Gravity

[Little Bang]

If the the Earth were a black hole what would the force on a one gram object at it's event horizon be as compared the same object at the Earth's present mean radius?

[sanctus]

Force will always be F=GMm/r^2; M is the earth mass or bh mass but since it is the same it depends only the distance from the center r....

 

Tormod cleared that up once for me, if earth becomes a black hole the moon still orbits it as when earth was there. This is really intuitive, if you realise that just no mass is created and bh or not gravity is the same...

[CraigD]

If the the Earth were a black hole what would the force on a one gram object at it's event horizon be as compared the same object at the Earth's present mean radius?

No.

 

The radius of the event horizon of an Earth-mass black hole is about 0.09 m. The mean radius of the Earth’s surface is about 6373000 m.

 

Ignoring complicating modern physics considerations, since gravitational attraction [imath]= \frac{k}{r^2}[/imath], this means the force of gravity on a body at [imath]r = 0.09[/imath] is about [imath]5 \times 10^{15}[/imath] (5 quadrillion) times greater than on the same body at [imath]r = 6373000[/imath].

[Qfwfq]

If the the Earth were a black hole what would the force on a one gram object at it's event horizon be as compared the same object at the Earth's present mean radius?

This actually is a matter of a quite subtle kind which, not being of much interest to astronomers, is not usually worked out properly by general relativists. A main pitfall is that one must define the matter appropriately in order to remove a lot of arbitrarity of coordinate choice. People ususlly tend to work it according to Schwartzschild coordinates without questioning the meaning of the values in play. Even worse, there are them who, like Sanctus and Craig, use Newtonian approximations all the way down to [imath]r_{\rm S}[/imath].

 

I would interpret your question as being the force necessary to hold a pointlike unit mass at a constant radius [imath]r[/imath] in the limit of this approaching [imath]r_{\rm S}[/imath], but translated into coordinates which are locally inertial and Minkowskian at the same point (and in which it is at rest). This is arguably the relevant choice of coordinates for discussing what happens to a "small enough" object that is placed on a solid sphere of extremely high density, if we imagine there being a strong enough material.

 

I did some scribblings a year or so ago, concerning this and related matters, but I kept getting into some sticky issues with the calculus and finding that these things aren't easy to tally up with each other. One thing that I didn't get worked out involved figuring whether an integral diverges or has a finite value. I'll see if I can find those scribbles and even make any sense of them.

 

However, I did get the one as defined above worked out. It is to be calculated as the acceleration of a pointlike body, immediately after being released from that constraint. I found what I had expected according to a qualitative argument: in the specified limit, the force increases toward infinity. This obviously implies that, by the time [imath]r[/imath] gets to be hardly more than [imath]r_{\rm S}[/imath], no material could be strong enough to form the solid sphere.

[modest]

Here's a fair link:

 

The Schwarzschild solution in detail

 

Stationary observers:

 

[...]

 

The magnitude of the actual acceleration felt by the observer is the magnitude of the spatial acceleration

 

[math]a= \sqrt{-a \cdot a} = \frac{1}{\sqrt{1-2m/r}}\frac{m}{r^2}[/math]

 

For large values of r this is essentially Newtonian, but as r->2m, the acceleration becomes infinite. In practical terms, in order to be stationary, the observer must somehow provide the acceleration which keeps him at a fixed location; as r->2m the magnitude of this acceleration increases without limit. As we'll see in a bit, the surface r = 2m is the event horizon of the Schwarzschild black hole...

 

~modest

 

EDIT--->

 

Forgot to give the link: http://www.math.ku.edu/~lerner/GR09/Schwarzschild.pdf

[Qfwfq]

I'm glad you found that, because it seems I threw my old scribbles out in a tidying frenzy.