# Energy of Big Bang

cansay27
I have recently been studying cosmology in my free time, but I came across something that strikes me as rather odd. From what I know, at the time of the Big Bang, there was an even amount of matter as there was antimatter. There was also a superpartner for every particle. This would cause the cancellation of energy that allows the universe to be created from "nothing".

What I don't understand is if there was more matter than antimatter, wouldn't that mean that there must've been energy needed to cause the Big Bang? If so, how much energy was it?

I know there's probably a flaw in my logic, so please point it out. This is bugging me alot.

## Answers and Replies

There are several unresolved questions, which you've touched upon. The origin of the big bang (as opposed to how it developed) is one example. Another is the disappearance of the antimatter.

cansay27
There are several unresolved questions, which you've touched upon. The origin of the big bang (as opposed to how it developed) is one example. Another is the disappearance of the antimatter.

Yes, but is the amount of energy needed in order for the Big Bang to occur calculated already? I read somewhere that it was a rather minute amount as compared to the energy the Big Bang created.

I have recently been studying cosmology in my free time, but I came across something that strikes me as rather odd. From what I know, at the time of the Big Bang, there was an even amount of matter as there was antimatter. There was also a superpartner for every particle. This would cause the cancellation of energy that allows the universe to be created from "nothing".
No, these are different concepts. Anti-matter has positive energy just like normal matter, so if you want to create an electron/positron pair, you need at least twice the mass energy of the electron to do this.

The conservation of energy is a separate phenomenon from the conservation laws that lead to there being an equal (at first) amount of matter and anti-matter. This is a very good look at energy conservation in the context of cosmology:
http://www.phys.ncku.edu.tw/mirrors/physicsfaq/Relativity/GR/energy_gr.html

Basically, there are two ways of looking at it.
1. Because energy conservation only occurs in a system that has certain properties that do not change in time, but the expansion of the universe breaks this, energy simply isn't conserved in an expanding universe.
2. The above analysis only takes into account the energy in matter fields. If you also take into account gravitational potential energy, it turns out that the gravitational potential energy exactly cancels the energy in matter fields, such that an expanding universe, whether full of matter or completely empty, always has zero energy.

cansay27
2. The above analysis only takes into account the energy in matter fields. If you also take into account gravitational potential energy, it turns out that the gravitational potential energy exactly cancels the energy in matter fields, such that an expanding universe, whether full of matter or completely empty, always has zero energy.

Oh ok. But does dark energy affect this, seeing as how it cancels the effect of gravity in the universe? I know dark energy is causing the acceleration of the universe, but is it considered antigravity?

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Oh ok. But does dark energy affect this, seeing as how it cancels the effect of gravity in the universe? I know dark energy is causing the acceleration of the universe, but is it considered antigravity?

Isn't the missing bit of information here that matter/antimatter annihilations produced the flood of radiation that is the CMB? So that is where most of the positive energy ended up.

Oh ok. But does dark energy affect this, seeing as how it cancels the effect of gravity in the universe? I know dark energy is causing the acceleration of the universe, but is it considered antigravity?
Dark energy has no impact on this result. The result holds no matter what the makeup of the universe, and dark energy, as far as General Relativity is concerned, is just a form of matter with negative pressure.

The only caveat, which I didn't mention, is that the "total energy = 0" only holds for a closed universe. Infinities creep into the calculation that prevent it from being done for flat or open universes, if I remember correctly.

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2. The above analysis only takes into account the energy in matter fields. If you also take into account gravitational potential energy, it turns out that the gravitational potential energy exactly cancels the energy in matter fields, such that an expanding universe, whether full of matter or completely empty, always has zero energy.
I don't think this is quite right, at least not in the exact form in which you stated it. In standard GR, the total energy in a cosmological solution is not a well-defined notion. There is a good discussion of this in MTW on p. 457. GR does not have any general definition of mass-energy that applies to all spacetimes.

