Distances between planet and observer near BH

[sergiokapone]

Imagine that we have a system that consists of a massive black hole, and the asteroid revolving around it on a stable orbit. What method can determine the distance to these asteroids observer who is still close to the event horizon?
The first thing that came to mind is to determine the signal propagation time to the asteroid and back. This time is equal to:
##\Delta \tau_{observer} = \sqrt{1-\dfrac{r_g}{r_{observer}}}
\int\dfrac{dr}{1-\frac{r_g}{r}}
##
But I do not know ##r_{observer}## and ##r_{asteroid}##
Need a different way.

 

[DrGreg]

But I do not know ##r_{observer}## and ##r_{asteroid}##
.

You'll need to tell us what you do know.

 

[sergiokapone]

Suppose, the observer knows that the orbit is circular and also knows the asteroid's rotation period determined by proper clock. A black hole is not rotating, i.e. we can use the Schwarzschild metric.

 

[TWest]

Suppose, the observer knows that the orbit is circular and also knows the asteroid's rotation period determined by proper clock. A black hole is not rotating, i.e. we can use the Schwarzschild metric.

NASA Disagrees?

http://www.nustar.caltech.edu/

 

[Bill_K]

To calculate r_obs and r_ast from the observations of the orbits you need to use Kepler's Law. A truly remarkable fact about the Schwarzschild solution is that, when expressed in terms of the Schwarzschild coordinates r and t, Kepler's Law is completely unmodified from its Newtonian form.

In more detail, the angular velocity of a test particle in a circular orbit about a Schwarzschild mass is (dφ/dτ)² = (GM/r³)(1 - 3r_s/r)^-1. Here τ is the particle's own proper time. Hopefully you recognize the leading factor (GM/r³) as Kepler's expression. Also, dφ/dτ is related to the period T of the orbit (expressed in proper time) by dφ/dτ = 2π/T.

The proper time is related to Schwarzschild coordinate time t by dt/dτ = (1 - 3r_s/r)^-1/2, so (dφ/dt)² = (dφ/dτ)²(dτ/dt)² = GM/r³. (The relativistic factors exactly cancel!) Thus we are left with the familiar "period squared proportional to distance cubed". By measuring the period of an asteroid in Schwarzschild time we can immediately calculate from this the radius of its circular orbit.

EDIT: Corrected some algebra!

 

[TWest]

To calculate r_obs and r_ast from the observations of the orbits you need to use Kepler's Law. A truly remarkable fact about the Schwarzschild solution is that, when expressed in terms of the Schwarzschild coordinates r and t, Kepler's Law is completely unmodified from its Newtonian form.

In more detail, the angular velocity of a test particle in a circular orbit about a Schwarzschild mass is (dφ/dτ)² = (GM/r³)(1 - 3r_s/r)^-1/2. Here τ is the particle's own proper time. Hopefully you recognize the leading factor (GM/r³) as Kepler's expression. Also, dφ/dτ is related to the period T of the orbit (expressed in proper time) by dφ/dτ = 2π/T.

The proper time is related to Schwarzschild coordinate time t by dt/dτ = (1 - 3r_s/r)^-1/2, so dφ/dt = (dφ/dτ)(dτ/dt) = GM/r³. (The relativistic factors exactly cancel!) Thus we are left with the familiar "period squared proportional to distance cubed". By measuring the period of an asteroid in Schwarzschild time we can immediately calculate from this the radius of its circular orbit.

Yes but does that Explain a black hole transforming into a supermassive one then into a blazar?

 

[sergiokapone]

Bill_K, thank you. There is a simple way to get such an expression (dφ/dτ)² = (GM/r³)(1 - 3r_s/r)^-1/2 from dτ²=(1-r_g/r)dt² - r²dφ² ? Where can I get the equation in order to exclude dt² ie dt/dτ = (1 - 3r_s/r)^-1/2?

 

[TWest]

Use Calculus, but what I don't wanna?

 

[Bill_K]

Bill_K, thank you. There is a simple way to get such an expression (dφ/dτ)² = (GM/r³)(1 - 3r_s/r)^-1/2 from dτ²=(1-r_g/r)dt² - r²dφ² ? Where can I get the equation in order to exclude dt² ie dt/dτ = (1 - 3r_s/r)^-1/2?

The key word is "circular". You have to solve the equations of motion for a test particle in a Schwarzschild field, in particular for a circular orbit.

 

[sergiokapone]

Oh, right. I will try.

 

[sergiokapone]

So, we need to do as in the case of Newton's theory, ie we need to have the solution of the field equations and the equations of motion. The expression for the metric is only a solution of the field equations.

 

[sergiokapone]

I have another question.
To determine the distance between two points in Minkowski space, it was necessary to send a light signal forth and back. Knowing time between sending and receiving, we could determine the distance by multiplying time/2 by the speed of light. In the case of the Schwarzschild metric, it is enough for us to know the time of sending and receiving the signal to determine the distance between two points?

 

[pervect]

One approach for circular orbits is to look at the geodesic equations directly. See https://www.physicsforums.com/showpost.php?p=4354272&postcount=12 for some detailed calculations. (I think I've found most of the typos, but I might have missed a few.)

There's other ways as well, finding the valley in the effective potential, for instance.
http://www.fourmilab.ch/gravitation/orbits/