> I had to guess what you might mean by "propagate at -c", and I guessed you
> had in mind something consistent with this procedure: take some radiative
> solution, transform it via t -> -t, and compare with the original.
That is not what I had in mind
> you saying that you think that in Maxwell's theory of EM, reversing time
> in a null field results in a solution which carries -negative- energy?
Defintely not. The energy of light is positive whichever way you look
at it. We do not need Maxwell's theory to confirm this. We just need
the law of conservation of energy.
> That you think that in Einstein's theory of gtr, reversing time in a type
> N vacuum solution results in a solution which carries -negative- energy?
Defintely not. It is not necessary to reverse time to produce negative
energy gravitational fields. The Newtonian gravitational field of the
Sun, for example, has negative energy at all points in space except
(theoretically) at infinity.
> Gee. Well, I guess I must not understand what you mean by "propagating at
> a speed of -c".
We have been discussing, in effect, the theoretical possibility that,
when a gravitational field is modified, the effects at progressively
greater distances, occur at progressively earlier times. Although this
does appear bizarre, it is a theoretical possibility for gravity. It
might also be a theoretical possibility for light, but empirical
observation rules that out. We have no comparable empirical observation
for gravity, as far as I can tell. If we are to apply the rigors of the
scientific method to gtr, then I think we should, at least, examine
whether there is any evidence to confirm the validity of one
theoretical possibility over the other.
>And how are you defining/computing "energy"?
In the present scenario, simply by application of the law of
conservation of energy to what we can observe astronomically. If we
observe that binary pulsars lose energy, apparently through
gravitational radiation, then there are only two possible explanations
that are consistent with the law of conservation of energy. A negative
speed negative energy wave, or a positive speed positive energy wave.
This is an either or scenario, on energy conservation grounds.
> For example, consider a linearly polarized plane wave, in either theory.
> What would it mean for this to "propagate at a speed of -c"?
It would mean that its point of detection is earlier in time than its
point of emission, in a specified reference frame.
> To have
> "negative energy"?
I refer you to Einstein's and Newton's usage of gravitational potential
> Can you write out a computation?
Yes, ridiculously simply.
Let body A have energy E until time 2, at which time it loses energy
del E by gravitational radiation.
Let body B have energy E until time 1, at which time it gains kinetic
energy del E by that gravitational excitation.
The total energy of the system before time 1 is 2E (as is the total
energy of the system aftter time 2).
The total energy of the bodies is 2E + del E between time 1 and time 2.
Therefore the energy of the gravitational wave must be minus del E
between times 1 and 2, under the law of conservation of energy.
Can you now write out a mathematical proof that a massive beam in outer
space, which generates a negative energy gravitational field by virtue
of its mass, will start reversing this behaviour when spun end over
end, by generating positive energy gravitational waves? (I think this
is believed to be one of the simplest examples of a gravitational wave
generator, within the theory.)