I have heard people speculate about the idea that there might be some theory beyond GR, e.g., a theory of quantum gravity, and that in such a theory the total energy of the universe might be both well defined (which it isn't in plain GR) and zero. I think the motivation is that then the big bang could be considered as the mother of all vacuum fluctuations. But this is definitely not something that can be addressed in straight GR.

The only caveat, which I didn't mention, is that the "total energy = 0" only holds for a closed universe. Infinities creep into the calculation that prevent it from being done for flat or open universes, if I remember correctly.
This may be a correct statement about some theory, but it definitely isn't a correct statement about standard, plain-vanilla GR. If you look at p. 457 of MTW, you'll see that the case of a closed universe is exactly the one they focus on, and they explain why things like the total charge and total energy are not well defined.

I don't think this is quite right, at least not in the exact form in which you stated it. In standard GR, the total energy in a cosmological solution is not a well-defined notion. There is a good discussion of this in MTW on p. 457. GR does not have any general definition of mass-energy that applies to all spacetimes.
It's a well-known result from using the Hamiltonian formulation of General Relativity.

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What is "MTW"? (From internet search, it seems to be a textbook, but I can't find the title.)

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It's a well-known result from using the Hamiltonian formulation of General Relativity.

After googling a little and trying to figure out what you mean by this, I think I may understand what you mean.

On the face of it, your statement would seem to contradict MTW p. 457. This would raise the question of what you consider to be the error in the logic on p. 457 of MTW. It would also raise the question of what definition of mass-energy are you saying is used in this well-known result. It can't be Komar mass, ADM mass, or Bondi mass, since closed cosmological solutions aren't stationary or asymptotically flat.

When I googled, one of the top hits I found was a post you made here in November:
A well-known result from taking the Hamiltonian formalism of General Relativity is that the total energy (matter energy + gravitational potential energy) for a closed, homogeneous universe is identically equal to zero. If you want more detail, this is an authoritative paper on the subject:

The paper you linked to uses a specific mass-energy pseudotensor. So this seems to answer some of the questions above. MTW p. 457 is assuming that the quantity we're going to talk about must be a tensor. What you consider a "standard result" assumes a different definition of mass-energy, which is not a tensor.

If I've got this right, then, OK, what you're saying is fine except that IMO you're vastly oversimplifying and overselling it. Pseudotensors have some very serious problems. There is no widely accepted measure of mass-energy in GR, so IMO it's very misleading to suggest that "the" total energy of a closed universe is zero. Here is a discussion of the objections to pseudotensors: http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

Note that the "standard result" you're referring to is in a paper by Berman, using a certain mass-energy pseudotensor defined by Berman. I'm sure Berman considers his own mass-energy pseudotensor to be "the" best definition of energy, but that doesn't mean that it's universally accepted.

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You can naively estimate the total energy of the big bang by plugging the total mass of the universe into e=mc^2 then adding the radiation content. We actually have numbers for this - just don't expect mind boggling precision. The net zero answer for total energy content of the universe is more complicated. It only computes, IF, the universe is dead flat - which is not a bad bet [re: google on 'Lawrence Krauss universe from nothing']. Be warned, however, creation ex nihilo can be an emotional topic.

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I can't resist a footnote, as I find the net zero energy universe concept very intriguing. In an eternally expanding universe, all matter [even supermassive black holes] and energy will eventually be smeared out into an infinitely thin, homogenous soup - a virtual Milne universe. Such a condition looks just right for a putative quantum creation ['big bang'] event. If the 'big rip' hypothesis is correct, it may not even take a gazillion years for the universe to reinvent itself.

After googling a little and trying to figure out what you mean by this, I think I may understand what you mean.

On the face of it, your statement would seem to contradict MTW p. 457. This would raise the question of what you consider to be the error in the logic on p. 457 of MTW. It would also raise the question of what definition of mass-energy are you saying is used in this well-known result. It can't be Komar mass, ADM mass, or Bondi mass, since closed cosmological solutions aren't stationary or asymptotically flat.

