# Black holes slowing as they pass each other?

1. Apr 8, 2012

### cephron

I understand that a gravity well bends space-time such that, close to it, time passes "slower" than further away. The archtypical example is that if a chronometer were dropped into a black hole, we would observe it to tick slower and slower as it approached the event horizon.

Based on this, I have a couple questions...

1. Suppose two black holes were passing close by each other. They will be changing velocity all the time due to each other's gravity, but let's abstract this away, and worry only about time dilation. Would an observer at infinity, stationary relative to the system's center of mass, see them as slowing down as they passed each other? My understanding is that a slowing would be observed.

2. Suppose two black holes are in concentric orbits around a heavier gravity source. Consider the inner one passing by the outer one, as bodies with smaller orbits around their primary are given to do, but let us ignore the perturbations in their orbits. Would their movement past each other be slowed as in the above case? My understanding is that it would.

Now for the part that I find weird...would the orbital speed of the inner black hole appear to decrease while they passed each other? Would the orbital speed of the outer black hole appear to increase while they passed each other? These seem strange, but they seem to follow from the above. If I'm making a conceptual error, please point it out to me--many thanks!

2. Apr 8, 2012

### Nabeshin

Well an observer won't see anything, black holes are black. Seriously though, if there is matter in orbit around one or both of them emitting radiation indeed it will do the normal time dilation thing. When the holes are far apart it will be the usual story for a single black hole, but when close together the nonlinearities of GR make it complicated to say anything too general, but there will be more time dilation.

With Re: to the rest of your question, if two black holes are sitting in a much larger potential well, they will be primarily subject to the time dilation effects caused by this large well. If they are close enough to significantly alter the geometry around the individual holes, the orbits will be severely perturbed. You can't really have it both ways. But in some small-field sense where you have a newtonian-like gravitational potential which adds linearly, then yes the orbiting bodies will experience some dilation.

3. Apr 8, 2012

### yuiop

Not sure if this helps, but if a stationary local observer were to measure the period of a satellite orbiting around a black hole as T, then a very distant observer would measure the orbital period as $T*\sqrt{1-R_s/R}$ so the distant observer sees the orbital speed as slower. Oddly enough the orbital period according to the distant observer is in exact agreement with the Newtonian Kepler prediction even in GR.

4. Apr 8, 2012

### Naty1

Change 'black hole' to 'star' or 'planet' or 'asteroid'....what would be different??

In all cases the objects accelerate......change direction.....

5. Apr 8, 2012

### cephron

The difference (as I understand it, at least) is that the gravity wells of stars and planets do not cause significant time dilation effects, while a black hole's does.

Maybe I can put this another way...

Suppose we drop an object in a highly elliptical orbit around a planet. We drop it from the farthest point in the orbit (I think it's called the "aphelion"), and measure the time it takes to complete one orbit. Now, if we were to attempt to calculate the time the orbit should take using Newton's law of universal gravitation, that should agree pretty closely to the measured time.

But suppose we did the same thing with a black hole, and the elliptical orbit brought the dropped object close to the event horizon (let's assume it could withstand the tidal stresses). Would the calculated orbit time of the be shorter than the actual time, due to the effects of time dilation?

6. Apr 8, 2012

### Passionflower

You are mistaken, whether something is (going to be) a black hole or not does not depend on its mass but on its density. Consequently one can have very massive black holes but also black holes with very little mass.

Last edited: Apr 8, 2012
7. Apr 8, 2012

### Naty1

the calculation should match the observation if done correctly....but of course there is some clock slowing, like near the surfaceof the earth compared with a distance of an orbiting satellite, for example....

that's an unfair inappropriate comparison....anyway, it depends on the mass of the black hole....the super massive ones at the center of our galaxy do not have especially strong gravitational fields outside their huge event horizon....very little tidal gravity....The earth's 'Schwarszchild radius' is only about 9 mm....2Gm/c2

Let's say you compressed a marble into a black hole...now that would have a very highly curved spactime...

edit: I see passionflower posted while I was composing....he correctly expresses another characteristic....

