Calculation of gravity near a black hole.

[avito009]

Newtons Universal law of gravity equations are an excellent approximation when dealing with low velocities (i.e., velocities whose magnitude is much smaller than the speed of light) and when dealing with weak gravity fields (such as those found on Earth or around low-mass stars). The approximation fails when you're dealing with speeds close to that of light, or gravity fields around black holes, at which point, you switch to general relativity.

So to calculate gravity close to a black hole you have to use equations from general relativity. I tried finding these equations on google but I wasnt able to find any.

Can somebody tell me the equations to find gravitational force near a black hole using general relativity?

 

[ShayanJ]

Look at here and here!

 

[HallsofIvy]

You won't find equations of the type you are used to, except, possibly, as simplified approximations.

The "equation of gravity" in general relativity is $$G_{\mu\nu}= 0$$. G here is Einstein's gravitational tensor. Its entries are derived from derivatives of the metric tensor so each entry is a differential equation for the entries of the metric tensor. Once you have found those, you can use them in the equation for the geodesic curves- the lines of motions of objects freefalling in that gravitational field.

 

[pervect]

Newtons Universal law of gravity equations are an excellent approximation when dealing with low velocities (i.e., velocities whose magnitude is much smaller than the speed of light) and when dealing with weak gravity fields (such as those found on Earth or around low-mass stars). The approximation fails when you're dealing with speeds close to that of light, or gravity fields around black holes, at which point, you switch to general relativity.

So to calculate gravity close to a black hole you have to use equations from general relativity. I tried finding these equations on google but I wasnt able to find any.

Can somebody tell me the equations to find gravitational force near a black hole using general relativity?

Well, what I think you probably want to know is the answer to one of the two following questions.

The first question is "What is the amount of proper acceleration a (as measured by an onboard accelerometer) that a rocket would need to hover at a Schwarzschild coordinate r for a black hole of mass M."

The answer to that question turns out to be

$$a = \frac{GM}{r^2\sqrt{1-\frac{2GM}{rc^2}}}$$

Note that the circumference of a circle at a Schwarzschild coordinate of r is, by definition $2 \pi r$. However, the "distance" to the event horizon is in general not equal to that number.

The second question is "What would this observer measure for the distance to the event horizon, assuming they use fermi-normal coordinates to define and measure the distance". If you have some other procedure in mind to measure the distance, you'd have to specify what it was for me to be able to answer the question. Offhand I would guess you don't have an exact procedure for measuring the distance in mind, so I am applying what I think is a "reasonable" default. It still may or may not match what you really intended, but there's no cure for that other than you to specify exactly how you intend to measure the distance :(. This is trickier than it sounds - a radar signal would take an infinite time to reach the event horizon and come back. However, the coordinate speed of light in the Rindler coordinate system approaches zero, so in the end the procedure I suggest comes up with a finite number.

I don't have a formula for a small black hole to answer the second question exactly, but for a large black hole, you can ignore tidal forces and approximate the black hole by the Rindler metric of a uniformly accelerating observer. With this approximation, the horizon is always located at the Rindler horizon, a distance of $\frac{c^2}{a}$ . This means that if you are accelerating at 1g to hold station away from a large black hole, the event horizon will be about 1 light year away (since c is approximately 1 light year/ year^2). Another way to put this : the (proper) acceleration required to hold station is $c^2/d$ for a large black hole, d being the distance away from the event horizon in the fermi-normal coordinates of the hovering observer.

 

[DrGreg]

Note that the radius of a circle at a Schwarzschild coordinate of r is, by definition $2 \pi r$.

I'm sure pervect meant to say "circumference", not "radius".

 

[pervect]

I'm sure pervect meant to say "circumference", not "radius".

Ooops, yes, I was able to edit and fix that typo.