When I googled, one of the top hits I found was a post you made here in November:

The paper you linked to uses a specific mass-energy pseudotensor. So this seems to answer some of the questions above. MTW p. 457 is assuming that the quantity we're going to talk about must be a tensor. What you consider a "standard result" assumes a different definition of mass-energy, which is not a tensor.

If I've got this right, then, OK, what you're saying is fine except that IMO you're vastly oversimplifying and overselling it. Pseudotensors have some very serious problems. There is no widely accepted measure of mass-energy in GR, so IMO it's very misleading to suggest that "the" total energy of a closed universe is zero. Here is a discussion of the objections to pseudotensors: http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

Note that the "standard result" you're referring to is in a paper by Berman, using a certain mass-energy pseudotensor defined by Berman. I'm sure Berman considers his own mass-energy pseudotensor to be "the" best definition of energy, but that doesn't mean that it's universally accepted.
Well, I think the answer is more that it's due to a rather specific configuration of the space-time which allows us to make this definition. It's obviously not a general result that holds for any potential space-time. But it just so happens that this particular configuration (closed FLRW) is very likely applicable to our universe.

I should mention, though, that I don't think there is anything particularly special about this result. Energy simply isn't a conserved quantity in General Relativity in the general case, because as you mention, there is no well-defined definition of the energy. There's the additional point that from Noether's Theorem, energy conservation is a consequence of time-invariance of a system. The expansion of the universe breaks this time invariance, which eliminates the conservation of energy.

For other readers, what we do have in General Relativity instead is the conservation of the stress-energy tensor, a construct which includes not only energy, but also momentum, pressure, and stresses. Conservation of this larger construct often forces energy to change, though exactly how energy changes often depends on the numbers you use to describe your system.

Tanelorn
Chronos, do we not also need some kind of cause and effect, especially for something as sudden and energetic as the big bang?

cansay27
These responses are helpful, but I can't quite understand what it all means yet.

Is there a universal understanding on this topic, or has it not been studied enough in order to have a solution?

These responses are helpful, but I can't quite understand what it all means yet.

Is there a universal understanding on this topic, or has it not been studied enough in order to have a solution?
I would say it's pretty well-understood, it's just that the concepts are sometimes difficult to explain. The basic punchline is that because of how General Relativity works, there is no problem with lots of matter being made out of essentially nothing in the early universe.

cansay27
I would say it's pretty well-understood, it's just that the concepts are sometimes difficult to explain. The basic punchline is that because of how General Relativity works, there is no problem with lots of matter being made out of essentially nothing in the early universe.

Ok thank you. Does it also explain exactly where all of this matter and energy AND dimensions came from?

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You can naively estimate the total energy of the big bang by plugging the total mass of the universe into e=mc^2 then adding the radiation content.

Yes, you could do that if you knew the total mass of the universe. But the total mass of the universe isn't well defined, not even for a closed universe.

For other readers, what we do have in General Relativity instead is the conservation of the stress-energy tensor, a construct which includes not only energy, but also momentum, pressure, and stresses.

The stress-energy tensor is a *density*. It isn't the total amount of some conserved quantity in some region. Since it isn't a total, it isn't conserved. The total isn't well defined because Gauss's theorem doesn't apply to nonscalar fluxes in curved spacetime.

Conservation of this larger construct often forces energy to change, though exactly how energy changes often depends on the numbers you use to describe your system.
There is no larger construct that is conserved, and there is no standard sense in which it forces energy to change, because total energy is not well defined in GR, in any general, widely accepted sense.

The basic punchline is that because of how General Relativity works, there is no problem with lots of matter being made out of essentially nothing in the early universe.
No, this is totally wrong, or else you're expressing yourself extremely unclearly.

Ok thank you. Does it also explain exactly where all of this matter and energy AND dimensions came from?
Well, exactly where all of the matter came from is another matter. That's mostly down to the specific initial conditions and the nature of physics at energies far higher than we've been able to test in the lab so far. All that our knowledge of General Relativity tells us is that there is no obvious problem with creating a lot of matter in the early universe. It doesn't tell us how this might have occurred.

