Does mass increase in SR mean higher gravity in GR?

[pervect]

The gravitational attraction of a body with a given (rest-, or invariant-) mass is always the same, no matter how fast it's moving with respect to anything else.

- Warren

What makes you say this? No disrespect, but I think this is incorrect. Certainly, F=K q1 q2 / r^2 does not work for the force between a relativistically moving charge and a stationary one (see my previous post) - why would one think that the analogous equation works for gravity under the same conditions?

Furthermore, it is difficult if not impossible to explain how pressure causes gravity if velocity does not change the gravitational force between particles. If we use the "swarm of particles" model for a perfect fluid, the difference between a fluid without pressure and a fluid with pressure is just the fact that the particles are moving. If the motion of the particles had no effect, the gravitational field of a pressureless fluid would be the same as that of a fluid with pressure - but it is not, the pressure of the fluid contributes to the stress energy tensor.

 

[Creator]

...the behavior of an electric field - the component of the field E in the direction of a boost is unchanged by the boost, i.e. Ex is unchanged by a boost (velocity change) in the x direction.

.

Thanks for your response, Pervect, to my Aikelburg-Sexl comments, and for the reference.
Before we go on to address the origin of your differences with that of Pete's derivation (the discussion of which I find very interesting), let me first ask you a question:

1st. Aikelburg & Saxl (whose authority you seem to accept) state that (as determined in both the linearized and exact solutions) ..."Physically the gravitational field of a rapidly moving particle shows the same characteristic behavior as its electromagnetic field: it is...compressed in the direction of motion".

2ndly, Your own link (http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_15.pdf )
verifies the validity of this statement and in the diagram given clearly shows shortened field lines in the direction of motion; (look at it closely if you missed it).

Why then do you insist on saying, " ... the behavior of an electric field - the component of the field E in the direction of a boost is unchanged by the boost, i.e. Ex is unchanged by a boost (velocity change) in the x direction."??
Respectfully, on this point then, I think you are possibly in error.
If it is your derivation that has led you to such a conclusion, then it is not only contrary to that of Pete's but also to the above referenced items.

Creator

 

[pervect]

Thanks for your response, Pervect, to my Aikelburg-Sexl comments, and for the reference.
Before we go on to address the origin of your differences with that of Pete's derivation (the discussion of which I find very interesting), let me first ask you a question:

1st. Aikelburg & Saxl (whose authority you seem to accept) state that (as determined in both the linearized and exact solutions) ..."Physically the gravitational field of a rapidly moving particle shows the same characteristic behavior as its electromagnetic field: it is...compressed in the direction of motion".

2ndly, Your own link (http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_15.pdf )
verifies the validity of this statement and in the diagram given clearly shows shortened field lines in the direction of motion; (look at it closely if you missed it).

Why then do you insist on saying, " ... the behavior of an electric field - the component of the field E in the direction of a boost is unchanged by the boost, i.e. Ex is unchanged by a boost (velocity change) in the x direction."??
Respectfully, on this point then, I think you are possibly in error.
If it is your derivation that has led you to such a conclusion, then it is not only contrary to that of Pete's but also to the above referenced items.

Creator

I probably should have posted a link to the predecessor article

http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_14.pdf

Note that at the same location in space-time Ex, measured in the O frame, is the same as Ex', measured in the O' frame.

This is a consequence of how the electromagnetic field transforms under a boost

this link

might be useful in confirming that fact (or it might be an unneeded complication at this point, I dunno).

So why is the field clearly shorter in the diagram? It's shorter not because the field at the same point in space-time is different because of the boost - it's because the field is expressed in different variables. x' is different from x by a factor of gamma, and when one is moving towards the source, the distance r' is diffrent from the distance r by a factor of gamma.

As far as the non-electrostatic problem goes, the summary of my calculations are at

http://www.geocities.com/pervect303/movingmass.pdf

I feel fairly happy about the results in 3), but not very happy about what happens when I "boost" back to the coordinate system in 5). Interestingly, the steps needed to go from 3 to 5 are fairly simple. The results in 5) are the ones the most comparable to Pete's results, and the pictures I've presented, and what I want to find out - the fields that a stationary observer would see as a large mass "whizzes by" him.

I don't see at this point how to reconcile the results with Pete's. I have some ideas on things I want to try and look at, but I haven't gotten around to doing any more serious work, in part because of a nasty flu that I've come down with.

I'm also still suspicious of the fact that Pete's results are the gradient of a scalar function.

 

[Chronos]

Respects to pervect and creator. Good technical points all. Anyways, in defense of Chroot [like he needs it] he is correct in the real universe. The mass of an accelerated object does increase as it gains speed, but the system mass does not. [a self propelled object actually bleeds off mass entropically due to efficiency losses in the propulsion system]. This is a classic fallacy in SR. While it is true a mass possessing body accelerated to the speed of light acquires infinite mass, it requires an infinite amount of energy to accelerate it to that speed. If you dutifully obey the laws of thermodynamics and do the math [e=mc^2], you will find the net effect is zero in a self propelled system. Paradoxes only occurs when you import 'free energy' from nowhere to power the acceleration. I suggest that a similar energy cutoff [albeit gigantic] should be considered in any real universe solutions to field equations. The universe may be infinite, but, the causally connected fragment we reside in, is not.

 

[pervect]

It's not necessarily unrealistic to envision a spaceship with an external power source - rockets are so bad for reaching relativistic velocities (except perhaps for anti-matter ones) that a laser driven light-sail is quite competitive. It's extremely unrealistic to imagine accelerating a planet to light speed, but significantly less so to ask what the gravity of a planet would "look like" as seen from such a viewpoint.

