Good point, even for the massless case, a spherical uniform distribution of light directions becomes non-uniform under a Lorentz transformation.
So there is a comoving frame for this part of the universe. It sounds like (from reading this) that the comoving frame would be different if I moved...
Yes I agree that 'no privileged inertial frame' does not decide what a particular distribution of proper velocities should be.
But I would assume that the very early universe was unbiased. If its particles all had proper velocities within 1000m/s of mine then that is not an arbitrary starting...
Maybe it is clearer to use proper velocities, as I didn't make it clear I was talking about particles of the same rest mass.
What would a very simple toy universe with particles in it be like? Any unbiased distribution of particles proper velocities would be unbounded because there is no...
Since there is no privileged inertial frame, I would have expected the first particles in the universe to have no particular bias in their momenta. Relative to an observer I would expect the distribution to be uniform and unbounded. The mean momentum of the initial particles relative to an...
That makes sense. Maybe the first three are somehow Kruskal–Szekeres coordinates.
Anyway, thanks for the information, I think that is useful enough for my level of understanding.
In order:
https://plato.stanford.edu/entries/spacetime-singularities/lightcone.html
http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/black_holes/index.html
http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/black_holes_picture/index.html...
Many diagrams show light cones tipping over when closer to a black hole singularity, such that emitted light can have a downwards (negative time) component in the distant observer coordinate frame. e.g this diagram:
or this one:
or this one:
However, other diagrams show that the light...
It seems quite likely I would be over-constraining the vacuum equations, in which case I would need to provide fewer constraints, or allow for the solution closest to a vacuum rather than pure vacuum. But basically I'm after a solver that doesn't solve the usual initial value problem forward in...
Hi, I am interested in simulating the vacuum field equations, but solving a full boundary value problem rather than the initial value problem. i.e. I might have boundary conditions in all spatial and temporal extents/extremes, rather than just an initial 3D surface.
Does anyone know any free...
Thanks, I agree with you on these points. I think I'll read up some more before any attempt to rephrase the question. And perhaps in my studying I might find the answer. It would be better to get a cleaner and more correct question as a new post anyway.
Yes, Baez talks about the volume of a ball of test particles. But if there is no matter present there can be no particles. Never-the-less, if it is clearer then let's talk about some imaginary test particles that have zero mass (or as mass tends to zero if that is preferred). Instead of a ball...
Sorry, badly (and wrongly) written question, I will try to be clearer.
Let us say we have a Ricci-flat 3+1D spacetime, but which is not Minkowski space. In other words it has some curvature but is a vacuum spacetime.
Now at a particular event e=(t, x, y, z) we look at a small 4-volume \delta...
If we have a small 4-volume of empty spacetime of boxlike dimensions t, x, y, z, then according to the vacuum field equations the change of the shape of this box with respect to time is (I think): \frac{d (xyz)}{dt}=0 or equally: yz\frac{dx}{dt}+xz\frac{dy}{dt}+xy\frac{dz}{dt}=0
in other words...
The wikipedia page says:
"Simple examples of Einstein manifolds include:
Calabi–Yau manifolds admit an Einstein metric that is also Kähler, with Einstein constant {k=0}."
I understand your comment that Calabi-Yau manifolds seem to always be described as Riemannian (rather than...
Hi,
Wikipedia lists about 10 vacuum solutions of the Einstein Field Equations.
However, if I look for topologies of Einstein space, there are many different families, which include Calabi-Yau manifolds, of which abelian, Enriques, Hyperelliptic and K3 surfaces are subsets. Within K3 surfaces...