Equations of motion for an observer falling radially into a blackhole

[Buzz Bloom]

I understand that the coordinate system (CS) for a distant observer O_d is different than that for an observer O_f who is falling radially toward the event horizon of a non-rotating black hole (BH). Using the Schwarzschild metric, I would like to understand the transformation equations that calculate the change in observations of time, t, and radial position, r, as seen by both O_d and O_f.

Here is the scenario I have in mind.

Each observer has a one of two identical clocks, syncrhonized at time t₀.

Starting at a given radius, r₀, which is the distance from the center of the BH (in O_d's CS at time t₀), O_d observes as O_f falls towards the BH. O_d is maintained stationary at r₀ by a suitable repulsion engine. t₀ is the time a which O_f begins to fall radially toward the BH.​

Using repulsion engines, there are stationary marker objects that can be observed by both O_d and O_f. As O_f passes the marker m_r at radius r (in O_d's CS), both observers can measure the value of t at O_d's radius r (no doubt getting different values.

O_d and O_f can communicate with each other with light speed messages. The messages they exchange enable each observer to know:

a) when the other (using the other's clock) saw O_f passing m_r, and
b) when a message sent by one observer was received by the other.​

In this scenario, both O_d and O_f measure t(r) for the falling O_f and both can calculate the two respective r(t) functions: t_d(r) and t_f(r).

Questions:
1) What are the functions r_d(r) and r_f(t)?
2) Are either/both of these two functions different as measured/calculated by O_d and O_f? If so, in what way?

I am looking forward to a discussion answering these two questions, as well as a description of how the answers were developed.

 

[Mentz114]

https://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates

I understand that the coordinate system (CS) for a distant observer O_d is different than that for an observer O_f who is falling radially toward the event horizon of a non-rotating black hole (BH). Using the Schwarzschild metric, I would like to understand the transformation equations that calculate the change in observations of time, t, and radial position, r, as seen by both O_d and O_f.
...

Questions:
1) What are the functions r_d(r) and r_f(t)?
2) Are either/both of these two functions different as measured/calculated by O_d and O_f? If so, in what way?

I am looking forward to a discussion answering these two questions, as well as a description of how the answers were developed.

Perhaps this is a good place to start https://en.wikipedia.org/wiki/Lemaître_coordinates

and https://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates

If you want to compare fallers with static observers it is done here

https://en.wikipedia.org/wiki/Frame_fields_in_general_relativity

 

[PeterDonis]

I understand that the coordinate system (CS) for a distant observer Od is different than that for an observer Of who is falling radially toward the event horizon of a non-rotating black hole (BH).

This is not correct as you state it. Any observer can adopt any coordinates he wants to. The distant observer could adopt Painleve coordinates, or the infalling observer could adopt Schwarzschild coordinates. No physics depends on the choice of coordinates, so if you're thinking about the physics primarily in terms of coordinates, you're doing it wrong.

Using the Schwarzschild metric

I assume here you mean the geometry of spacetime around a black hole, the Schwarzschild geometry, since you've just pointed out that there are multiple coordinate charts that can be used in this geometry.

I would like to understand the transformation equations that calculate the change in observations of time, t, and radial position, r, as seen by both Od and Of.

What does "as seen by" mean? Does it mean "as calculated using particular coordinates"? Or does it mean "as actually seen by receiving light signals"? The latter is a question about physics; the former is a question about coordinate conventions. Which are you interested in? (I would recommend the latter, as above.)

Each observer has a one of two identical clocks, syncrhonized at time $t_0$.

Synchronized how? By being spatially co-located and setting both readings to be the same? By exchanging light signals? (Note that these are questions about how, physically, the clocks are actually synchronized, not about coordinate conventions.)

Starting at a given radius, $r_0$, which is the distance from the center of the BH

No, it isn't. It isn't physically, and it isn't in either of the two coordinate systems you are implicitly using (Schwarzschild or Painleve). The $r$ coordinate is defined such that the surface area of a 2-sphere at coordinate $r$ is $4 \pi r^2$. In flat spacetime, this $r$ is also the distance from the center of the 2-sphere to its surface; but Schwarzschild spacetime is not flat spacetime. In Schwarzschild spacetime, the concept "distance from the center" has no meaning, because the "center" (the singularity at $r = 0$) is not a place in space; it's a moment of time.

