From: hal@finney.org
Date: Sun Aug 22 1999 - 19:26:21 MDT
Robert J. Bradbury, <bradbury@www.aeiveos.com>, writes:
> This is related to a question I've wondered about the maximum density
> of light. Say I have a number of really large lasers (powered by
> stars) arranged in a large sphere, perhaps like a globular cluster.
> Say I beam all of those lasers at a single point in space. What
> happens? Do I generate matter? Do the beams simply pass through
> each other?
I don't really know the answer to this, but I speculate below.
> A related question would appear to be -- since a mass density
> above a certain level will generate a black hole, will an
> energy density equivalent to that mass density (by e=mc^2)
> do the same thing?
Technically, gravitational curvature is not generated by mass density
as such, but by what is called the stress-energy tensor, T. This can
be thought of as a symmetric 4 by 4 matrix, which represents the local
mass/energy density, the local momentum flow, and the internal stresses
in the matter which occupies the local region of space. The mass/energy
density is just the upper left term of the matrix, although this is
the one which is most important for weak gravitation like we normally
deal with.
The distinction becomes important when dealing with questions like, if you
accelerate an object so that its "relativistic mass" becomes large enough,
would it form a black hole? The answer obviously has to be no, otherwise
you would have given meaning to absolute motion, which is contrary to the
principle of relativity. In terms of T, the coordinates may be large in
some particular coordinate system, but that is fundamentally a projection
effect and does not indicate that the absolute magnitude of T is large.
Matters are different if you have a relativistic dust or fluid, which
has many particles all moving in different directions and all at very
high relative velocities. Now any given coordinate system is going to
measure high values of some components of T, and its absolute magnitude
truly is large.
The simplest case in between would be two masses heading towards
each other at relativistic speed, such that in the c.o.m. frame their
relativistic masses would sum to be large enough to form a black hole.
I think in this case it does happen. What happens is that there is
an analog of magnetism in the gravitational field, a component which
depends on the velocity of the attracted particle. You could have
two masses moving along near each other with the same large velocity,
and almost no gravitational interaction between them even though their
relativistic masses are large. This can be explained by saying that there
is actually a large gravitational field, but because the test particle is
moving rapidly there is a "magnetic" component which causes a repulsion
between the two particles and that balances the gravitational attraction.
Now when you send in a test particle going in the opposite direction,
the magnetic effect is reversed, and it sees a very strong gravitational
field. Each particle going in the opposite direction sees the other as
having a large gravitational field, and when its own mass is added the
geometry is such that the black hole forms.
In the case of your photons, probably something similar happens.
A simpler case would be two intersecting EM beams, such that the
mass-energy in the intersecting region (E^2 + B^2 times a constant) is
large enough to form a black hole. I don't think single beams can form
black holes because the BH would have to be going at the speed of light,
and I don't think this can happen. However with two crossing photon beams
some components of the momentum cancel, and you are left with a black hole
moving at less than the speed of light, of mass equal to the sum of the
two mass-energies of the beams in the intersecting regions. I guess if
your lasers were outputting continuous energy then you'd actually create
a train of black holes heading outwards, but that is pretty speculative.
> If so, wouldn't this be a way of generating small black holes
> that science fiction writers and some physicists use for
> various purposes?
I think the energy densities are too high to be practical.
> And finally, if laser light to black hole conversion is feasible,
> does anyone have any idea how much we would have to juice the
> laser at Lawrence Livermore to create black holes (of say
> neutron mass or larger).
I don't think it works like this; the problem is that the necessary size
to pack the photons into is too small. A neutron of mass 1 amu or 1E-27
kg corresponds to an energy of 1E-10 J or 1E9 eV, which unfortunately
implies a black hole size of 1E-54 meters! So you need to pack your
photons into an area that small in order to form a neutron-sized black
hole, which is really impossible.
Hal
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