As for the number of dimensions, from what I understand the best answer we have for this so far is that no other number of dimensions is habitable. So beings like ourselves will always see themselves in a 3+1 dimensional space-time, just because any other number of dimensions is uninhabitable.

The stress-energy tensor is a *density*. It isn't the total amount of some conserved quantity in some region. Since it isn't a total, it isn't conserved. The total isn't well defined because Gauss's theorem doesn't apply to nonscalar fluxes in curved spacetime.
The way you conserve energy in terms of a density is to have the energy flowing into the region match the change over time of the energy within a region, that is, a continuity equation. Even this does not work in General Relativity. And it isn't because there isn't a good definition of overall energy, because a continuity equation is a differential form that exists only in terms of the density.

There is no larger construct that is conserved, and there is no standard sense in which it forces energy to change, because total energy is not well defined in GR, in any general, widely accepted sense.
Um, it's called the stress-energy tensor.

No, this is totally wrong, or else you're expressing yourself extremely unclearly.
I don't understand where your difficulty is. This is just an overly-simplistic description of cosmic inflation.

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bcrowell said:
The stress-energy tensor is a *density*. It isn't the total amount of some conserved quantity in some region. Since it isn't a total, it isn't conserved. The total isn't well defined because Gauss's theorem doesn't apply to nonscalar fluxes in curved spacetime.
The way you conserve energy in terms of a density is to have the energy flowing into the region match the change over time of the energy within a region, that is, a continuity equation. Even this does not work in General Relativity. And it isn't because there isn't a good definition of overall energy, because a continuity equation is a differential form that exists only in terms of the density.
OK, let's take this piece by piece.
"The way you conserve energy in terms of a density is to have the energy flowing into the region match the change over time of the energy within a region, that is, a continuity equation." This is correct.
"Even this does not work in General Relativity." This is incorrect. It does work in GR. Mass-energy is locally conserved in GR.
"And it isn't because there isn't a good definition of overall energy,[...]" This is incorrect. There isn't a good definition of overall energy. For example, see the discussion on p. 457 of MTW.
"[...] because a continuity equation is a differential form that exists only in terms of the density." This is correct.

bcrowell said:
There is no larger construct that is conserved, and there is no standard sense in which it forces energy to change, because total energy is not well defined in GR, in any general, widely accepted sense.
Um, it's called the stress-energy tensor.
No, the stress-energy tensor is not a conserved quantity, as I explained in #20.

bcrowell said:
Chalnoth said:
The basic punchline is that because of how General Relativity works, there is no problem with lots of matter being made out of essentially nothing in the early universe.
No, this is totally wrong, or else you're expressing yourself extremely unclearly.
I don't understand where your difficulty is. This is just an overly-simplistic description of cosmic inflation.
OK, I would agree that it was overly simplistic. If you want to try to frame it more rigorously, that would be fine.

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We can only talk about the mass content of the observable universe without risk of invoking metaphysics. For estimates see:
http://hypertextbook.com/facts/2006/KristineMcPherson.shtml
Needless to say, there is some disparity.

Discussing the mass of the observable universe is different from discussing the mass of the universe.

Looking at the list of values tabulated in the hypertextbook.com link, I think it's important to note that several of the sources appear to be cranks.

The first sentence of the Wahlin reference is "My God is a Goddess."

The first sentence of the Nielsen reference is "In the following I shall show that there is an intimate - holistic - context between microcosmos - the atomic world - and macrocosmos - the Universe as a whole." Reading further shows that it's definitely crank material.

If you look at the ones that are not crank material, it's important to realize that they are not necessarily claiming to estimate the total mass-energy of the universe, which is not well defined in standard GR. For example, the Immerman reference is talking about the sum of the rest masses of all atoms in the observable universe. This is completely different from claiming to estimate the total mass-energy of the observable universe.

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I had already written up a brief FAQ entry on this topic, which I've now updated with more detail based on this discussion. Here's the present version.

FAQ: How does conservation of energy apply to cosmology? What is the total mass-energy of the universe?

