I With gravitational lensing, which way is down?

[.Scott]

TL;DR Summary

I would think that gravity has no choice than to propagate through the same space that light uses.

Let me construct a scenario.

My space craft is due south of a black hole - or other gravitational lens. But I have arranged my thrusters to precisely counter the effects of its gravity. So, I am hovering light years away.

Now I add a third item to my universe. A powerful star almost directly behind the lens. Because of the gravitational lensing, that star appears further to the side of the lens than is "really" is.

Because I have added a new object to my universe, this new gravitational source will cause me to drift away from my original hover location.

Question: In which direction will I fall? Directly towards the star - as if the lens was not there? Or directly towards the image of the star? Or something else?

My answer is that I fall directly towards the image. After all, space is really bent and gravity cannot propagate through anything other than space.

If that is right, I have some follow-on questions.

 

[Dale]

almost directly behind

I don't know what to do with the "almost" here. If it is almost directly behind then why would you expect that directly towards the star is even a possibility? Is the "almost" a mistake or is that meaningful to the question?

 

[PeterDonis]

Now I add a third item to my universe.

You can't. Objects can't just magically appear out of nowhere.

Remember that GR is a theory of spacetime. Whatever objects are going to be there for all time need to already appear in your model. You can't start with a model that doesn't contain some object and then just add it.

You need to rethink your entire scenario in the light of the above.

 

[.Scott]

@Dale regarding "almost":

If it was directly behind the lens, I would get an Einstein ring. But for this scenario, I don't want an Einstein ring. I just want the apparent direction of the star displaced - as is expected of gravitational lensing. So, "almost" means not so perfectly aligned that it forms a ring, but enough to show a noticeable deflection.

 

[.Scott]

You can't. Objects can't just magically appear out of nowhere.

You need to rethink your entire scenario in the light of the above.

OK. It was there all along - but I am only applying enough thrust to balance out the direct gravitational effects of the lens (ie, the galaxy, black hole, whatever).

 

[Dale]

@Dale regarding "almost":

If it was directly behind the lens, I would get an Einstein ring. But for this scenario, I don't want an Einstein ring. I just want the apparent direction of the star displaced - as is expected of gravitational lensing. So, "almost" means not so perfectly aligned that it forms a ring, but enough to show a noticeable deflection.

OK, but then we can immediately discard any possibility that it falls directly towards the star.

 

[.Scott]

OK, but then we can immediately discard any possibility that it falls directly towards the star.

If "L" is the lens, "S" is the star, and "C" is my spacecraft, then without "S", C hovers stationary (by thrusters) at some distance from "L". But because of "S" it moves - it is pulled in some direction.
I proposed two answers: 1) in the same direction it would had "L" and the thrusters not been included in the scenario (which could be described as "directly"). Or 2) in the direction where "S" visibly appears to be.

Are you saying that we can immediately discard option 1?

In other words, the thrusters are there to null out the direct gravitational effects of the lens. I am only looking for the gravitational contribution of the star.

 

[PeterDonis]

I am only applying enough thrust to balance out the direct gravitational effects of the lens (ie, the galaxy, black hole, whatever).

You can't, at least not the way you appear to be thinking.

You can calculate what magnitude and direction of thrust would be sufficient to hover above the lens object at a given altitude, if the lens object were the only gravitating object present, and apply that thrust. But that doesn't mean your ship now behaves as if the lens object wasn't there. The lens object is still affecting the spacetime geometry, and therefore it's still affecting the worldline of your ship.

So my short answer to your OP question is "something else".

 

[.Scott]

You can't, at least not the way you appear to be thinking.

You can calculate what magnitude and direction of thrust would be sufficient to hover above the lens object at a given altitude, if the lens object were the only gravitating object present, and apply that thrust. But that doesn't mean your ship now behaves as if the lens object wasn't there. The lens object is still affecting the spacetime geometry, and therefore it's still affecting the worldline of your ship.

So my short answer to your OP question is "something else".

OK, glad to hear option 1 is out.
I didn't expect my ship to act exactly the same. For one thing, as soon as "C" is displaced by the gravitational contribution of the star ("S"), the pull from the lens on "C" will begin changing.

But you are saying that the gravitational contribution of "S" on "C" will not be precisely in the apparent (ie visible) direction of "C" - even in the moment when "C" is in a stationary hover ??

