Observing a Beacon Impact a Neutron Star/Event Horizon

[Grinkle]

If I am observing a pinging beacon free-falling into a neutron star from a distance far enough away that I am in approximately flat spacetime, I think I observe the pings redshifting as the beacon gets deeper into the gravity well, in other words I see the clock of the beacon slowing with respect to my clock and finally, I see the beacon impact the neutron star.

I never see the beacon itself moving more slowly with respect to me, I always see it accelerate with respect to me. What I do see is the beacons clock slowing down as it gets closer to impact even as the beacon itself is moving faster and faster with respect to me.

Is that correct, and if so can I replace "neutron star" with "event horizon" in the above and the only change I need to make is that I lose sight of the beacon before I can ever observe it impact cross the event horizon?

 

[Nugatory]

I never see the beacon itself moving more slowly with respect to me

How do you "see" the speed of the beacon?
That's not a flippant question - you're observing some physical phenomenon (the angle subtendsd by the probe, shrinking as the beacon moves away from you? The time taken for a radar signal return?) and associating it with the coordinate distance between you and the beacon. How you do that defines what you're calling the speed of the beacon.

 

[Grinkle]

the angle subtendsd by the probe

I was picturing that when I wrote the question. Thinking about it, radar return vs watching from the side seem equivalent if I am watching reflected light. If I could watch by the beacon occluding some hypothetical thing in the background that would be the only other way I can think of to see the speed.

 

[Nugatory]

I was picturing that when I wrote the question. Thinking about it, radar return vs watching from the side seem equivalent if I am watching reflected light. If I could watch by the beacon occluding some hypothetical thing in the background that would be the only other way I can think of to see the speed.

OK, and with which if any of these methods do you see the beacon increasing its speed relative to you?

 

[Grinkle]

OK, and with which if any of these methods do you see the beacon increasing its speed relative to you?

Each method reduces to photons coming out of the gravity well, so I expect either all of them do or none of them do.

Without knowing what the formula for gravitational shifting looks like, my thinking is that I would calculate the speed of the beacon as accelerating away from me after accounting for gravitational shifting - there would be some red-shift component left over that is due to the acceleration towards the star.

 

[PeterDonis]

I never see the beacon itself moving more slowly with respect to me, I always see it accelerate with respect to me.

You don't "see" the speed of the beacon at all. In flat spacetime, you could calculate it directly from the redshift of its light, but you can't do that in curved spacetime. Strictly speaking, in curved spacetime, the speed of an object distant from you, relative to you, is not even well-defined.

You can define a "speed" by choosing coordinates; note carefully that @Nugatory said "coordinate speed". But such a speed might not behave as your intuition says it will. For example, the coordinate speed of an object free-falling into a black hole, relative to a far distant observer, in Schwarzschild coordinates (which are the ones that most people intuitively adopt), does not continue to increase as the object gets closer and closer to the horizon; it reaches a maximum value and then decreases, approaching a limit of zero as the object approaches the horizon. If the central object is a neutron star and is compact enough, the coordinate speed of an object free-falling radially inward, in Schwarzschild coordinates, will also reach a maximum value and then start to decrease before the object hits the star.

 

[Grinkle]

@PeterDonis @Nugatory

Thanks very much. I was bothered that my thinking about massive classical objects is counter to what I know at a B level to be how objects behave near black holes.

such a speed might not behave as your intuition says it will.

A good thing to remember.

 

[Matterwave]

You can define a "speed" by choosing coordinates; note carefully that @Nugatory said "coordinate speed". But such a speed might not behave as your intuition says it will.

Another place this issue shows up over and over again is in cosmology where coordinate speeds of distant observers can exceed the speed of light. This often leads to endless discussion of "does GR violate SR?" and the like.

 

[Justin Hunt]

What about the time between pings? time dilation from gravitational difference and possibly relative ones too depending on what speed it reaches relative to us the observer would make the time increase between pings. But, if we don't know the actual speed or distance the beacon is from us how can we take the changing distance into account? if the only thing we can record is the time elapse between pings (which would be constant from the perspective of the beacon itself) wouldn't the changing distance make a difference?

 

[PeterDonis]

What about the time between pings?

If you mean the time on the distant observer's clock between receiving successive pings, which are sent at constant intervals by the beacon's clock, this time will increase as the beacon falls.

 

[Ibix]

But, if we don't know the actual speed or distance the beacon is from us how can we take the changing distance into account?

We deploy some markers whose redshift does not change. These would be powered buoys hovering at fixed Schwarzschild r coordinates. As the infaller passes each hovering marker, it can record the (locally measured) relative velocity, and it's also obvious that distance from us is increasing.

This is just me choosing a definition of distance and time, a coordinate system attached to a set of physical objects. I could make another choice and come up with a different idea about the distance and the velocity, but I would still end up with the same redshift.