Re: Conservation of momentum (and a weird space drive)

From: Tom Roberts (tjroberts_at_lucent.com)
Date: 10/07/04 

Date: Thu, 07 Oct 2004 01:31:21 GMT

Joe wrote:
> I have a question about general relativity. [...]

Let me approach your scenario from below, via simpler gedankens.

First start in the context of Newtonian mechanics. Imagine an isolated
planet with mass M having a tall tower. A small mass m is initially at
the base of the tower. It is carried up the tower and then dropped back
to the base (bring the carrying mechanism back to its original state).
Initially both M and m were at rest in some inertial frame, and
afterward they are again at rest at the same position in that same
inertial frame.

Now let's consider this in GR. If we assume that M is "small" (no larger
than the sun, say), then for this gedanken GR is indistinguishable from
Newtonian mechanics, and the same conclusion holds.

Now instead of a small mass m, let's consider a flashlight which starts
charged up at the base of the tower, is carried to the top, shines down
onto the surface of the planet (and discharges), and is then carried
back down (after discharging). Let me assume the light beam is
completely absorbed by the surface, and is turned into heat. As we are
still in the weak-field approximation, gravitational radiation can be
neglected, as can other things like the variations in stress of the
tower. The above conclusion is then unchanged -- the position and
inertial frame of the planet are the same beforehand as afterward.

It would be interesting to be able to abandon the weak-field limit here,
and discuss the full predictions of GR. But AFAIK nobody knows how to do
that. I can guess that for strong fields (large m and larger M) that the
emission of gravitational radiation during the carrying and falling will
be important -- but I don't know for sure if the momentum from radiation
during carrying will cancel the momentum from radiation during falling
(both are outgoing so the system's energy cannot be conserved, which
loosely implies that its momentum may not be).

        Note to near-experts: it is tempting to invoke the translation-
        invariance of the initial system and conclude momentum is
        conserved and therefore the system cannot move. But this
        manifold is not translation-invariant (i.e. there is no
        spacelike Killing vector), due to the gravitational radiation.
        So there is no momentum conservation; and I'm pretty sure that
        applies to the obvious pseudo-energy tensor.

In your scenario you forgot to include the change of internal energy in
the light source, and the implications of that. In my discussion above,
if one does not carry the flashlight up/down and only considers the time
during which it discharges, then there is clearly energy transfer from
the top of the tower to the bottom. That cannot be neglected, and tends
to move the planet+tower+source in the opposite direction of the effect
you do mention (blueshift of the light).

Bottom line: if one can neglect gravitational radiation, then this
gedanken cannot be used to propel the planet through space. If
gravitational radiation cannot be neglected, then I don't think anybody
knows the answer. But as a practical means of propulsion this is a
non-starter....

> It is based off of the
> Feynman Lectures. Suppose you have a tower on a planet. And at the
> top of the planet you drop a photon to the Earth. As it leaves, the
> photon kicks the tower in one direction and then speeds to the
> surface. At the surface, the photon strikes with an increse in
> energy, a higher frequency, and thus a higher momentum. It kicks the
> tower/Earth in the opposite direction than before. But this kick is
> stronger than the one at the tower when it left. So there is a net
> momentum transfer in the opposite direction. But this doesn't make
> sense. you could make the Earth more compact, so that the difference
> in kick gets more and more. Until finally you could build a space
> ship where all you do is shine a bit of light in one direction, and go
> careening off into space somewhere. But this net "motion exnihilo" is
> weird. Clearly I'm missing something.

See above. You are missing the change in internal energy of the light
source. And other small effects on the same order as the effect you
mention (e.g. variations of stress in the tower).

        In GR, unlike Newtonian Gravitation, all types of energy
        contribute to gravitation, including light and stress.

> You have the same thing in
> Newtonian mechanics. Where on the tower you drop a ball and it falls
> to the earth, striking with greater momentum than when it left. But
> when you look at this from a non-inertial reference frame, you see
> what's happening. The momentum picked up by the ball is exactly
> counterballanced by the momentum picked up by the Earth/tower.

Actually you want to look at this in the center-of-mass inertial frame
of the planet and ball. Looking at it in the non-inertial frame of the
planet or ball is quite confusing....

> This
> is because the force acting on the ball is exactly equal in magnitude
> to the force of the ball on the Earth, but opposite in direction, and
> the impulse of force over time is the same, and they both cancel out.
> But with a photon, you don't have that luxury. For a photon to
> "accelerate" the Earth upwards, there would have to be some "advanced
> warning" that the photon was coming, travelling faster than light.

You are omitting the internal stresses in the planet and tower, the
speed of sound in each, etc. The "photon" need not "accelerate the Earth
upwards", as these stresses and sound waves will do that. They, of
course, are generated by the depletion of energy in the light source.

> But looking at it from a non-inertial frame, momentum is conserved
> here (isn't it?).

In Newtonian mechanics, momentum is conserved, but only when referenced
to an inertial frame. But that's OK as you really meant the inertial
frame of the center-of-mass.

In GR, conservation of momentum is problematical. In the weak-field
limit there's no serious problem here. But for strong fields I don't
think anyone knows the answer (due to the emitted gravitational radiation).

Tom Roberts tjroberts@lucent.com

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