Centrifugal Forces and Black Holes

[starthaus]

Nope, that should be:

\frac{d^2r}{ds^2}= -\frac{M}{r^4}(r^2-\frac{H^2r}{M}+3H^2)

Same difference, you now know how to do the analysis right.

 

[stevebd1]

It seems there is some agreement that the relativistic equation for centripetal acceleration in Schwarzschild metric is-

a_c=\sqrt{1-2M/r}\,\frac{v^2}{r}

which is supported by the rearranging the Kepler equation for stable orbit and Kev's derivation, (what's slightly odd is that while the equation for gravity in Schwarzschild metric is considerably well known, you literally have to dig to find the equivalent equation for centripetal acceleration). The whole notion of centrifuge becoming negative at the photon sphere may be nothing more than a mixing of coordinates.

Another equation I found that I thought was of interest is the redshift for an orbiting object, which in Schwarzschild metric is-

A=\sqrt{g_{tt} - \Omega^2 g_{\phi \phi}}

where g_{tt}=(1-2M/r), g_{\phi \phi}=r^2 and \Omega is angular velocity (the equation is reduced from a more complete version which is relative to Kerr metric- A=\sqrt{(g_{tt} + 2\Omega g_{\phi t}-\Omega^2 g_{\phi \phi})}).

The equation for a stable orbit (in Schwarzschild metric) for \Omega is-

\Omega_s=\frac{\sqrt{M}}{r^{3/2}}

If \Omega_s is used in the equation for A then A\equiv\sqrt{(1-3M/r)} and becomes zero at the photon sphere.

The relativistic equation for a_c in Schwarzschild metric can also be derived using \Omega_s where-

v_s=\frac{\Omega_s r}{\sqrt{(1-2M/r)}}

which again, is reduced from a more complete equation relative to Kerr metric.

 

[stevebd1]

According to a number of sources, the equation for specific angular momentum of an object in stable orbit is-

L/m=\frac{\sqrt{Mr}\,r}{\sqrt{r^2-3M}}

which has been reduced from an equation relative Kerr metric (http://www.tat.physik.uni-tuebingen.de/~kokkotas/Teaching/Relativistic_Astrophysics_files/GTR2009_4.pdf" page 21). This equation can be rewritten-

L/m=\frac{\Omega_s\,g_{\phi \phi}}{A}=\frac{\sqrt{M}\,r^2}{r^{3/2}\sqrt{1-3M/r}}

where \Omega_s,\ A and g_{\phi \phi} are as the equations in post #52. The equation can also be written as-

L/m=\frac{v_s r}{\sqrt{(1-(v_s/c)^2)}}

where v_s is as the equation in post #7

In all cases, the equations are equivalent and approach infinity at 3M.


Another equation I see a lot for specific angular momentum is-

l=-\frac{\Omega\,g_{\phi \phi}}{g_{tt}}

where g_{tt}=(1-2M/r) which again is reduced from an equation relative to Kerr metric (http://luth2.obspm.fr/fichiers/seminaires/straub-seminar.pdf" page 13-14). When \Omega=\Omega_s, this isn't equivalent to the other equations and approaches infinity at 2M. It's possible there is a reduction factor that needs to be included for, possibly relative to \Omega, in order to make it match. I'd be interested to hear other peoples opinions.

 

[stevebd1]

It appears the factor that brings the equation for l in post #53 in line with the other equations for angular momentum is the equation for conserved energy per unit mass for an object in stable orbit, which for Schwarzschild metric is-

E/m=\frac{r^2-2Mr}{r\sqrt{r^2-3Mr}}

http://www.tat.physik.uni-tuebingen.de/~kokkotas/Teaching/Relativistic_Astrophysics_files/GTR2009_4.pdf" page 21

The equation for angular momentum for an object in stable orbit can be written as-

L/m \equiv (E/m)\frac{\Omega_s\,g_{\phi \phi}}{g_{tt}}

_____________________________________________________________________

Incidentally E/m in Schwarzschild metric is equivalent to the product of the gravitational redshift and the Lorentz factor relative to the orbiting object-

E/m \equiv \frac{\sqrt{1-2M/r}}{\sqrt{1-(v_s/c)^2}}

and A is equivalent to the gravitational redshift divided by the Lorentz factor-

A=\sqrt{1-2M/r}\cdot \sqrt{1-(v_s/c)^2}The above only applies to an object in a stable orbit.

