You might find this old blog post regarding various types of radial infalling plungers into a BH in various reference frames of interest (velocity is towards the bottom) -
https://www.physicsforums.com/threads/properties-of-in-falling-radial-plungers.768338/
While this thread is marked basic, you still might find this old library entry regarding black hole thermodynamics useful-
https://www.physicsforums.com/threads/black-hole-thermodynamics.762982/
I've not seen the equation for stable orbit explicitly expressed in Schwarzschild terms, only in Kerr terms (probably due to the fact that rotating black holes are more realistic than absolutely static ones). I simply took the Kerr version and removed the spin parameter a which gives you the...
It's worthwhile working through this yourself but you might also find this post (and the thread) useful-
https://www.physicsforums.com/threads/orbit-velocity-in-schwarzschild-metric.963831/#post-6117557
The equation for gravity for an object with tangential velocity around/near a static BH in...
While PeterDonis has answered the question, you can see this at work in the Reissner–Nordström metric which is the vacuum solution for a charged black hole. Gravity for a charged black hole is-
a_g=\frac{M}{r^2\sqrt{1-\frac{2M}{r}-\frac{Q^2}{r^2}}}
where M=Gm/c^2 (mass in geometric units) and...
While you may be looking at elliptical orbits, the following is the equation for stable orbit velocity in Schwarzschild metric-
v_s=\sqrt{\frac{M}{r\left(1-\frac{2M}{r}\right)}}
where M=Gm/c^2 (note the answer would be a fraction of the speed of light, multiply by c for S.I. units). The above...
The first law of thermodynamics states that energy can neither be destroyed or created, hence there are two scenarios when virtual particles are pulled apart near the event horizon. In the case of say an electron and an anti-electron (a positron) being pulled apart, if the positron was to be...
It's worth pointing out that the max realistic spin a Kerr black hole can have is a/M=0.998 simply due to the CMB, photons, other radiation and dark matter that falls into a black hole, stopping the bh from approaching too close to maximal. Kip Thorne mentions this in relation to the bh in...
Thanks for the input, while this is more up to date, unless I'm missing something, this doesn't show how the distribution of dark energy and matter has changed over the age of the universe which is what I wanted to point out.
While you are talking about change in density, you still might find these two links of interest which look at the change in distribution of dark energy and matter over time-
http://www.einsteins-theory-of-relativity-4engineers.com/cosmocalc_2010.htm
adjust the redshift to see how the...
While it's worthwhile working this out for yourself, the following could be used as a check. Wheeler and Turner would refer to an object falling from rest at a specific r as the 'drip' frame (an object falling from rest at infinity would be rain frame and an object hurled inward at speed from a...
Without going into a great detail, to clarify, the Kerr black hole has two horizons, the outer and inner (sometimes referred to as the Cauchy) where the coordinate radii of these horizons are defined by r_+=M +\sqrt(M^2-a^2) and r_-=M -\sqrt(M^2-a^2) respectively (where M is mass and a is spin)...