As $O_f$ passes the marker $m_r$ at radius $r$ (in $O_d$'s CS), both observers can measure the value of $t$ at $O_d$'s radius $r$

How can $O_f$ make this measurement? He is spatially separated from $O_d$.

 

[Buzz Bloom]

Hi Mentz and Peter:

I much appreciate both your responses to my questions. I see that I will likely have to rephase my questions to make them physically meaningful. I made an effort to be clear so that such issues that you have rasied would obvious to knowledgeable readers, although not yet to me. I think it will take me a while to digest both your posts and the items Mentz cited before I can ask some clear follow-up questions.

Thanks for your posts,
Buzz

 

[pervect]

I did some posts that I think address some (though not all) of your questions. See for instance

https://www.physicsforums.com/threa...inite-fall-toward-the-eh.656805/#post-4185014
https://www.physicsforums.com/threa...inite-fall-toward-the-eh.656805/#post-4185039

I make use of ingoing Eddington-Finklestein coordinates to "simplify" the analysis. Another part of your question could be addressed by considering outgoing Eddington-Finklestein coordinates. It would be possible to approach the whole problem without using these coordinates, but I didn't take that approach.

I only analyzed the fall from infinity in these posts. I think I had some more where I considered falls from a starting radius other than infinity, but they're significantly more complex mathematically and don't add that much insight. Anyway, I'm not going to try to find them unless there is some interest.

 

[pervect]

Let me add a quick summary of the second linked above without the supporting math, just to try and informally explain what the results mean and clarify some of the issues involving the geometric units used.

We consider a black hole with a Schwarzschild radius of 4km - this would be slightly higher mass than our sun (this corresponds to M=2 in the geometric units used, because r=2M in geometric units). We further assume we had an infalling observer who starts from at infinity - and also assume we have a radio/laser broadcast shining radially inwards with some fixed frequency $f_0$ (as judged by the stationary observer at infinity) then:

The frequency measured by the infalling observer as a function of r will be lower than the transmitted frequency $f_0$, just as it would be in the Newtonian case. The Newtonian case would only have doppler effects, the GR case has doppler effects + time dilation. The observed incoming frequency as a function of r will be $f_0 \frac{\sqrt{r}}{\sqrt{r} + 2}$, where we measure r in kilometers. Note though that even though we measure r in km it is not is not a radial distance. Rather, he circumference of a circle of radius r is $2 \pi r$ km, which is how the Schwarzschild r coordinate is defined. The Schwarzschild metric coeficient $g_{rr} = 1 / (1-r_s / r)$ gives us the relationship between the change in proper distance ds and changes in the r-coordinate by the relationship $ds^2 = g_{rr} dr^2$

As to how this relates to your original question - the doppler shift factor is the ration of the received frequency to the transmitted frequency , but is also the ratio of the transmitted period to the received period since period = 1 / frequency. Letting $t_s$ be the time of transmission and $t_r$ be the time of reception, we can say that $d t_s$ = (doppler factor) * $d t_r$. So integrating the doppler factor gives us $t_s$ as a function of $t_r$.

BTW, I get for the integral $r - 4 \sqrt{r} + 8 ln (\sqrt{r} + 2) + C$, where C is some constant.

 

[m4r35n357]

I have to confess to not understanding the questions in the body of your post, but as to the title, there is a very exhaustive treatment of the subject here. The results he derives within are widely quoted in the literature without proof, even in MTW.

 

[PAllen]

A little asterisk to Pervect's last post: the proper distance he mentions is distance for the family of stationary observers. The radial free faller would have a radically different notion of proper radial distance (per, e.g. Fermi-Normal coodinates). Consistent with normal usage, 'proper distance' is an invariant, given simultaneity hypersurface. Any coordinates may be used, given the hypersurface. However, different families of observers will have different notions of what a simultaneity surface is.

 

[Buzz Bloom]

Hi pervect, m4r35n357, and PAllen:

My post #4 thanking Mentz and Peter for their posts applies equally well to you.

Thanks for your posts,
Buzz