When a mass enters or leaves a wormhole mouth is momentum transferred

[harrylentil]

Was that title possibly truncated?

*When a mass enters or leaves a wormhole mouth is momentum transferred?*

Also, does mass/energy passing through an event horizon generally or a black hole's event horizon transfer momentum?
That's three questions, but the title question is what interests me here.

 

[PeterDonis]

Was that title possibly truncated?

Apparently it was, I've fixed it.

 

[PeterDonis]

The general answer to your questions is to ask, "momentum transfer to what?"

In the case of a black hole (at least a classical one--quantum effects add complexities), the hole is vacuum--it is just spacetime curvature, there is no matter or energy present. (There was long ago when some massive object collapsed to form the hole, but there isn't long after that.) So there's nothing for momentum to be transferred to.

In the case of a wormhole, there is "matter" present--but it's very unusual matter, called "exotic matter", because it has properties that no ordinary matter has, and that many physicists believe are not physically possible. But it could be that momentum could be transferred to and from such matter; I don't know that anyone has studied the question.

 

[harrylentil]

If the answer is 'yes' then the wormhole mouth would react by moving, similar to the reaction of a solid body to an impact. It would either move in the direction of the impinging mass or in the opposite direction. If the answer is 'no' the wormhole mouth would not move because of momentum transferred to it from the mass, though it might move for another reason.

 

[PeterDonis]

If the answer is 'yes' then the wormhole mouth would react by moving

Only if the object going through the wormhole actually "hit" the exotic matter that is holding the wormhole open. Physics discussions of wormholes generally assume that this doesn't happen when objects traverse the wormhole, just as ordinary objects going through, say, a tunnel don't normally hit the sides of the tunnel. As I said, I don't know that anyone has ever studied the question of what would happen if an object did hit the "side" of the wormhole, assuming that that's possible given the fact that the "sides" are made of exotic matter.

 

[harrylentil]

If a massive object falls from rest with respect to a star both these objects accelerate toward one another, and the net change in momentum is zero. This is also true if the 'star' is a black hole. If the massive object is fired at high speed into the star it will transfer momentum to it and the star will recoil. I think this is true of a black hole also. If so, the recoil happens as the massive object asymptotically approaches empty space - the event horizon. Granted, the situation is different with a wormhole mouth: it's not an event horizon, but the precedent is there: momentum can be transferred to empty space. Now, if a massive object moving at speed traverses a wormhole mouth, and before it can hit the wormhole wall, will an observer near the wormhole mouth see it recoil?

 

[pervect]

Was that title possibly truncated?

*When a mass enters or leaves a wormhole mouth is momentum transferred?*

Also, does mass/energy passing through an event horizon generally or a black hole's event horizon transfer momentum?
That's three questions, but the title question is what interests me here.

If you assume that the wormhole mouth is in an asymptotically flat space-time (this also works for the black holes with the same assumption) there is an effective energy and momentum of the wormhole mouth (or the black hole) "at infinity" that can be found via an appropriate integral of the metric and/or it's derivatives. Specifically, one is interested in the 1/r terms of the metric.

There's an argument that says that the total energy and momentum measured this way can't change, so the effect is that a wormhole mouth (or a black hole) has an effective "mass", which can be regarded as the result of an effective energy and momentum that transforms as a 4-vector in the asymptotically flat space-time.

To be a bit more specific, we can find the ADM energy and momentum in an asymptotically flat space-time for a wormhole mouth, or black hole, by considering the metric "at infinity". Or, if we're willing to keep tract of gravitational radiation, we might instead compute the Bondi mass, energy, and momentum instead of the ADM quantities. In the Bondi case, though, we have to consider the energy and momentum carried away from the wormhole / black hole by gravitational radiation, something that we don't need to worry about in the ADM case.

According to this picture the total energy and momentum don't change, so when an object passes through a wormhole mouth (or into a BH event horizon), the energy and momentum are transferred to the wormhole (or black hole) mouth from the infalling object.

The later case is discussed in Visser's "Lorentzian wormholes". There's been an occasional thread on this on PF in the past.

 

[PeterDonis]

If a massive object falls from rest with respect to a star both these objects accelerate toward one another, and the net change in momentum is zero.