If not, how far off will it be? Will it be close?

 

[Dale]

If "L" is the lens, "S" is the star, and "C" is my spacecraft, then without "S", C hovers stationary (by thrusters) at some distance from "L". But because of "S" it moves - it is pulled in some direction.
I proposed two answers: 1) in the same direction it would had "L" and the thrusters not been included in the scenario (which could be described as "directly"). Or 2) in the direction where "S" visibly appears to be.

Are you saying that we can immediately discard option 1?

In other words, the thrusters are there to null out the direct gravitational effects of the lens. I am only looking for the gravitational contribution of the star.

OK, I misunderstood. The "almost directly" refers to the path from C to L, and the "directly" refers to the path from C to S. I thought both referred to the path from C to L.

I don't know the answer to your question directly for gravity, but I do know the answer for electromagnetism. The equations are the same for weak gravity, so they may be applicable, but maybe not.

For electromagnetism if you have a charged L, S, and C, and in addition L and S are emitting light, the force from S acting on C may come from a different direction than the light from S to C. The light will come from the retarded position, but the force will come from the retarded position plus the retarded velocity times the difference between the current and the retarded time. In your specific scenario that is the same, but not in all scenarios.

 

[.Scott]

I am not surprised that there is an "option 3" which is the accepted answer.

If option 3 is anything similar to option 2, then you would get interesting effects as you approached the photon sphere of a Schwarzschild black hole. The gravitational effect of everything outside the photon sphere would be pulling you up or to the side - even if it was on the opposite side of the black hole.

Then, after dropping below the photon sphere, it might be valid to model the mass of the black hole as existing only on its event horizon. If you do that, and follow the "light line" paths of that mass to your space craft, I believe you get something very different than an ever-increasing gravitational field.

In other words, I was hoping to make things as simple for me to understand as possible - and to try to avoid the worse of the messy black-hole arithmetic.

 

[PeterDonis]

the gravitational contribution of "S" on "C"

Strictly speaking, there isn't one; GR is nonlinear. You can't separate out the contribution of one particular source.

Note also that the system is not stationary; "S" and "L" will be moving relative to each other. And for "S" to make a difference to the worldline of "C", it has to be pretty close to "L"--if it's far away, its contribution is negligible. And "C", by your scenario, is pretty far away from both "L" and "S"; light years, you said. So from "C"'s viewpoint, it's going to be very difficult to even tell, gravitationally, that there isn't just one source of gravity, "L" and "S" combined.

As for any actual calculation of this scenario, it would have to be numerical. I'm not aware of any such calculation in the literature, but perhaps there is one. In the absence of such a calculation, I don't think we can say much more about the answer than has already been said.

 

[.Scott]

Strictly speaking, there isn't one; GR is nonlinear. You can't separate out the contribution of one particular source.

It had to be non-linear. For option 2, I was trying to work with first order estimates.

I haven't been able to grasp how gravity "propagates" or "communicates" as the photon sphere of a Schwarzschild black hole is approached and passed.

What about the gravitational pull of "everything else in the universe"? Certainly the combination of that and the black hole gravity combine non-linearly. How could the net of all the "everything else" be anything other than to pull up?

 

[.Scott]

As for any actual calculation of this scenario, it would have to be numerical. I'm not aware of any such calculation in the literature, but perhaps there is one. In the absence of such a calculation, I don't think we can say much more about the answer than has already been said.

That's the exact storyline I was trying to avoid.

 

[Dale]

I haven't been able to grasp how gravity "propagates" or "communicates"

For that sort of thing I think that the electromagnetic analogy in the weak field limit is instructive. In EM the direction of the force does not always coincide with the direction of the light. That must also be true in the weak field limit of gravity.

Although I cannot answer for black holes, I would very surprised to learn that strong gravitational fields act to make the direction of the “force” always coincide with the direction of the light when that is not true of weak fields.

 

[PeterDonis]

For option 2, I was trying to work with first order estimates.

That only works in the regime where the higher order terms are negligible. That can be true in the local region around your "C", but it can't be true for the region containing your sources, "L" and "S".

I haven't been able to grasp how gravity "propagates" or "communicates" as the photon sphere of a Schwarzschild black hole is approached and passed.

This is a different question from the one we've been discussing, and should be discussed in a new thread.

How could the net of all the "everything else" be anything other than to pull up?

