Recent content by Tertius

I figured out where my approach was wrong. The boundary conditions for this are a little tricky because it is not a bound system. So I discretized it manually and just solved $$H\psi = E\psi$$, where E is the kinetic energy of the incoming particle. E in this case is just a single eigenvalue...

I am doing this to have my own solution for customization and understanding. I also want to manually check the WKB approximation accuracy at various energies against this static solution. I've split the problem into 3 regions and am solving it in 1D, but am having problems with how to define...

I realized the problem was quite simple. The covariant derivative is "correcting" for changes to the metric. The changes in the FRW metric look volumetric because ##\sqrt{g}=a^3##.

I'm studying Carroll's section on covariant derivatives, which shows that the covariant divergence of a vector ##V^\mu## is given by $$\nabla_\mu V^\mu = \partial_\mu V^\mu + \Gamma^\mu_{\mu\lambda}V^\lambda$$. Because ##\Gamma^\mu_{\mu\lambda}=\frac{1}{\sqrt{g}}\partial_\lambda \sqrt{g}## we...

@PeterDonis, @vanhees71 I see. I neglected the negative sign, if you can get different answers with different metric conventions then it can't be physical. The quantity I am really after is the matter density that is timelike. From my understanding timelike matter should always have a...

@PeterDonis Ok, I've had some time to evaluate my thinking here. Using Noether's theorem results in a stress energy momentum tensor that is defined using translational symmetries. These are not only usually non-existent in curved spacetime, but there are other symmetries, such as gauge...

That's fair. Just didn't want the thread to die. I'll do the maths and hopefully have a more detailed thread later on.

I hadn't thought of it in terms of needing isolation before. But I see what you are saying. I should have specified my flat space 4-momentum comment was just an example (a mostly not precise relationship) of what I am trying to understand about curved spacetime. To be precise, I want to find...

Right, so the trace with a perfect fluid is the expected $$-\rho + 3p$$ For dust, we use ##p=0##, and for photons (traceless) we use ##\rho=3p## So a timelike KVF, if it exists, can give us the total energy. A spacelike KVF, which does exist in FRW spacetimes, can give us momentum. So in FRW...

I see. I originally thought about the trace as the invariant mass density of the system because of this paper: https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-38/issue-4/The-connection-between-the-energy-momentum-tensor-and-the-tensor/cmp/1103860087.pdf But I...

I should have specified 'the spatial curvature changes' I was referring to a Schwarzschild metric, which has a timelike KVF, but has spatial curvature. From my understanding, the trace is a coordinate independent quantity, and should be observer independent because if any timelike observers...

GR has very limited situations in which a total mass-energy can be defined. The Komar mass, for example, requires the presence of a timelike killing vector field and an asymptotically flat spacetime. Basically, if the metric change with time or it's spacelike curvature does not flatten out...

Makes sense. I suppose in this case then it wouldn't be possible to have ##P=\rho/3##.

From a simple calculation, in order for a homogeneous, isotropic scalar field to have that equation of state, the ##V[\phi]## function would have to be equal to ##1/4 \dot{\phi}^2##. It seems this would be a very strange potential function because it is of the exact same form as the kinetic...

Yes, that was a typo.