Yes, the net change in momentum is zero. Which means that, viewed from infinity (i.e., from very far away), the system in question is a single system with an unchanging momentum. The fact that the system starts out, viewed internally, as an object and a star, that then fall together and become a slightly larger star, is irrelevant to the total momentum (and energy) as viewed from infinity.

The case of a black hole is somewhat different, because a black hole is not an ordinary object like a star. As I said before, the hole is made purely of spacetime curvature, and spacetime curvature doesn't "recoil" the way an ordinary object does. You can still assign a momentum to the hole as viewed from infinity, and still look at things from infinity as you could in the case of the object falling into a star, and compute a total momentum for the system that doesn't change. But viewed internally, this will not look like the hole "recoiling" from the impact of the object, the way a star would. It will look like a configuration of spacetime curvature that has a sort of "bend" in it (it's hard to express this in ordinary language).

According to this picture the total energy and momentum don't change, so when an object passes through a wormhole mouth (or into a BH event horizon), the energy and momentum are transferred to the wormhole (or black hole) mouth from the infalling object.

But in the case of the wormhole, the energy and momentum come out the other end. So, providing nothing hits the "walls" of the wormhole while passing through, it seems to me that the wormhole's energy and momentum (defined as seen from infinity using the asymptotically flat assumption you describe) would be unchanged, at least between the start and end of the process. Possibly during the process, i.e., while the object was inside the wormhole, the wormhole's energy and momentum might appear larger. I haven't tried to calculate it. I'll look up the Visser reference you give when I get a chance.

 

[harrylentil]

Only if the object going through the wormhole actually "hit" the exotic matter that is holding the wormhole open. Physics discussions of wormholes generally assume that this doesn't happen when objects traverse the wormhole, just as ordinary objects going through, say, a tunnel don't normally hit the sides of the tunnel.

But in the case of the wormhole, the energy and momentum come out the other end. So, providing nothing hits the "walls" of the wormhole while passing through, it seems to me that the wormhole's energy and momentum (defined as seen from infinity using the asymptotically flat assumption you describe) would be unchanged, at least between the start and end of the process.

This makes a strange scenario possible. A rocket ship is in solar orbit, blasting out high velocity, high mass exhaust, but it is not accelerating. In fact it is behaving as though there was no exhaust. Investigating astronauts push it around and and measure its mass. They also measure the rocket exhaust thrust and conclude the rocket ship should be accelerating at one G. It appears to be an impossible object! It is ejecting mass but its mass does not change. It is violating fundamental principles! Then it starts an acceleration which increases to one G, then decreases back to zero, but the rocket exhaust thrust does not change, nor does the mass of the rocketship, nor is there a significant dynamic influence from the space near it. More impossible behavior! The explanation is this. When the rocketship is not accelerating the exhaust is flowing through a wormhole, out a wormhole mouth inside the rocket hull, and angled directly out the 'nozzle'. During acceleration the wormhole mouth is deflected and repositioned to reflect the stream off a plate in the direction of the nozzle, when it becomes a propellant. When acceleration reduces to zero the wormhole mouth and plate are moving until the stream is again flowing without touching the rocket ship, when it stops being a propellant. None of these internal manouvers are observable from outside the rocket ship (the movement of weights can be disguised by counterweights).

Is this possible in principle, or would the investigating astronauts be right to think fundamental principles have been violated?

 

[PeterDonis]

Is this possible in principle

I don't think so. You are leaving out a crucial piece of the scenario: what is producing the rocket exhaust? Whatever it is will be some kind of engine, which will be made of material parts, and the rocket exhaust will push against those material parts and thereby push the engine. That is inherent in the operation of the engine. If the engine is attached to the rocket, the push gets transferred to the rocket, regardless of what happens to the exhaust after it exits the engine, wormhole or no wormhole. If the engine is not attached to the rocket, then the engine will move when the rocket doesn't and will separate from it (or punch through it and damage it).

