Recent content by Kostik

The narrative in #2 has now been added in LaTex.

That makes sense, I will endeavor to use LaTex in the future. Obviously it’s much faster for me to take a snapshot of my own notes (made with MSWord).

@PeterDonis These are my own notes. If I have made an error, please critique it!

I think I found the (or an) answer. The relations $$\delta g^{\mu\nu} = -g^{\mu\rho}g^{\nu\sigma}\delta g_{\rho\sigma} \quad , \qquad \delta g_{\mu\nu} = -g_{\mu\rho} g_{\nu\sigma} \delta g^{\rho\sigma} \qquad(*)$$ appear to indicate that ##\delta g_{\mu\nu}## and ##\delta g^{\mu\nu}## are not...

From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by ##g_{\alpha\beta}## to get $$\delta g_{\beta\nu} = -g_{\alpha\beta}...

The photoelectric effect is essentially the observation that light below a certain frequency cannot ionize an atom, no matter how large its intensity. Einstein explained this in 1905 by postulating that light consists of particles (photons) with energy proportional to their frequency. However...

@anuttarasammyak Excellent! This was hugely helpful. I wish Dirac had been a little less cryptic. It's not often that one consults Landau & Lifshitz for a less terse explanation!

@anuttarasammyak Very helpful, thanks. I should have looked there, because Dirac's book takes a lot from LL. In my copy of LL, this discussion is on pp 283-284. Anyway, I will give it a careful read.

You mean, consider a coordinate change that moves the "front" surface of the hypercylinder from ##t = a## to some other value, leaving the rear surface at ##t = b##? Well, yes, I suppose we could cook up such a "stretching" coordinate transformation, but I'm not sure that the integral remains...

Apologies, I will try to dig in and understand this better. My GR knowledge is based on reading Dirac, Landau-Lifshitz Vol. 2, Weinberg, and Ohanian-Ruffini. So, I have never seen it the "abstract" way.

Something else bothers me here, unrelated to the discussion above. Dirac says on p. 62: "It is not possible to obtain an expression for the energy of the gravitational field satisfying both the conditions: (i) when added to other forms of energy the total energy is conserved, and (ii) the...

The energy momentum tensor in contravariant form is constructed in a straightforward way for a dust, perfect fluid, etc. See for example Schutz. Let’s consider a simpler example. The ##x^k## momentum contained in a volume is $$\int mv^k \, dV$$ where ##v^\mu## is the 4-momentum. In flat space...

The conservation law ##{M^{\mu\nu}}_{,\nu} = 0## requires that ##\nu## be in a contravariant position.

The Dirac-Einstein pseudotensor is definitely not symmetric (with the indices in any position). So this seems a good reason to prefer the LL pseudotensor.

I meant the Landau-Lifshitz pseudotensor is symmetric, while the Dirac-Einstein pseudotensor is not. I wasn't referring to the energy-meomentum tensor of the matter-energy fields -- which is symmetric.