Conservation of energy doesn't apply to cosmology. General relativity doesn't have a conserved scalar mass-energy that can be defined in all spacetimes.[MTW] There is no standard way to define the total energy of the universe (regardless of whether the universe is spatially finite or infinite). There is not even any standard way to define the total mass-energy of the *observable* universe. There is no standard way to say whether or not mass-energy is conserved during cosmological expansion.

Note the repeated use of the word "standard" above. To amplify further on this point, there is a variety of possible ways to define mass-energy in general relativity. Some of these (Komar mass, ADM mass [Wald, p. 293], Bondi mass [Wald, p. 291]) are valid tensors, while others are things known as "pseudo-tensors" [Berman 1981]. Pseudo-tensors have various undesirable properties, such as coordinate-dependence.[Weiss] The tensorial definitions only apply to spacetimes that have certain special properties, such as asymptotic flatness or stationarity, and cosmological spacetimes don't have those properties. For certain pseudo-tensor definitions of mass-energy, the total energy of a closed universe can be calculated, and is zero.[Berman 2009] This does not mean that "the" energy of the universe is zero, especially since our universe is not closed.

One can also estimate certain quantities such as the sum of the rest masses of all the hydrogen atoms in the observable universe, which is something like 10^54 kg. Such an estimate is not the same thing as the total mass-energy of the observable universe (which can't even be defined). It is not the mass-energy measured by any observer in any particular state of motion, and it is not conserved.

MTW: Misner, Thorne, and Wheeler, Gravitation, 1973. See p. 457.
Berman 1981: M. Berman, unpublished M.Sc. thesis, 1981.
Berman 2009: M. Berman, Int J Theor Phys, http://www.springerlink.com/content/357757q4g88144p0/
Weiss and Baez, "Is Energy Conserved in General Relativity?," http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html
Wald, General Relativity, 1984

OK, let's take this piece by piece.
"The way you conserve energy in terms of a density is to have the energy flowing into the region match the change over time of the energy within a region, that is, a continuity equation." This is correct.
"Even this does not work in General Relativity." This is incorrect. It does work in GR. Mass-energy is locally conserved in GR.
Nope. I mean, you can write something that looks very similar to the classical continuity equation, but it has extra terms. It is those extra terms that mean that classical energy conservation, where the change in energy density at a point is equal to the energy flow in/out of that point, cannot hold.

"And it isn't because there isn't a good definition of overall energy,[...]" This is incorrect. There isn't a good definition of overall energy. For example, see the discussion on p. 457 of MTW.
I didn't say there was a good definition of overall energy. I just said that you don't have local energy conservation in General Relativity.

No, the stress-energy tensor is not a conserved quantity, as I explained in #20.
How is it not conserved? Its covariant derivative is identically zero.

OK, I would agree that it was overly simplistic. If you want to try to frame it more rigorously, that would be fine.
Don't think that would add anything but confusion.

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Nope. I mean, you can write something that looks very similar to the classical continuity equation, but it has extra terms. It is those extra terms that mean that classical energy conservation, where the change in energy density at a point is equal to the energy flow in/out of that point, cannot hold.
OK, maybe we're just talking past each other by not defining our terms carefully and by not writing out equations to show what we mean. When I say that energy-momentum is exactly locally conserved, I mean that $\nabla_aT^{ab}=0$. The interpretation is that the net flow of energy-momentum in and out of any infinitesimal volume is zero.

bcrowell said:
No, the stress-energy tensor is not a conserved quantity, as I explained in #20.
How is it not conserved? Its covariant derivative is identically zero.
As I explained in #20, it's not conserved because it's a density. What is conserved (locally) is the energy-momentum four-vector. Compare with E&M: what's conserved is charge, not current density. The fact that the divergence of the current density vanishes doesn't tell you that current density is a conserved quantity, it tells you that charge is a conserved quantity.

