The Big Bang time paradox

[Bill3000]

Originally posted by KrazyHorse

Ok, stupid questions as I know I've heard an explanation before. How does this relate to the 1st law of thermodynamics? When the expansion of the universe alters the proportion of say, Radiation and hot matter to the rest of the "stuff" in the universe, won't the total energy of the universe change as well in at least the case of a finite universe?

Global energy conservation is gone. Please see my post to kuci.

*mourns the loss of global energy conservation*

[Kuciwalker]

Yes it does; in fact, that's how we found out it exists, through its effect on the shape of galaxies.

edit: xpost

[KrazyHorse]

The spatial density of DE is generally assumed to be uniform. This is not necessary (except if it's simply a cosmo constant), but it's a good approx on cosmo scales.

We know much less about DE than we do about DM. Since its presence was unimportant until very recently (like redshifts of order one) we have limited information about its history. What we basically have is a brief clip that tells us it's there and what overall density it has currently/in the recent past, but not much about anything else.

[KrazyHorse]

Originally posted by Kuciwalker
Yes it does; in fact, that's how we found out it exists, through its effect on the shape of galaxies.

edit: xpost

No, that's DM.

[Kuciwalker]

Originally posted by Bill3000
*mourns the loss of global energy conservation*

...

[Geronimo]

Originally posted by KrazyHorse


No. Think about it as work done on the Universe. If there's negative pressure P then if you expand a tiny bit dV the work done on the Universe is -PdV, so the Universe has less "total" energy than before. So vice versa, if your component drops faster than 1/(a^3) (a the scale factor) then it exerts a negative pressure, i.e. a deceleration (like radiation)

If there's positive pressure then the work done on the universe is PdV, i.e. the Universe gains energy.

Your problem is that you're thinking of the Universe "creating" this energy and applying some sort of conservation of energy equation, where the expansion rate (derivative of a wrt time) plays the role of the kinetic term. This is not applicable to GR. Throw total energy conservation out the window. Locally energy is conserved, but there's no real consistent way to define total energy in curved space-time.

I'm sure you don't mean to say that total energy is undefinable in general in curved space-time. Is it only undefinable for the entire universe in curved space time? Can total energy be defined for any region smaller than the entire universe even in curved space time?

[KrazyHorse]

It can be defined locally on short time scales. Globally it cannot be.

The larger the curvature, the bigger the volume and the longer the time (for expansion) the worse the approximation.

[Bill3000]

I'm sure you don't mean to say that total energy is undefinable in general in curved space-time. Is it only undefinable for the entire universe in curved space time? Can total energy be defined for any region smaller than the entire universe even in curved space time?

From Wikipedia, for the simple sake of doing it before KH explains it:

n general relativity, the partial derivatives given above are actually covariant derivatives. What this means is that the continuity equation no longer implies that the energy and momentum expressed by the tensor are absolutely conserved. In the classical limit of Newtonian gravity, this has a simple interpretation: energy is being exchanged with gravitational potential energy, which is not included in the tensor, and momentum is being transferred through the field to other bodies. However, in general relativity there is no way to define physical quantities corresponding to densities of gravitational field energy and field momentum; any "pseudo-tensor" purporting to define them can be made to vanish locally by a coordinate transformation. In the general case, we must remain satisfied with a partial "covariant conservation" of the stress-energy tensor.

In curved spacetime, the spacelike integral now depends on the spacelike slice, in general. There is in fact no way to define a global energy-momentum vector in a general curved spacetime.

EDIT: NOOOOOOOOOO [/Darth Vader]

[KrazyHorse]

The Killing vectors from special relativity don't apply, in general, to GR.

Conservation laws fly out the window.

[Geronimo]

thanks!

Originally posted by KrazyHorse
Dark matter is assumed to have the same "equation of state" as regular matter. What the equation of state tells you is how much "pressure" is exerted by the component in question. Matter exerts 0 pressure. This is basically equivalent to saying that the "total amount "of matter is independent of the scale factor (except insofar as matter is annihilated or pair-created). More properly, it tells you that the density of matter goes down as the cube of the scale factor. Think of the Universe as a box. If you double the scale factor you increase its volume by 2*2*2 = 8. If you had 1kg of matter per meter cubed before doubling then after doubling you will have 1/8 kg per meter cubed. Both dark matter and regular matter are hit by the same scale factor cubed dependence, so to a first approximation the amount of dark and regular matter in the universe remain in the same proportion.

Radiation (light) and hot matter (stuff that's moving ultrarelativistically) are hit by an extra scale factor (from redshift), so they drop off quicker than matter. They also exert negative pressure on the Universe's expansion.

I'm having a bit of trouble seeing how the density of matter moving relativistically would change at a different rate from matter which is moving slower as the universe expands. Is this a difficult concept to explain?

[KrazyHorse]

No. It's actually easy.

The "density of matter" is more properly "the energy density in matter". For non-relativistic matter, most of that energy density is in the mass itself (mc^2). For ultrarelativistic matter most of that energy density is in the kinetic energy (gamma-1)mc^2. Total energy is gamma*mc^2. Gamma is a quantity which depends on velocity. It's 1 at v = 0 and goes to infinity at v = c.

Kinetic terms in the energy density are hit by a redshifting factor. If the mass is nonrelativistic then this redshift has minimal effect on the energy density, since the largest chunk (mc^2) is unaffected, so we see a 1/a^3 dependence. If the mass is ultrarelativistic then we can basically ignore the mc^2 bit and concentrate solely on the (gamma-1)mc^2 part. This part scales as 1/a^4 (like radiation).

These are the simple cases. For matter with (gamma-1) of order 1 (i.e. the kinetic energy is roughly the same as the rest mass) the behaviour is slightly more complicated. It behaves somewhere in between the two extremes above.

[Ogie Oglethorpe]

Just for the record at what fractions of c is matter considered relativistic or ultrarelativistic?

[Jon Miller]

I think that above gamma = .5 is generally considered relativistic.

JM
(edit: that should be inverse gamma...)

[Bill3000]

Originally posted by Ogie Oglethorpe
Just for the record at what fractions of c is matter considered relativistic or ultrarelativistic?

I'm not exactly sure, but you could alwys look at the gamma factor as a good measure. 1/sqrt(1-v^2/c^2) - factor commonly used in special relativity; relativistic when it significantly differs from 1.

[KrazyHorse]

Originally posted by Ogie Oglethorpe
Just for the record at what fractions of c is matter considered relativistic or ultrarelativistic?

You could probably apply the following:

gamma < 1.5, v < 0.75c => nonrelativistic
gamma > 3, v > 0.94c => ultrarelativistic

in between is relativistic but not ultrarelativistic

That should generally get you a decent approximation.

If more precision is required then you broaden the area which requires the full treatment (in between non- and ultra-relativistic)