimpy.
 

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