[flr (OP)]

 The mass increases with the speed, as: m=gamma*m0.

 As two objects travels faster and faster, their mass increases, but does that means they attract gravitationally stronger?

Let's consider we observe two electrons moving parallel and accelerated to such a speed that their relativistic mass is equal to that of planet (say) Jupiter. Will these electrons attract gravitationally each other as two objects of the size of Jupiter? 

Mod edit: Rephrased the post subject as a question.

[lightarrow]

The mass increases with the speed, as: m=gamma*m0.

No, mass is invariant. Relativistic mass is not used anylonger from most of physicists.

http://www.thenakedscientists.com/forum/index.php?topic=16789.0
http://www.thenakedscientists.com/forum/index.php?topic=21363.0
http://www.thenakedscientists.com/forum/index.php?topic=8298.50

[graham.d]

It is unfortunate you picked electrons. Yes, from your frame of reference they will have such gravitational attraction but they will also have a much, much larger magnetic repulsion. They will gradually diverge as they would if you were in the same frame as they are but at a rate slowed by the time dilation effect. Electromagetic effects are fully consistent with Special Relativity so there can be no possibility of the electrons doing anything different from moving away from each other as they would if you were in their frame of reference.

[lightarrow]

It is unfortunate you picked electrons. Yes, from your frame of reference they will have such gravitational attraction but they will also have a much, much larger magnetic repulsion.

[] You mean "attraction"?
Anyway the magnetic attraction can never be greater than the coulombian repulsion, so don't see your point.

[lightarrow]

does that means they attract gravitationally stronger?

Yes, remember that it's not only mass which generates curvature: energy and momentum too (through the "stress energy tensor").

Let's consider we observe two electrons moving parallel and accelerated to such a speed that their relativistic mass is equal to that of planet (say) Jupiter. Will these electrons attract gravitationally each other as two objects of the size of Jupiter? 

Instead of answering to your question, I want to give you a consideration on which you can (if you want) think (a long) about:

in this exact moment, you and your friend close to you (immagine there is one) are both moving at ultra relativistic speed with respect to some ultra relativistic particle of cosmic rays. Are you attracting each other gravitationally as two (big) Jupiters?

[graham.d]

It is unfortunate you picked electrons. Yes, from your frame of reference they will have such gravitational attraction but they will also have a much, much larger magnetic repulsion.

[] You mean "attraction"?
Anyway the magnetic attraction can never be greater than the coulombian repulsion, so don't see your point.

I was trying to show the effect as perceived from a stationary frame. I think that flr was trying to reconcile that view with the simpler view of someone moving with the electrons where the situation is clearly trivial. I agree that this cannot be fully answered without resort to GR, but I don't think this would aid understanding without a much greater knowledge of differential geometry.

[flr (OP)]

in this exact moment, you and your friend close to you (immagine there is one) are both moving at ultra relativistic speed with respect to some ultra relativistic particle of cosmic rays. Are you attracting each other gravitationally as two (big) Jupiters?

That does not change the fact that the cosmic ray will see us as two Jupiters, right? Will then the cosmic ray see us (me and my (girl-)fried) collapsing as two Jupiters?
From my perspective I am sure we are not.

[graham.d]

The cosmic ray (if it could) would not see you attract each other (I mean gravitationally) more than you would normally, because of time dilation. When it comes to gravitational collapse, and even in the lesser case we are talking about, special relativity is insufficient to describe the events. It is fair to say that nothing should be observed by the cosmic ray than would be expected in the frame of you and your friend (however "attractive" she may be).

In General Relativity the concept of mass is not so well defined. I think this is what lightarrow was alluding to.

[lightarrow]

The cosmic ray (if it could) would not see you attract each other (I mean gravitationally) more than you would normally, because of time dilation. When it comes to gravitational collapse, and even in the lesser case we are talking about, special relativity is insufficient to describe the events. It is fair to say that nothing should be observed by the cosmic ray than would be expected in the frame of you and your friend (however "attractive" she may be).

In General Relativity the concept of mass is not so well defined. I think this is what lightarrow was alluding to.

Yes. But I also intended more: spacetime curvature is frame invariant, so the cosmic ray particle would not measure any curvature; a real massive object, instead, DOES generate curvature.
This is another reason for which relativistic mass is not a "real" mass.

[yor_on]

Flr, you have quite a nice imagination

Now i will jump in too, if we assume that you mean that those two is non-uniformly accelerating you will get one result, if you assume they are uniformly accelerating you will get another. With uniform acceleration (1G) you and your partner are in a state of rest

There is no difference between that state and being stationary on Earth according to the principle of equivalence. In that frame you will as far as I understand it, also being at rest relative each other, not experience anything different from being on Earth. As for an 'stationary observer' of you both the effect will be as if it was him traveling from your point of view. in fact you and your friend will be what is called 'Rindler Observers'. What you will notice though that will differ you from your stationary observer is that the universe will become more energetic the further your acceleration have taken you, and as I understand it, even at some 'time and place' observe what your stationary observer would call 'virtual photons' as being constantly existing, also I've seen references to what's called the 'heat bath' which is part of the hidden energy in a vacuum that now will appear for you.

If we on the other hand assume that you are accelerating in a non-uniform way the equivalence principle shouldn't hold? At least I don't think so? But I'm not sure, it comes down to 'frames of reference' and how you define them, and your 'system', inside times arrow it seems to me? A very confusing concept to me. Where do I draw the limits? Can I by stating that from now on, in my personal 'frame of reference and time' a second will be one quackiljont of a second, sort of stretch out my periods of 'uniform accelerating' inside my rockets tries to vary that acceleration into a non-uniform one f.ex? That one hurts my head, the first one assumes uniform 'quanta' of 'speed' uniformly served and the second one? But as long as you have the exact same acceleration you will still be at rest relative each other and depending on the logarithm of your non-uniform acceleration become more 'attracted' to each other, I think

And then you have a third possibility, that you stop accelerating at some velocity and from there on are 'free falling'. As soon as you have done that your gravitational 'forces' will be gone but you should still observe time dilation, blue-shift etc and your invariant mass will bend SpaceTime and keep you together. As I think, quite nice in fact, considering it just was valentine day, united but rather cold..

So Wha'daya'tink, am i reasonable here or just bugging the he* out of you:)

[flr (OP)]

....What you will notice though that will differ you from your stationary observer is that the universe will become more energetic the further your acceleration have taken you, and as I understand it, even at some 'time and place' observe what your stationary observer would call 'virtual photons' as being constantly existing, also I've seen references to what's called the 'heat bath' which is part of the hidden energy in a vacuum that now will appear for you.

The observation of a more energetic universe is an effect of acceleration or only of speed?
Let's say the acceleration took me at high enough speed that I see what you called "heat bath". Then I suddenly stop the acceleration (made it zero) and I move at constant speed. Am I still seeing an energetic universe and the "heat bath" which I was seeing it before cutting out the acceleration? 

[yor_on]

Yep, you will do just fine

As I understands it it is a direct result of his uniformly accelerating. And the idea seems to have come from Hawking's work on Black Hole radiation. Take a look here Rindler observers and Unruh Radiation