Distances between planet and observer near BH

In summary, a black hole can be determined distance by knowing the time of sending and receiving a signal.
  • #1
sergiokapone
302
17
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.
 
Last edited:
Physics news on Phys.org
  • #2
sergiokapone said:
But I do not know ##r_{observer}## and ##r_{asteroid}##
.
You'll need to tell us what you do know.
 
  • #3
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.
 
  • #4
sergiokapone said:
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/
 
  • #5
To calculate robs and rast 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 - 3rs/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 - 3rs/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!
 
Last edited:
  • #6
Bill_K said:
To calculate robs and rast 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 - 3rs/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 - 3rs/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?
 
  • #7
Bill_K, thank you. There is a simple way to get such an expression (dφ/dτ)2 = (GM/r3)(1 - 3rs/r)-1/2 from dτ2=(1-rg/r)dt2 - r22 ? Where can I get the equation in order to exclude dt2 ie dt/dτ = (1 - 3rs/r)-1/2?
 
Last edited:
  • #8
Use Calculus, but what I don't wanna?
 
  • #9
sergiokapone said:
Bill_K, thank you. There is a simple way to get such an expression (dφ/dτ)2 = (GM/r3)(1 - 3rs/r)-1/2 from dτ2=(1-rg/r)dt2 - r22 ? Where can I get the equation in order to exclude dt2 ie dt/dτ = (1 - 3rs/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.
 
  • #10
Oh, right. I will try.
 
  • #11
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.
 
  • #12
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?
 
  • #13

FAQ: Distances between planet and observer near BH

How is the distance between a planet and an observer near a black hole determined?

The distance between a planet and an observer near a black hole is determined by measuring the time it takes for light to travel from the planet to the observer. This is known as the light-travel distance and it is affected by the strong gravitational pull of the black hole.

Can the distance between a planet and an observer near a black hole change over time?

Yes, the distance between a planet and an observer near a black hole can change over time. This is due to the effects of the black hole's gravity on the space-time fabric, which can cause the distance between objects to stretch or contract.

How does the distance between a planet and an observer near a black hole affect the planet's orbit?

The distance between a planet and an observer near a black hole can significantly affect the planet's orbit. The strong gravitational pull of the black hole can cause the planet's orbit to become distorted or even unstable, leading to irregularities in its movement.

Is the distance between a planet and an observer near a black hole constant?

No, the distance between a planet and an observer near a black hole is not constant. As the planet and the observer move through space, the distance between them can change due to the effects of the black hole's gravity and the planet's orbit around the black hole.

How does the distance between a planet and an observer near a black hole affect the appearance of the planet?

The distance between a planet and an observer near a black hole can greatly affect the appearance of the planet. Due to the strong gravitational pull of the black hole, light from the planet can become distorted, making it appear stretched or distorted to the observer. In extreme cases, the planet may even appear to be duplicated due to gravitational lensing.

Similar threads

Back
Top