- #1

j8hart

I studied Physics at university, but went on to become a computer consultant, and I have forgotten quite a bit.

Moreover I have not been able to find anyone willing to take the time to explain it.

The puzzle came to me whilst I was reading "A brief history of time". At one point in the book Hawkin talks about the possibility of manufacturing a Black Hole by accelerating a particle until it becomes massive enough.

I thought, OK, so we know that relativistic mass effects the inertia of a particle, but that does not necessarily bring gravity into it.

In short, does relativistic mass have a gravitational component? And if so, does it increase in proportion to the inertia?

My university course obviously covered Special Relativity, but we did not really cover General Relativity in depth.

So I devised a couple of thought experiments which I hopped did not require General Relativity.

First, imagine two spheres in empty space. Each has a mass of 1Kg, and they are 1m apart.

In this situation, we do not have to worry about General Relativity, we can calculate the force exerted by one sphere on the other using Newton’s law of Universal Gravitation.

F = GM2/r2

M = 1Kg and r = 1m, therefore F = G Newtons.

Lets say for the moment that this force is exactly counterbalanced by an electrostatic force of repulsion between the two spheres (i.e. also G Newtons).

The spheres therefore remain at the same relative distance apart (1m).

Now imagine an observer "o" at exactly the mid point between the two spheres. Also imagine that o observes the spheres as moving with a velocity close to the speed of light along a course exactly perpendicular to a line drawn between the two spheres and through o.

The diagram below shows the situation, o in the middle, the two spheres are represented by *, and the arrows show the direction on motion.

*-------------->

o

*-------------->

If the velocity of the spheres is chosen so that the relativistic mass is 10 times the rest mass, then we should still be able to use Newton’s law of Universal Gravitation to calculate the force exerted by one sphere on the other as observed by o.

To observer o, M' = 10Kg. r is still 1m (obviously no contraction, since it is perpendicular to the line of motion).

Thus theoretically the gravitational force (F') as observed by o is 100xG Newtons.

In other words F' = 100xF.

If an observer on the spheres sees no acceleration towards the other sphere because of the electrostatic force, then nether can o.

If therefore the Gravitational force increases 100 fold, then so must the electrostatic force.

This is probably explainable by Lorentzian contraction, since the field lines will be closer together, but I have never had enough time to think this through.

If we take away the electrostatic force, and instead assume that the spheres just start to accelerate towards one another at the moment they pass o, then an observer on one of the spheres would calculate the acceleration as:

a = F/M i.e. G Newtons/1Kg, or G m/s2.

The observer o would calculate:

a' = F'/M'

F' = 100 x F, and M' = 10 x M therefore:

a' = (100 x F)/(10 x M) = 10 x (F/m)

or

a' = 10 x a

This is not the result I would expect. Time for the observer o is running 10 times faster than on the spheres.

Since acceleration is dD/dT2 from a time dilation point of view I would expect:

a' = 100 x a

In other words, velocity along the line between the two spheres should be 10 times faster as seen by o, acceleration, being the rate of change of velocity with time should therefore be 100 times faster.

If I work the other way around and assume that I do not know how the gravitational force is affected by the relative velocities, however I assume that:

1) a' = p2 x a where p is the ratio of rest mass to relativistic mass (10 in this case) is correct.

2) F' = q x F where q is some factor relating the gravitational force at rest with the gravitational force in motion.

In 1) we can substitute a' = F'/M' = (q x F)/(p x M) (in this case M=Inertia, so M' = p x M). This gives:

(q x F)/(p x M) = p2 x a = p2 x (F/M)

If we divide both sides by F/M we get:

q/p = p2

or

q = p3

In this case p=10, so F' = 1000 x F, not 100 as I calculated earlier.

This does not look right either.

Can anyone shed some light on this?

Thanks in advance,

Jonathan