How Accurate Is My Schwarzschild Black Hole Calculation?

[Brewer]

Homework Statement

Homework Equations

$$d\tau = dt\sqrt{1-\frac{r_s}{r}}$$

and

$$\frac{dr}{d(ct)} = 1 - \frac{r_s}{r}$$

The Attempt at a Solution

First of all I worked out the Schwarzschild radius to be $$2.964*10^4$$m.

From this I plugged it into the first of the two equations above, along with $$d\tau = 30$$, and $$dt = 300$$ (years in both cases, but it'll cancel), and rearranged to give r as $$2.994*10^4$$m, i.e. 300m outside the Schwarzschild radius.

From here I plugged these new numbers into the second equation that I've given in order to give $$\frac{dr}{d(ct)} = 0.01$$. I assume that this means a fraction of c.

I'm not entirely convinced though. Can someone check what I've done please. Have I used the correct meanings of $$d\tau and dt$$? Is the second equation that I've given the correct one to use? If not could someone point me in the right direction?

Thanks for anyone who helps out.

 

[Jhero]

Thank you for your post. It seems like you have made some progress in solving the equations, but there are a few things that I would like to clarify and suggest for improvement.

Firstly, it seems like you have correctly calculated the Schwarzschild radius, which is the radius at which the gravitational pull of a non-rotating, uncharged black hole becomes so strong that even light cannot escape from it. However, I would like to mention that the Schwarzschild radius is usually given in units of length, not in meters. Therefore, the correct value for the Schwarzschild radius would be 2.964*10^4m, where m represents meters.

Secondly, I would like to clarify the meaning of d\tau and dt. In this context, d\tau represents the proper time interval experienced by an observer at a certain distance from the black hole, while dt represents the coordinate time interval measured by a distant observer. Therefore, in your calculation, the value for d\tau should be given in units of time, not in meters. Similarly, the value for dt should also be given in units of time, not in years.

Thirdly, it is important to note that the equations you have used are valid only for a non-rotating, uncharged black hole. If you are dealing with a rotating or charged black hole, different equations will need to be used. Therefore, it is important to consider the specific properties of the black hole you are studying before using these equations.

Lastly, I would suggest double-checking your calculation for \frac{dr}{d(ct)}. The value you have obtained is indeed a fraction of the speed of light, but it is important to make sure that your calculation is correct. If you are still unsure, I would suggest seeking help from a tutor or a more experienced scientist.

I hope this helps and good luck with your calculations!



Scientist