Think of a small hill (potential] but with very steep sides...a big gravitational 'curvature'....big gravitational gradient....very dense.....a LOT of tidal gravity.....now consider a mountain with a very gradual slope...much less steep curvature...huge mass, huge potential change, but less tidal gravity on its slope....and you get a crude picture....

A clock sitting in a dense planet or neutron star (gravitational time dilation and red shift) reflects the same effects as a clock hovering outside an event horizon, because they are the same phenomenon in GR....or

The acceleration you will need to hover just above the surface of a huge planet with enormous mass will be enormous, but tidal forces will be small (because radius is enormous). The acceleration is due to depth of gravitational well - a global feature; tidal forces due to gradient - a local feature.

Last edited: Apr 8, 2012
8. Apr 8, 2012

### pervect

Staff Emeritus
Imagine having one observe, observer, S, using rockets to hover above a black hole and/or planet at a constant height and constant angular coordinates, while another observer, observer O, orbits the black hole or planet.

There isn't as much difference between the two cases as you would think. On the Earth, we have to deal with the Earth's rotation, which does complicate things slightly. In our idealization, observer S will be at constant angular coordinates relative to the fixed stars for ease of exposition.

S and O are initially at the same location, and synchronize their clocks. After one orbit by observer O, the two observers will again be close to one another, and they compare clocks again. When they do, observer O's clock reads less time than observer S's clock, because of observer O's velocity.

The effects are easily detectable for satellite orbits around the Earth with modern clocks, though observer S's clock is generally calculated or inferred rather than measured.

9. Apr 8, 2012

### jartsa

Let's say two black holes move into almost same direction. Their distance decreases slowly. Does the speed they are moving ahead slow down? No it doesn't.

Let's say we drop a spinning top onto the pole of a spinning neutron star. Both objects spin the same way at the beginning. Does spinning of the top slow down? No it doesn't.

Let's say two identical black holes move into opposite directions at speeds 0.999999999999 c. Do their speeds slow down when they pass? No, that does not happen.

10. Apr 8, 2012

### Naty1

11. Apr 9, 2012

### yuiop

If we lower a very efficient flywheel quasi-statically into a deep gravitational well, I imagine that it WILL slow down from the point of view of a distant Schwarzschild observer.

12. Apr 9, 2012

### cephron

I believe I have refined my question.

For now, I only need to talk about one black hole and its effect on objects passing near it.

Consider the following diagram:

What we have is a black hole with two objects of negligible mass (clocks) orbiting it. Let's say both objects pass point A at the same time--and, at this instant, they have identical speed (obviously not identical velocity). The orbit of clock 1 takes it in a circular path, while the orbit of clock 2 takes it perilously close to the black hole--as close as it can go without being torn apart by tidal forces.

Now, we wait for the two clocks to make a complete orbit and end up back at point A.

Here are three "possible" outcomes:
1) The two clocks both make a complete orbit and arrive at point A at the same time. We read the clocks, and they show identical times. This is what we would expect if we were only considering Newtonian gravity and no effects of relativity.
2) The two clocks both make a complete orbit and arrive at point A at the same time. We read the clocks, and clock 2 is slightly late. This is of course due to the time dilation it experienced while near to the black hole...but the time dilation did not affect the time it took to orbit the black hole.
3) The two clocks both make a complete orbit, but clock 2 arrives at point A after clock 1 has already passed it. Both clocks showed the same time when they passed point A, but clock 2 is obviously late--it should have been showing that time earlier, when clock 1 was passing point A.

I'm pretty sure outcome #1 wouldn't happen in real life, since we know gravity isn't purely Newtonian. My question is, would outcome #2 or outcome #3 be the result if this experiment were carried out in real life?

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