Schwarzschild metric and BH mass

[pervect]

Steve Carlip
And harder to understand for those here who are less adept at this crazy math stuff.

Yes, but when you actually want to calculate things, Killing vectors are extremely handy. With different clocks ticking at different rates, what constitutes a "time translation" can get confusing without the formalism. With the formalism, this notion is made unambiguous and independent of various coordiante choices (covariant or contravariant coordinates, for instance).

The total energy of the gravitating source or for a particle moving in the field the source creates? I was speaking of the later.
Pete

I was speaking of the former

 

[Orion1]

Massless Metric...


'Massless' Photonic-Schwarzschild BH:

Radial solution for sphereically symmetric Schwarzschild metric:
$$r_b = \frac{2GM_b}{c^2}$$

Mass-Energy Equivalence principle for Schwarzschild BH:
$$M_b = \frac{E_b}{c^2} = \frac{r_b c^2}{2G}$$
$$E_b = \frac{r_b c^4}{2G}$$

Energy Equivalence for single photon:
$$E_p = \frac{\hbar c}{\overline{\lambda_p}}$$
$$\overline{\lambda_p}$$ - wavebar (photon wavelength)

General Relativity Mass-Energy Equivalence Principle:
$$E_p = E_b$$

$$\frac{\hbar c}{\overline{\lambda}} = \frac{r_b c^4}{2G}$$

Radial-Wavebar solution for spherically symmetric 'massless' Schwarzschild BH:
$$r_b \overline{\lambda} = \frac{2 \hbar G}{c^3}$$
$$r_b = \overline{\lambda}$$
$$r_b = \sqrt{\frac{2 \hbar G}{c^3}$$

This solution describes a 'Massless' Photonic-Schwarzschild BH composed of a single photon traveling at luminous velocity.

'Massless' Photonic-Schwarzschild BHs exist as a mathematical solution in General Relativity due to the General Relativity Mass-Energy Equivalence Principle. In Classical GR, BHs can be composed of both mass and/or energy.

Based upon the Orion1 equations, what is the radius and wavelength for a 'Massless' Photonic-Schwarzschild BH?

Based upon the Orion1 equations, what is the energy magnitude for this single photon?


if a black hole is made from photons, would it be massless and move at the speed of light?


The Orion1 solution descibes a classical GR 'massless' Photonic-Schwarzschild BH composed of a single photon traveling at luminous velocity.

 

[DW]

'Massless' Photonic-Schwarzschild BH:

Radial solution for sphereically symmetric Schwarzschild metric:
$$r_b = \frac{2GM_b}{c^2}$$

Mass-Energy Equivalence principle for Schwarzschild BH:
$$M_b = \frac{E_b}{c^2} = \frac{r_b c^2}{2G}$$
$$E_b = \frac{r_b c^4}{2G}$$

Energy Equivalence for single photon:
$$E_p = \frac{\hbar c}{\overline{\lambda_p}}$$
$$\overline{\lambda_p}$$ - wavebar (photon wavelength)

General Relativity Mass-Energy Equivalence Principle:
$$E_p = E_b$$

$$\frac{\hbar c}{\overline{\lambda}} = \frac{r_b c^4}{2G}$$

Radial-Wavebar solution for spherically symmetric 'massless' Schwarzschild BH:
$$r_b \overline{\lambda} = \frac{2 \hbar G}{c^3}$$
$$r_b = \overline{\lambda}$$
$$r_b = \sqrt{\frac{2 \hbar G}{c^3}$$

This solution describes a 'Massless' Photonic-Schwarzschild BH composed of a single photon traveling at luminous velocity.

'Massless' Photonic-Schwarzschild BHs exist as a mathematical solution in General Relativity due to the General Relativity Mass-Energy Equivalence Principle. In Classical GR, BHs can be composed of both mass and/or energy.

Based upon the Orion1 equations, what is the radius and wavelength for a 'Massless' Photonic-Schwarzschild BH?

Based upon the Orion1 equations, what is the energy magnitude for this single photon?


if a black hole is made from photons, would it be massless and move at the speed of light?


The Orion1 solution descibes a classical GR 'massless' Photonic-Schwarzschild BH composed of a single photon traveling at luminous velocity.

No it doesn't. This is yet another example why "relativistic mass" is a bad concept.

 

[pervect]

'Massless' Photonic-Schwarzschild BH:

Radial solution for sphereically symmetric Schwarzschild metric:
$$r_b = \frac{2GM_b}{c^2}$$

etc etc etc

This random assortment of equations appears to me to be "not even wrong".

 

[pmb_phy]

This random assortment of equations appears to me to be "not even wrong".

Seems to be that Orion1 is confusing the proper mass, M, which appears in the Schwazchild metric with the relativistic mass of a photon. So the lack of clarity of this distinction has take one more soul. (just kidding of course).

Pete

 

[Orion1]

Classical Radial solution for spherically symmetric Schwarzschild metric:
$$L = 0$$ - angular momentum
$$r_b = \frac{2GM_b}{c^2}$$

Relativistic Radial solution for spherically symmetric Schwarzschild metric:
$$L = 0$$ - angular momentum
$$n_v = \left( \frac{v}{c} \right)$$ - velocity number
$$\gamma = \sqrt{(1 - n_v^2)}^{-1}$$
$$r_b = \frac{2G \gamma M_b}{c^2}$$

$$r_b = \frac{2GM_b}{c^2 \sqrt{(1 - n_v^2)}}$$
$$n_v = .998$$

Based upon the Orion1 solution, what is the relativistic effect on the Schwarzschild metric for a near-luminous velocity Schwarzschild BH with $$L = 0$$ angular momentum?

 

[pervect]

I think I've said about all I need to say about my opinion of Orion's ramblings.

To move onto more positive matters, I will go back to an old issue. It turns out that when pmb was talking about E_0 being conserved, and tying this to a Lagrangian, there is a rigorous justification for this.

The geodesic deviation equations can be derived from a "least action" principle. In this sense, there is a rigorously justifiable "Lagrangian" and a so-called "Super-Hamiltonian" that can be talked about for geodesic motion. This also justifies MTW's remarks about how t^a and E_0 were "conjugate momenta" .

I thought this was sort of interesting, when I ran across it. It doesn't affect of course any of my remarks about the computation of system energy, but it is relevant to conserved quantites for systems following geodesic motion.

 

[Chronos]

Hmm, it seems a lot simpler to assume the net total energy of particles in a conserved energy field, such as gravity, is always zero.

 

[pervect]

Hmm, it seems a lot simpler to assume the net total energy of particles in a conserved energy field, such as gravity, is always zero.

Well, if you scroll back over the previous long discussion, you'll see that the conservation of energy in GR isn't, in general, very simple.

There are a few very simple quantites that are conserved in the Schwarzschild metric, though, that are very useful, and easy to explain without trotting out the whole dog_and_pony show. One of these is the covariant value of the energy component of the energy-momentum 4-vector, E₀.

While this only works in Schwarzschild coordinates, or other coordinates that have a unit timelike killing vector, it's quite handy when it can be used.