 

[PAllen]

People do speak of momentum of back holes. See link to recent paper. However for wormholes, I am not sure it makes sense for the reasons Peter describes. One addition is that exotic matter is only needed for a traversible wormhole. I believe the collision of a body with a non traversible wormhole is rather mundane - the wormhole collapses, and the matter has the same total momentum it had before. So the only interesting case is the one Peter described. In this case, the wormhole is really irrelevant - the issue is collision of matter and exotic matter. I have no idea what has been analyzed about such a hypothetical scenario.

https://arxiv.org/abs/1503.00728

 

[pervect]

Yes, the net change in momentum is zero. Which means that, viewed from infinity (i.e., from very far away), the system in question is a single system with an unchanging momentum. The fact that the system starts out, viewed internally, as an object and a star, that then fall together and become a slightly larger star, is irrelevant to the total momentum (and energy) as viewed from infinity.

The case of a black hole is somewhat different, because a black hole is not an ordinary object like a star. As I said before, the hole is made purely of spacetime curvature, and spacetime curvature doesn't "recoil" the way an ordinary object does. You can still assign a momentum to the hole as viewed from infinity, and still look at things from infinity as you could in the case of the object falling into a star, and compute a total momentum for the system that doesn't change. But viewed internally, this will not look like the hole "recoiling" from the impact of the object, the way a star would. It will look like a configuration of spacetime curvature that has a sort of "bend" in it (it's hard to express this in ordinary language).
But in the case of the wormhole, the energy and momentum come out the other end. So, providing nothing hits the "walls" of the wormhole while passing through, it seems to me that the wormhole's energy and momentum (defined as seen from infinity using the asymptotically flat assumption you describe) would be unchanged, at least between the start and end of the process. Possibly during the process, i.e., while the object was inside the wormhole, the wormhole's energy and momentum might appear larger. I haven't tried to calculate it. I'll look up the Visser reference you give when I get a chance.

I'll quote a bit from the Visser reference. There may be a few typos.

Now consider the case of a traversable wormhole that connects two otherwise separate universes. There are now two asymptotically flat rigions, and thus two ADM masses (which can be unequal, in generl, see p. 103). The convervation law normally deduced for the ADM mass carries through mutatis mutandis to a pair of conservation laws. Each ADM mass in each asympotically flat universe is conserved separate.y.

Consider the effect of mass $m_i$ that is initally far away from the wormhome mouth in the + universe. Suppose now that this object traverses the wormhole and eventually settles down far from the wormhole in the - universe. THen toe total ADM masses of both sides of the wormhole are:

$M^+_{total} = M_i^+ + m_i = M_f^+ \quad M^-_{total} = M^-_i = M_f^- + m_f$

Notation: Here $M^\pm_i$ denote the intial masses and $M^\pm_f$ denote the final masses of the two wormhole mouths in the + and - universes, respecitvely.

As far as black holes having momentum, the ADM momentum is defined in Wald by a formula in $11.2.

 

[harrylentil]

I don't think so. You are leaving out a crucial piece of the scenario: what is producing the rocket exhaust? Whatever it is will be some kind of engine, which will be made of material parts, and the rocket exhaust will push against those material parts and thereby push the engine. That is inherent in the operation of the engine. If the engine is attached to the rocket, the push gets transferred to the rocket, regardless of what happens to the exhaust after it exits the engine, wormhole or no wormhole. If the engine is not attached to the rocket, then the engine will move when the rocket doesn't and will separate from it (or punch through it and damage it).

Good point - something is being pushed while the rocket ship is stationary. Elsewhere in the universe a conventional rocket exhaust/mass driver is operating and that is being pushed - the engine on the other side of the wormhole. The exhaust then traverses the wormhole and exits the nozzle of the rocket ship in the solar system. That rocket ship does not have to be pushed by the exhaust unless it arranges itself to deflect the exhaust.

 

[harrylentil]

Thinking about the rocket ship example, apparently violating conservation of momentum and mass, maybe there are other conservation laws that could be apparently violated by the hidden presence of a wormhole.

 

[PeterDonis]

That rocket ship does not have to be pushed by the exhaust unless it arranges itself to deflect the exhaust.

And if it doesn't arrange to deflect the exhaust, then the exhaust itself will have detectable momentum in the solar system, because it never got transferred to the rocket.

 

[Sue Rich]

Has there ever been any verifiable proof of worm holes?

 

[PeterDonis]

Has there ever been any verifiable proof of worm holes?

No. In fact, many physicists do not believe they are possible, because they require exotic matter and many physicists believe exotic matter is not physically possible.

 

[Sue Rich]

Thank you.

 

[PeterDonis]

Thank you.

You're welcome!