I think I'm finally beginning to understand why most of the analyses I've seen covers only the low velocity case, though, and why I should probably do likewise.

The problem is fairly basic, but hard to see under all the math. Gravity acts on everything - so in order to determine the "acceleration of gravity", one needs an external reference system. With such an external coordinate system, one can impose the condition that an object moves "in a straight line with constant velocity", (this will be a different path than the natural geodesic that the object would follow on its own, of course), and then measure the acceleration required to force the object to follow this path. (One can measure this acceleration from any of several possible viewpoints, a convenient one would be an accelerometer mounted on the body itself).

Such a procedure requires that space be reasonablly close to being "flat", however. At first it is not obvious why high velocities should be an issue with respect to the flatness of space.

But if we look at the connection coefficients in more detail, some of the concerns become apparent.

Take the value of d^2 x / d tau^2 in a Newtonain gravity field (g_xx = g_yy = g_zz = 1+2U, g_tt = -1+2U, where U is the additive inverse of the Newtonian potential and assumed not to be a function of time).

Let the velocity be in the x direction. The connection coefficients that will influence this are

\Gamma_{xxx} ={\partial U}/{\partial x}}
\Gamma_{xxt} = \Gamma_{xtx} = 0
\Gamma_{xtt} = -{\partial U}/{\partial x}

Then
<br /> \frac{d^2x }{ d \tau^2} = g^{xx} ( \Gamma_{xxx}(\frac{dx}{d\tau})^2 + \Gamma_{xtt}(\frac{dt}{d\tau})^2) <br />

thus for a velocity \mbox{\beta} in the x direction

<br /> \frac{d^2x }{ d \tau^2} = g^{xx} ( \Gamma_{xxx} \frac{\beta^2}{1-\beta^2} + \Gamma_{xtt} \frac{1}{1-\beta^2})<br />


When \mbox{\beta} << 1, the time component of the curvature provides the largest contribution to the acceleration - but for large \mbox{\beta}, approaching 1, the space curvature term is just as important, and of opposite sign. Thus the notion that space is "flat" is suspect, because the contribution of the purely spatial curvature to the acceleration is comparable to the time curvature.

 

[pervect]

I'm pretty sure everyone's probably very tired of this thread by now, but I can't resist One More comment.

It seems to me that the best approach to considering what happens to gravity at high velocities is to go back to the idea of exploring "tidal gravity", the Riemann tensor, at high velocities, due to the aforementioned problems of dealing with the traditional notion of gravity as a "force".

This approach has the definite advantage that it can be done at a point - one does not need any reference to the outside world or dependence on an external global coordinate system to know what tidal forces one is experiencing, one can measure the tidal forces directly. (Except for the problem of rotation, which I'll get into).

The biggest stumbling block I have here is the issue of how to deal with rotation. Some relativly simple calculations can give the tidal forces on a body moving at relativistic velocities relative to a large mass. One needs to compute the Riemann tensor rather than the connection coefficients, then the tidal forces can be neatly summed up by the following matrix:

E^a{}_b = R^a{}_{bcd} u^b u^d
where u^x is the four-velocity of the moving object.

The main difficulty in making this approach rigorous is dealing with eliminating rotation from the coordinate systems used, so that "centrifugal forces" from a rotating coordinate system don't get confused with the actual components of the tidal force.

Thus, we can directly measure the tidal forces we experience due to passing close to a body moving at relativisitic velocities directly if, and only if, we have zero rotational angular momentum - the later is not a very stringent condition, but it does mean we can't quite ignore the rest of the universe, we have to pay enough attention to it to be able to say that we aren't rotating.

Another thing which one can calculate in principle is the total amount of momentum transferred to a body by an object "flying by".

So if one was initially at rest, and an object came whizzing by at ultra-relativistic speeds, then left for infinity again, it's reasonable to ask what velocity one has after the body has left. Space and space-time should be perfectly flat when the massive body has "passed through", so there shouldn't be any ambiguity in this question.

 

[GoodPR]

It was mentioned earlier in the thread that if you didn't account for the Energy required to reach that velocity you would violate thermodynamics.

What if this relative mass was seen as adding mass from one side, and subtracting mass from the other? So essentialy a fast moving body would pull things stronger then a rest mass body from one side and weaker, (or possibly negative) from the other side?

 

[GoodPR]

Also, from the first example, that would mean the ball on a stick would have no effect on the moon if it was being swung parallel to the moon. If the ball was being swung so that on one side of the orbit it was moving directly towards the moon and the other directly away, then the overall result would still be null because it would pull one direction and push the other.

 

[hcm1955]

Maybe this question has already been answered but: Assuming two particles (X1 and X2) moving at relativistic velocities relative to each other.

X1 --->
|
d
|
<---X2


Is there some distance d and some less than c velocity that X1 and X2 orbit each other?



Cheers,
Bert

 

[Dale]

Not due to SR effects.

 

[hcm1955]

Bert:

Maybe this question has already been answered but: Assuming two particles (X1 and X2) moving at relativistic velocities relative to each other.

X1 --->
|
d
|
<---X2


Is there some distance d and some less than c velocity that X1 and X2 orbit each other?

Cheers,
Bert



DaleSpam:
Not due to SR effects.



Bert:
What about GR effects increases mass increases the gravity?

 

[Dale]

What about GR effects increases mass increases the gravity?

Mass isn't the source of gravity in GR. The source of gravity is the stress-energy tensor:
http://en.wikipedia.org/wiki/Stress-energy_tensor

As you can see from the link, energy (proportional to mass) is only one component of 10 independent components. As you add energy to a particle by accelerating it not only do you increase the energy component, but you also increase the momentum component. The net effect is that you don't get increased gravity.