If the Universe’s Mass was 100 At the Moment of the Big Bang....

[Islam Hassan]

Supposing that the total mass + energy content of the Big Bang was 100 in energy-equivalent terms. What would we consider its energy-equivalent mass to be today? If the universe is today accelerating and that acceleration is not abating, then (naïvely perhaps...) its (dark-energy) potential energy is not reducing while its kinetic energy is increasing. Total energy-equivalent mass would be increasing, no?

First of all can we address this question or not given our present state of knowledge? If the present energy-equivalent mass is greater, then greater by how much? And what would be the (speculative?) mechanism by which such energy-equivalent mass increment was generated?

Finally, would we expect that the energy expended in inflation to have been pre-contained in the initial, pre-inflationary BB ‘stuff’, or would it somehow have been generated post t = 0?IH

 

[kimbyd]

Not really answerable, unfortunately. There are a few issues that make this difficult to compare:
1. Energy is not conserved in an expanding universe (this blog post has a good description, if you want an in-depth look).
2. In quantum mechanics, mass isn't conserved at all: it is possible for particles to annihilate or be created in matter/anti-matter pairs. We can certainly model what the average total particle mass would be at any given time based upon these models, but this kind of thing is neither useful nor accurate for the very early universe.

What we can say with reasonable certainty is that the total energy in a comoving volume in the early universe is, today, vastly less than it was in the very early universe, provided the time in the early universe chosen was after the end of inflation. The total energy in a comoving volume is growing today (due to dark energy), but the total mass is staying about the same. The total mass decreases very slowly as black holes evaporate (and potentially as protons evaporate). But those effects occur on time scales far, far longer than the current age of the universe (much greater than $10^{30}$ years).

 

[Islam Hassan]

Not really answerable, unfortunately. There are a few issues that make this difficult to compare:
1. Energy is not conserved in an expanding universe (this blog post has a good description, if you want an in-depth look).
2. In quantum mechanics, mass isn't conserved at all: it is possible for particles to annihilate or be created in matter/anti-matter pairs. We can certainly model what the average total particle mass would be at any given time based upon these models, but this kind of thing is neither useful nor accurate for the very early universe.

What we can say with reasonable certainty is that the total energy in a comoving volume in the early universe is, today, vastly less than it was in the very early universe, provided the time in the early universe chosen was after the end of inflation. The total energy in a comoving volume is growing today (due to dark energy), but the total mass is staying about the same. The total mass decreases very slowly as black holes evaporate (and potentially as protons evaporate). But those effects occur on time scales far, far longer than the current age of the universe (much greater than $10^{30}$ years).

Is the total energy today less than in the very early universe because matter was traveling at higher velocities then?

And if it is less today, where has this total energy decrease gone? Into work done in expanding the universe? But then dark energy is supposed to be responsible for that, if I understand correctly...IH

 

[phinds]

... where has this total energy decrease gone?

You are not listening to what you are being told. You think there is conservation of energy in an expanding universe but as has already been stated, that is not true. Please read the material in the link already provided to you.

 

[kimbyd]

Is the total energy today less than in the very early universe because matter was traveling at higher velocities then?

And if it is less today, where has this total energy decrease gone? Into work done in expanding the universe? But then dark energy is supposed to be responsible for that, if I understand correctly...IH

A combination of particles having lower momenta, and there being fewer high-mass particles around. As Phinds says, though, the energy didn't go anywhere. Sean Carroll breaks it down pretty well in the link I shared.

 

[Islam Hassan]

You are not listening to what you are being told. You think there is conservation of energy in an expanding universe but as has already been stated, that is not true. Please read the material in the link already provided to you.

My apologies...it is not easy to let go mentally of the principle of conservation of energy...it keeps coming back like in automatic reversion...will be more careful in the future...IH

 

[phinds]

My apologies...it is not easy to let go mentally of the principle of conservation of energy ...

