The threshold of general relativity gravity

[victorvmotti]

In Padmanabhan's Theoretical Astrophysics by defining a ratio for comparing gravitational potential energy with rest-mass energy it is shown that if massive objects with M=10^33 gm are confined to small regions with R= 1km then we cannot use Newtonian gravity because the system has general relativity effects.

I wonder if we do not use an energy ratio and instead use only Schwarzschild metric or Einstein field equations can we infer the same ratio between mass and radius?

I already see that Schwarzschild vacuum solution is asymptotically flat because the ratio of Schwarzschild radius over radius vanishes if the radius coordinate approaches infinity.

But establishing the above mentioned ratio of mass and radius for the threshold of general relativity without using the energy argument is not immediately clear.

 

[Bill_K]

In Padmanabhan's Theoretical Astrophysics by defining a ratio for comparing gravitational potential energy with rest-mass energy it is shown that if massive objects with M=10^33 gm are confined to small regions with R= 1km then we cannot use Newtonian gravity because the system has general relativity effects.

I wonder if we do not use an energy ratio and instead use only Schwarzschild metric or Einstein field equations can we infer the same ratio between mass and radius?

For a given mass M, the Schwarzschild radius is R_s = 2GM/c².

For m = 10³³ gm, R_s = 2(6.6x10^-8)(10³³)/(3x10¹⁰)² ≈ 10⁵ cm = 1 km.

 

[victorvmotti]

Is there any lower limit on Rs here?

I mean if we imagine a size almost equal to the hydrogen atom 0.5*10^-8 cm filled with a mass of 3.4*10^19 gm then what? Should we call this a candidate of a micro black hole?

 

[Bill_K]

Is there any lower limit on Rs here?

In principle there is: https://en.wikipedia.org/wiki/Micro_black_hole#Minimum_mass_of_a_black_hole.

But since we don't have a theory of Quantum Gravity, at present this is just a plausible guess.

 

[piareround]

Is there any lower limit on Rs here?
I mean if we imagine a size almost equal to the hydrogen atom 0.5*10^-8 cm filled with a mass of 3.4*10^19 gm then what? Should we call this a candidate of a micro black hole?

Well one thing you should realize is there is a lower limit that we can apply Rs is one of debates of String Theory vs. Quantum Gravity.

For example, string theory would make the lower limit the "length of a string", but quantum gravity makes it a much smaller Plank length.

Now if you believe in Hawking Radiation is the dominant form of radiation from black holes(*), then a black hole the size of a hydrogen atom would only last for a couple of picoseconds or nanoseconds.*https://en.wikipedia.org/wiki/Eddington_luminosity anyone? ... darn tough crowd tonight for making Star Trek 2009 astrophysics jokes.

 

[victorvmotti]

What about an upper limit on Rs?

Can we use the same ratio to determine if the whole universe is indeed inside a black hole?

What would be the mass here for the ratio and the related Rs?

Can we use the current epoch energy density parameter, that is $\Omega = 1$, then obtain the observable universe mass and see if the radius of the observable universe is less than Rs?

 

[Bill_K]

What about an upper limit on Rs?
Can we use the same ratio to determine if the whole universe is indeed inside a black hole?

No, this is a common misconception. There is no similarity at all between the universe we observe and a black hole.

A black hole, after all, has an inside and an outside. At any point it has a preferred direction, namely the radial direction, and consequently at that point the collapse is not isotropic.

By contrast the universe we observe is uniform, apparently flat and infinite, and expanding at the same rate in all directions.

 

[bcrowell]

Well one thing you should realize is there is a lower limit that we can apply Rs is one of debates of String Theory vs. Quantum Gravity.

For example, string theory would make the lower limit the "length of a string", but quantum gravity makes it a much smaller Plank length.

I'm not a specialist in quantum gravity, but I don't think this is right. String theory *is* supposed to be a theory of quantum gravity, and the length of the strings in string theory is presumed to *be* the Planck length.

 

[Bill_K]

I'm not a specialist in quantum gravity, but I don't think this is right. String theory *is* supposed to be a theory of quantum gravity, and the length of the strings in string theory is presumed to *be* the Planck length.

Ben is right. (And he survived the Earthquake!)

 

[bcrowell]

(And he survived the Earthquake!)

But I think my terrier is going to need therapy.