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

From: Joe (jhelfand_at_umd.edu)
Date: 10/08/04 

Date: 7 Oct 2004 18:35:08 -0700

Crap!@ I meant INERTIAL fram not NON-INERTIAL. I should have said
NON-NON-INTERTIAL! I have posted this question multiple times (and
elsewhere too!), all with the same mistake!

Okay anyway thanks for posting. Of all the crazy and obscure posts
I've made, this is the most important to me.

> 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.

Okay. But this is because the force on the falling ball is the same
and opposite to the force on the planet. If it weren't, this wouldn't
happen. And also are M and m in the same position? This is like two
people throwing a ball in a canoe. The canoe moves! But then it
stops. The center of mass is in the same position. But the canoe
isn't!

> 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.

I follow so far.

> 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.

Already here again I'm confused. This is like the solar sail
question. You have a solar sail, and you shine a light on it, if all
the energy of the light gets converted into heat, how does it move?!
The energy in a coherent bulk motion in one direction isn't there,
it's converted to random thermal motion of the sale. You don't need
gravity to make a space drive then. You just need to be in a box in
outerspace, where one wall is such that light incident on it is
directly and ONLY converted into heat. Then you can just stand on one
side, shine light in one direction, it kicks you and the box in that
direction, and on the other it gets converted into heat. Is this idea
right? It can't be! So there must be some maximum fraction of the
energy that can be turned into heat. At the same time, the energy
can't completely be turned into bulk motion, because that violates
conservation of momentum. This can be determined if it is completely
absorbed. (How the hell does this work out for particle physics?!
Never mind.) I'm rambling, because of course you didn't say
completely turned into heat. But it led me to think about this, which
is irrelevant...

>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's good that we can neglect this GR radiation, because I don't know
what the hell it is! But this is where I'm lost, and you have to go
slower. Why is it the same? In fact, why isn't there a net velocity
of transfer in one direction! This is where my greatest confusion is.

> 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).

I don't follow this. But if you say nobody can, I'm not afraid!

> 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.

I'll make sure I won't do that ;)!

> 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).

Now this is interesting! Do you mean that if just considering a
tower+planet+source all stationary, and you're at the top, and you
shine the light towards the planet, there is a net motion in the
direction of the light? A net CONTINUED motion? In otherwords the
tower+planet+source are now travelling in one direction?! But this is
exactly what I'm asking! But then carrying it up to the top must
produce a net continued motion in the opposite direction to make your
statement consistent. (In that the return trip doesn't produce a net
motion). Besides, doesn't this net motion produce a new motion of the
center of mass?!

> 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....

But didn't you just say that if you assume your at the top, and shine
light down, youo can? I must be reading you wrong!

> 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.

How do I take into account the internal energy of the light source.
And other small effects of variations of stress in the tower? Do
these contribute a net pull in the opposite direction of the incresed
kick from the falling photon? Boy, this other question about "all
types of energy contribute to gravitation" raises even more confusion
that I have that I won't mention here! [Yet.]

> 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.

Interesting. Do these neglections contribute the missing extra kick
of the photon on the planet?

> 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).

So momentum conservation might not hold for strong fields? Do other
theorists believe this?

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