 

[yuiop]

Your trajectory will not be a straight line but an ellipse, a parabola or a hyperbola. You don't know which until you find out and solve the equations of motion.

.. you need to be able to arrive to the equations of motion and solve them. This way, you will get to a discussion of the different possible trajectories. Putting in dr/ds=0 by hand will not do it.

No, it doesn't. The correct analysis is to write:

\frac{d^2r}{ds^2}= -\frac{M}{r^4}(r^2-\frac{H^2}{M}r+3H^2)/\sqrt{...}

You can then do an elementary discussion based on the discriminant:

\Delta=\frac{H^4}{M^2}-12H^2=H^2(\frac{H^2}{M^2}-12)

Are you suggesting that we can tell if the path is elliptical, circular, parabolic or hyperbolic simply by analysing this discriminant? If so I think you are barking
up the wrong tree.

 

[starthaus]

Are you suggesting that we can tell if the path is elliptical, circular, parabolic or hyperbolic simply by analysing this discriminant? If so I think you are barking
up the wrong tree.

No, I was simply pointing out that in addition to your making calculus mistakes you are also making glaring elementary algebra mistakes. That's all.

 

[yuiop]

No, I was simply pointing out that in addition to your making calculus mistakes you are also making glaring elementary algebra mistakes. That's all.

This is just a smear campaign with no substance. Let's have a look at your algebra.

\frac{d^2r}{ds^2} = -\frac{M}{r^2} + \frac{H^2(r-3M)}{r^4}

From the above it easy to see that when r<3m the last term on the right changes sign for any real value of H and this is where the conclusion that reactive centrifugal acceleration reverses below r=3M originates from. ...

No, it doesn't...

If H and r are positive or negative non-zero real numbers and M is a positive non-zero real number, then the term:

+ \frac{H^2(r-3M)}{r^4}

is positive when r>3M and negative when R<3M. If you believe otherwise (and your disagreement indicates you do) then it is you that is making a glaring elementary algebra mistake.

I also demonstrated that you made some very basic algebra blunders in https://www.physicsforums.com/showpost.php?p=2754412&postcount=50" of this thread.

Ever heard the expression "people who live in glass houses don't throw stones"?

 

[starthaus]

This is just a smear campaign with no substance. Let's have a look at your algebra.If H and r are positive or negative non-zero real numbers and M is a positive non-zero real number, then the term:

+ \frac{H^2(r-3M)}{r^4}

is positive when when r>3M and negative when R<3M. If you believe otherwise (and your disagreement indicates you do)

No, I don't believe otherwise, I simply see that you are unable to analyze the sign of the full algebraic expression correctly, that's all. I really don't understand why you keep arguing, I gave you the correct expression and sign analysis long ago, in post 48.

 

[hcgpi]

I thought this may be an isolated idea but on doing a search on the web, there seems to be a common interest in the idea that centrifugal force reverses near a black hole. Below are a couple of links-

http://www.npl.washington.edu/av/altvw55.html"

http://articles.adsabs.harvard.edu//full/1990MNRAS.245..720A/0000720.000.html"

http://arxiv.org/abs/0903.1113v1"

The same subject was mentioned in https://www.physicsforums.com/showthread.php?t=10369" (post #4).

According to most sources, it appears that the reactive centrifuge becomes zero at the photon sphere, my question is, how does this fit into the centripetal acceleration equation? I had a look the relativistic equation for the tangential velocity required for a stable orbit in Kerr metric and reduced it for a Schwarzschild solution (see https://www.physicsforums.com/showthread.php?t=354583"). Based on a_c=a_g (where a_c is centripetal acceleration and a_g is gravity), the only way the equations would work is if centripetal acceleration reduced in accordance with the redshift, becoming zero at the event horizon and negative beyond the EH.

This works also with the Kerr metric where frame dragging increases exponentially within the ergoregion, without a_c being reduced, it would appear that objects would tend to be thrown out of the ergoregion before crossing the EH but if a_c reduces in accordance with the redshift, then the object is overcome by gravity regardless of it's tangential velocity (relative to infinity) and crosses the event horizon.

I wanted to comment on stevebd1's post of aug13-08 but can't get back to it. Can stevebd1tell me if there is any credible theory to support the notion that an initial Planck sphere of Planck radius at Planck time could have contained the sum of all components of the observable universe or say, simply, the universe? His '08 discussion of Planck density (to which there were no comments) has intrigued me for many months, but I sense I may have missed the point.