Yep. One of many things that are counter-intuitive when you get into Cosmology and Quantum Mechanics. Welcome to the world of headaches

 

[Ibix]

Yep. One of many things that are counter-intuitive when you get into Cosmology and Quantum Mechanics. Welcome to the world of headaches

The thing I find interesting is that energy conservation works locally. For an infinitesimal volume, stress-energy in equals stress-energy out. And (I believe that) I know that it doesn't apply macroscopically because we can't work out a way to integrate the at-a-point statement, except in special cases like stationary spacetimes. I don't understand if there's a physical interpretation for why that should be.

 

[phinds]

The thing I find interesting is that energy conservation works locally. For an infinitesimal volume, stress-energy in equals stress-energy out. And (I believe that) I know that it doesn't apply macroscopically because we can't work out a way to integrate the at-a-point statement, except in special cases like stationary spacetimes. I don't understand if there's a physical interpretation for why that should be.

I think it's because you can't even DEFINE the energy of a far distant object in terms relative to us so you certainly can't say that something that you can't define should be conserved. Wouldn't make sense. For example, an asteroid moving towards Earth has a certain energy relative to us. Absent any forces on it (yeah, I know, gravity and all) it will continue to have that energy and when you figure in the effect of whatever forces DO act on it, energy is conserved.

BUT ... what is the energy of an asteroid in a galaxy 10 billion light years away from us?

 

[Ibix]

BUT ... what is the energy of an asteroid in a galaxy 10 billion light years away from us?

Ok - so you could answer that in a stationary spacetime by picking the timelike Killing vector field (an element of the geometry) to define your timelike direction. No such luck with a spacetime that doesn't have a timelike KVF. But FLRW spacetime does pick out everywhere a timelike vector field - the co-moving observers - by symmetry. Why can't we use that?

 

[phinds]

Why can't we use that?

calling @PeterDonis

 

[PeterDonis]

calling @PeterDonis

Yes?

FLRW spacetime does pick out everywhere a timelike vector field - the co-moving observers - by symmetry. Why can't we use that?

Because it's not a Killing vector field; the spacetime geometry is not unchanged along its integral curves. Comoving observers always see the universe as homogeneous and isotropic, but they do not always see its spacetime curvature as being the same.

How you translate this into more intuitive language depends on what you want to emphasize. If you want to say that each comoving observer marks out a "point in space", then you have to say that "space" is not unchanged--it's expanding, because the comoving observers are moving apart. And that means you can't use these "points in space" to define a conserved energy.

If, OTOH, you want to pick out a family of observers who aren't moving apart, then at most one of them can be comoving, so their worldlines don't line up with the vector field you picked out; they don't match the symmetry that you were using.

 

[Ibix]

How you translate this into more intuitive language depends on what you want to emphasize. If you want to say that each comoving observer marks out a "point in space", then you have to say that "space" is not unchanged--it's expanding, because the comoving observers are moving apart. And that means you can't use these "points in space" to define a conserved energy.

If, OTOH, you want to pick out a family of observers who aren't moving apart, then at most one of them can be comoving, so their worldlines don't line up with the vector field you picked out; they don't match the symmetry that you were using.

I understand that (or, if "understand" is too strong, at least I know it). I don't see why it leads to it being impossible to convert a differential statement of energy conservation into an integral form. Surely there's a clear way to define a volume element in terms of co-moving light clocks differential distances apart?

I must confess that this is something I have not done much reading on - it just seemed an apposite time to ask. So happy to accept "shut up and read MTW" or whatever, although slightly more precise directions would be much appreciated.

 

[PeterDonis]

I don't see why it leads to it being impossible to convert a differential statement of energy conservation into an integral form.

It's not that you can't do the integral; of course you can, you can always pick a spacelike slice of constant comoving time and integrate the stress-energy tensor over it. But its result won't be conserved, and arguably won't have any physical meaning.

(Technically, there is a way to construct an integral whose value will be "conserved"--but only because it vanishes identically for any spacetime whatsoever, and therefore tells you nothing useful at all.)

Surely there's a clear way to define a volume element

Sure, it's just $\text{d}^4 x \sqrt{-g}$. But that doesn't help with the other issues I described above.