If the everything else is distributed around the region you're interested in in a spherically symmetric manner (which is not a bad approximation in many cases), then its net effect on the spacetime geometry inside the region you're interested in is zero. This is the GR version of the shell theorem (or at least one part of it).

If the everything else is not distributed spherically symmetrically around the region you're interested in, then there can be some "pull" from it--but it's not at all clear that the obvious direction for that "pull" will be "up". (Indeed, a Newtonian "pull" might not even be the best way to picture the effects.)

 

[PeterDonis]

That's the exact storyline I was trying to avoid.

Unfortunately I don't think you can. There is no exact solution for your scenario, so we have no analytical equations we can use.

 

[.Scott]

If the everything else is not distributed spherically symmetrically around the region you're interested in, then there can be some "pull" from it--but it's not at all clear that the obvious direction for that "pull" will be "up". (Indeed, a Newtonian "pull" might not even be the best way to picture the effects.)

To avoid ambiguous wording, confusion, let me set up a scenario.

Say we are being lowered to the north pole of a black hole and we have reached the photon sphere. And say that there is a significant mass (SM) on the opposite side (ie, south pole) of that black hole. A direct path from that SM to us would go through the "interior" of the black hole - which is not a real path. If the gravity reaches us through the same mechanism as light - ie, through curved space - then we would see light from the SM coming to us mostly from the sides. If gravity followed the same path as the light, it might just as well pull us up (away from the event horizon) as down. I'm not saying that this would happen, but if you had subscribed to option 2, I would have been asking for clarification. In contrast anything mass behind us would tend to simply pull us up.

Attempting to follow option 2 once you've passed into the photon sphere is even more interesting - because now you need to communicate the black hole mass from within the event horizon through even curvier space to us. If you parse up the event horizon into pieces, and try to determine which direction each piece is pulling on us through curved space (integration), it is very hard for me to see how the gravitation forces could combine in any way similar to the way it does with a Euclidean model.

 

[PeterDonis]

If the gravity reaches us through the same mechanism as light

Light is radiation. "Gravity" in the sense you are using it here is not radiation. You can't treat the two as though they're the same kind of thing.

ie, through curved space

GR is a theory of spacetime, not space. Thinking of gravity as "propagating through curved space" is not going to work well.

Unfortunately, I think you are trying to work with a conceptual scheme that is not suited to the scenarios you are posing.

I'll save further comments on your "slowly lowering to the photon sphere" scenario for a separate post.

 

[PeterDonis]

Say we are being lowered to the north pole of a black hole and we have reached the photon sphere. And say that there is a significant mass (SM) on the opposite side (ie, south pole) of that black hole.

First, as I said before, this scenario will not be stationary. If the SM and the hole are at rest relative to each other at some instant of time, they will fall towards each other and merge. If the SM and the hole have some sideways motion relative to each other, they'll either be in mutual orbits about each other or they'll be in some kind of flyby scenario where each escapes to infinity relative to the other. None of these scenarios can be treated the way you appear to be thinking, as if the SM is at rest on the opposite side of the hole from the object being slowly lowered.

Second, as far as any slow lowering process is concerned, the photon sphere makes no difference whatever. All that happens during the slow lowering process is that the force that has to be exerted on the object being lowered gradually increases. There is no discontinuity or change in behavior at the photon sphere. The force (at the object) increases without bound as the hole's horizon is approached (if we only consider the hole and not the SM).

 

[Dale]

If the gravity reaches us through the same mechanism as light - ie, through curved space - then we would see light from the SM coming to us mostly from the sides.

I don’t think this reasoning is correct. In weak fields the force and the light can come from different directions even though both come through the same spacetime.

 

[.Scott]

I don’t think this reasoning is correct. In weak fields the force and the light can come from different directions even though both come through the same spacetime.

I am not surprised.
If the force can come from a different direction than its effect, it suggests that the field has orientation(s). But that orientation still has to survive spans of curved space. I wonder how well it does.

 

[Dale]

If the force can come from a different direction than its effect

What does that mean?

 

[.Scott]

In weak fields the force and the light can come from different directions even though both come through the same spacetime.

Responding to "What does that mean?"
I was interpreting this as meaning that the field was coming from the same direction as the light - but that its force was directed along that orientation.

 

[Dale]

I don’t know how you can define the direction that the force comes from independent of the direction of the force.