OK, maybe we're just talking past each other by not defining our terms carefully and by not writing out equations to show what we mean. When I say that energy-momentum is exactly locally conserved, I mean that $\nabla_aT^{ab}=0$. The interpretation is that the net flow of energy-momentum in and out of any infinitesimal volume is zero.
It's not quite that simple in GR, though, because GR doesn't necessarily deal solely with point-like objects. When you have things like fluids or other extended objects, you can't reduce the conservation of the stress-energy tensor to conservation of energy-momentum in the usual sense.

But in any event, that's somewhat of an aside. The point is that if you consider, say, an expanding universe full of radiation, the energy of a comoving volume decreases over time. Most people would consider this to be a situation where energy isn't conserved. Similarly, with inflation, you can take a region of the universe much smaller than a proton that has rather high energy density but very small size, and within a fraction of a second expand that region to be many light years across, with an energy density that has changed very little. Most would also call that a situation where energy isn't conserved, and this is pretty critical to the topic at hand because it allows huge amounts of matter to come from very little.

Tanelorn
Is this equivalent to energy instability or runaway like a cigarette turning into a huge forest fire or like the beating of a butterfly's wings eventually turning into a hurricane?

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Is this equivalent to energy instability or runaway like a cigarette turning into a huge forest fire or like the beating of a butterflies wings eventually turning into a hurricane?
Well, that's a very different sort of situation. Those things happen because some highly complex, non-linear systems can have changes which are out of proportion to the inputs that caused said changes. These sorts of changes have the property that they are very hard to predict, for instance, precisely because the eventual state of two systems with only very slightly different inputs can be so dramatically different.

Inflation is a bit of a different beast, because even though you're causing a massive ballooning of the universe, it's a highly deterministic one. You can make small changes to the starting configuration, and it really doesn't make much of any difference as to how inflation progresses (provided the changes you make don't stop inflation from occurring altogether). In fact, if you're willing to believe the no hair theorems as they apply to inflation, the initial conditions can't have anything whatsoever to do with the eventual state of the universe: all that is important is that inflation begins and lasts long enough to produce a large universe.

Tanelorn
Thanks Chalnoth, do we know what happened to stop inflation when it did? Assuming it did actually stop!

Thanks Chalnoth, do we know what happened to stop inflation when it did? Assuming it did actually stop!
The simplest models of inflation have inflation being driven by a field that has some potential energy, but is away from the minimum of the potential. As inflation progresses, the field slowly progresses towards the minimum of the potential. This progression is slowed by the extremely rapid expansion, which is what allows the universe to inflate in the first place. Once the field hits the minimum, however, it oscillates about that minimum, which causes the field to decay into standard model particles. This process is known as "reheating".

Once that has occurred, the universe is dominated by radiation energy density, and the expansion slows down dramatically. What follows is basically standard big bang theory.

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It's not quite that simple in GR, though, because GR doesn't necessarily deal solely with point-like objects. When you have things like fluids or other extended objects, you can't reduce the conservation of the stress-energy tensor to conservation of energy-momentum in the usual sense.
I think you're trying to describe the difference between global and local conservation of energy-momentum in GR, but this formulation doesn't work. The local conservation of energy-momentum described by the zero-divergence property of the stress-energy tensor has nothing to do with whether the stress-energy tensor describes a fluid or a collection of pointlike objects. It is exact in either case.

But in any event, that's somewhat of an aside. The point is that if you consider, say, an expanding universe full of radiation, the energy of a comoving volume decreases over time. Most people would consider this to be a situation where energy isn't conserved. Similarly, with inflation, you can take a region of the universe much smaller than a proton that has rather high energy density but very small size, and within a fraction of a second expand that region to be many light years across, with an energy density that has changed very little. Most would also call that a situation where energy isn't conserved, and this is pretty critical to the topic at hand because it allows huge amounts of matter to come from very little.
Yes, that's an example of the kinds of problems you get with defining *global* energy conservation in GR. It has nothing to do with the type of *local* energy conservation described by the zero-divergence property of the stress-energy tensor.

Have you looked at the section in MTW that I pointed you to? Do you have a copy of the book? If not, I might be able to find a reference that would be more accessible to you. This is really just standard stuff about the interpretation of mass-energy in GR.