Infinite mass = infinite gravity ?

[annatar]

Infinite mass = infinite gravity !?

Someone asks me this:

There is an object (in free space) attracted by a black hole and accelerates towards it. Can the object be accelerated to light-speed^-1？

I instinctively replied that "By Special Relativity, the mass will approach infinity as it moves closer to light speed, therefore it won't work."

However, that person argues that as the mass approaches infinity, the G. force on the obj. also approaches infinity! How can this be answered?

P.S. Additional thoughts: the mass and gravity actually never reaches infinity, right？
So...maybe the rate of increase of the former is much higher than the latter?

 

[ZikZak]

The concept of a speed-dependent mass ("relativistic mass") is outdated and deprecated, and for several very good reasons. This is one of them. General Relativity is NOT F=GMM/r^2 with M replaced by "relativistic mass." Not even close. In a relativistic situation, you cannot merely rely on Newton's formulas for gravity which simply do not apply. Kinetic energy does not gravitate in the same way that rest mass does.

No observer local to the falling body will ever observe it traveling at c. Only massless bodies travel at c. However, an observer holding station just over the event horizon will indeed observe it falling in at speeds arbitrarily close to c.

 

[annatar]

The concept of a speed-dependent mass ("relativistic mass") is outdated and deprecated, and for several very good reasons. This is one of them. General Relativity is NOT F=GMM/r^2 with M replaced by "relativistic mass." Not even close. In a relativistic situation, you cannot merely rely on Newton's formulas for gravity which simply do not apply. Kinetic energy does not gravitate in the same way that rest mass does.

No observer local to the ...

Ok, so I will tell him : "By G.R. ...
Gravity will NOT become infinite, and it still cannot accelerate to light speed because of the energy requirement.” ?

Sounds not so convincing

 

[atyy]

Kinetic energy does not gravitate in the same way that rest mass does.

http://arxiv.org/abs/gr-qc/9909014

 

[Fredrik]

A convincing argument would have to include the GR calculation, and I have a feeling that neither you nor your friend would understand it. I'm not even sure I would. You should however be able to understand that there is no gravity in SR. (How would you include it in the theory when Newton's law of gravity implies instantaneous transfer of information?) So even if you find the GR argument uncovincing because you haven't seen the details and wouldn't understand them anyway, you should still find the SR+Newton argument extremely unconvincing.

 

[atyy]

In GR, the motion of a small object in the gravitational field of a very much larger mass is independent of the small object's mass. The large mass sets up spacetime curvature, and the small object moves on a timelike geodesic of the curved spacetime. In comparison, light paths are null geodesics of the curved spacetime. If you take any very small patch of spacetime and compare the speed of the small object with the speed of a nearby ray of light, the speed of the small object will always be less than that of the nearby ray of light. We use small patches and nearby objects/rays of light because in curved spacetime, there is no unique definition of the relative speeds of distant objects.

 

[atyy]

If you want to compare two large point masses moving in GR - well that is a black hole collision - I have never followed the calculations in detail - but here's some references:

http://www.aip.org/png/2006/256.htm
http://arxiv.org/abs/gr-qc/0511048


http://arxiv.org/abs/gr-qc/0612117

 

[annatar]

How would you include it in the theory when Newton's law of gravity implies instantaneous transfer of information?

Ah ha! So this is our misconception.
Funny, that neither of us would understand the true answer

Actually, he also tried to formulate the situation using conservation of energy. i.e. the object's P.E. got converted to K.E. as it accelerates towards the B.H. The P.E. equation he used is derived from the Newton's Law of U.G. ,

By reasons mentioned in the above replies, he is completely wrong - at least when the object is close to the B.H. - is it?

 

[annatar]

If you want to compare two large point masses moving in GR - well that is a black hole collision - I have never followed the calculations in detail - but here's some references:

http://www.aip.org/png/2006/256.htm
http://arxiv.org/abs/gr-qc/0511048


http://arxiv.org/abs/gr-qc/0612117

Very interesting indeed!
Thank you

 

[ZikZak]

http://arxiv.org/abs/gr-qc/9909014

An interesting article that I will read carefully later, but I did qualify my statement about gravitation of KE with "in the same way" as rest mass. Even in the article, it discusses the well-known result that tangentially, a light ray "falls" at twice the rate given by a quasi-Newtonian prediction but the same is not true radially.

 

[Ich]

Actually, he also tried to formulate the situation using conservation of energy. i.e. the object's P.E. got converted to K.E. as it accelerates towards the B.H. The P.E. equation he used is derived from the Newton's Law of U.G.

I have N.I. what U.G. means, but C.o.E. is a good prompt. Total energy is kind of conserved even in GR. That means that any gain in K.E. is fed by a corresponding loss in rest energy.
Example: If you stop an infalling body short of the E.H. of the B.H., convert its K.E. to E.M. raditation and send it out to some far away receiver, the receiver will get, say, 95% of the initial rest mass of the object. The mass of the B.H. (including surroundings) increases by 5% of said rest mass.

 

[Fredrik]

Ah ha! So this is our misconception.
Funny, that neither of us would understand the true answer

Actually, he also tried to formulate the situation using conservation of energy. i.e. the object's P.E. got converted to K.E. as it accelerates towards the B.H. The P.E. equation he used is derived from the Newton's Law of U.G. ,

By reasons mentioned in the above replies, he is completely wrong - at least when the object is close to the B.H. - is it?

I actually didn't read the OP carefully. I just glanced at it quickly and thought it was the same question that has been asked here many times before. I didn't realize that you had said that the cause of the acceleration is a black hole. That makes everything much more complicated,

We still can't use Newton's law of gravity, for the same reason as before. And now we really have to think about what it means to say that the object is "accelerating". You probably meant with respect to another object which is held at a constant distance from the black hole. To be more specific, we should consider an object at constant spatial Schwartzschild coordinates. We can associate a local inertial frame with its motion, and now we can consider the other object's velocity in this frame. This velocity will be increasing, and that means that its energy in this frame is increasing too. But why would that increase its gravitational pull on other objects? GR doesn't say that the source of gravity is the energy in the local inertial frame of an object at fixed spatial Schwarzschild coordinates. It says that the source is the stress-energy tensor of all the matter in the universe. The falling object certainly contributes to that, but it's very difficult to see how exactly.