# Dark Energy, the Cosmological Constant, General Relativity and you

So there has been a few threads over the last few days where the whole issue of dark energy and/or the cosmological constant has been discussed in the context of whether it is a valid 'solution' to GR or if it violates some principle of relativity or even insults Einsteins 'legacy', whatever that is supposed to mean I thought it would be good to consolidate the discussion to a single thread and also explain this issue in a bit more detail.

I'd like to start at the start, with the Einstein Field equations as we know and love them. The simplest description goes like this (ignoring some factors of $$\pi$$ which I will do throughout for simplicity):

$$G_{\mu\nu} = T_{\mu\nu}$$

where the left hand side is the Einstein tensor, this is the 'geometric side' of the equation (I'll use this terminology later, so make note of it) while the right hand side is the stress-energy tensor, which describes the energy content.

We can expand the left hand side a bit by knowing that

$$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R$$

where $$R_{\mu\nu}$$ is the Ricci tensor, $$R$$ is the Ricci scalar and $$g_{\mu\nu}$$ is the metric tensor. Note that both the Ricci tensor and the Ricci scalar depend entirely on the metric tensor and hence the entire Einstein tensor can be described by just the metric tensor and its derivatives. Now, the conservation of energy and momentum is described by the Bianchi Identities such that

$$\Delta_{\mu}G^{\mu\nu}=0$$ and $$\Delta_{\mu}T^{\mu\nu}=0$$

Now, watch carefully as I pull a cosmological constant out of my hat....

If the LHS (the 'geometric' side) also has an additional term so that is becomes:

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R +\Lambda g_{\mu\nu}$$

then the Bianchi identities, and hence the validity of the solution, are unchanged, since the covariant derivative of the metric tensor vanishes by definition.

The oft repeated adage of GR that "matter tells space how to curve, the curvature of space tell matter how to move" is as applicable whether or not $$\Lambda$$ is zero or not. The equivalence principle or any other principle of relativity is unaffected.

In fact there is an error in suggesting the we can 'add' a CC term to the Einstein tensor. The truth is that mathematically, it is always there we can just choose to set the value of $$\Lambda$$ to zero, if that is what experiment tells us to do.

Now, lets review the history of this little beastie we call $$\Lambda$$. When Einstein first proposed the field equations he thought that the inevitable result of setting $$\Lambda$$ to zero was that the universe would either have to be expanding or contracting, but that it could not be static. Since at the time people believe the Universe was static he set $$\Lambda$$ to be non-zero since this allowed a static, though unstable solution.

Only a few years later Hubble discovered the expansion of the Universe and therefore there was no longer the need to have a static solution, hence the value of $$\Lambda$$ of zero was a better fit and simpler solution for the data as it stood at that time. Einstein suggested it was a blunder on his part to make $$\Lambda$$ non-zero to get a static solution instead of using a zero value to predict the universe should be expanding or contracting.

However the key point about the above was that regardless of the value you think is appropriate for the $$\Lambda$$ term to have, it was always known to be a perfectly valid term from a theoretical point of view. The 'blunder' related to fitting to data, but the existence of the $$\Lambda$$ term is not only allowed but suggested by the form of the Einstein Field Equations.

So, having established the $$\Lambda$$ is not a fudge factor, hack or addition to GR but fundamentally part of the theory lets now look at how dark energy relates to $$\Lambda$$.

We now need to turn to the right hand side of the Einstein Field equation, the stress energy tensor $$T_{\mu\nu}$$ defined as:

$$T^{\mu\nu} = (P + \frac{\rho}{c^2})u^{\mu}u^{\nu} + Pg^{\mu\nu}$$

where P is the pressure and $$\rho$$ is the density of the energy in the universe. We can further simplify this by defining the equation of state of the energy as

$$w = \frac{P}{\rho}$$

This gives us

$$T^{\mu\nu} = \rho(w + \frac{1}{c^2})u^{\mu}u^{\nu} + w\rho g^{\mu\nu}$$

For a homogenous and isotropic and infinite universe, the normalization of the four velocity for the fluid being at rest of

$$u^{\mu}u^{\nu} = diag[c^2,0,0,0]$$

simplifies this to:

$$T^{\mu\nu} = \rho(w+1) + w\rho g^{\mu\nu}$$

where we can just incorporate the factors of $$c^2$$ into the value of $$\rho$$

Now, we could now solve the field equations if we know the density and equation of state of the material in the Universe. The solution we get for this is known as the Friedmann equations. I'm not going to detail this solution as it's not necessary for this discussion, all we need to know is that the scale factor a(t) falls out of this solution and the form of this depends on the density and equation of state of the energy in the universe.

For matter, w=0, for radiation, w=1/3. But what would happen if say the equation of state was w=-1 for a component of the stress energy tensor?

The stress energy tensor for this component would become simply:

$$T^{\mu\nu} = -\rho g^{\mu\nu}$$

Notice now that this is precisely the form of the cosmological constant if we replace the energy density by $$\Lambda$$. If we took it over the other side (the 'geometric' side) of the field equations the solution is exactly the same.

Therefore as far as GR and cosmology is concerned, a non zero $$\Lambda$$ is identical to the presence of an energy with equation of state of -1. The interpretation of this is different, so if the term is on the right hand side we would call it 'vacuum energy' while if it was on the left it is just 'how gravity works', rather than being the effect of some energy.

Particle physicists would care about which side the term was on, since non-zero vacuum energy would need to be explained but a geometric term would leave the standard model of particle physics alone. But cosmologist don't care so much, observationally they are equivalent.

Now, the point about dark energy is that the current data suggests that there is a component of the universe with equation of state close to -1. However it does not have to be exactly -1 given current data. In fact the value may change with time. We call this component dark energy as a generic term, since we do not know the micro physical theory describing it. A specific candidate for dark energy would be something with w exactly =-1 for all time, in which case it might not be an 'energy' at all, just 'how gravity works' (geometric term) or vacuum energy (energy term).

The reason we introduce dark energy at all is because that is what the data says is there, we do not know how to fit the data we have in the framework of GR without having this energy component in the model.

So is Dark Energy a 'fudge factor'? Not at all! Ask yourself what is physics? It is the process of describing observations by mathematical models containing particles, fields and forces to explain what is observed. At one point in time we did not know about magnetism. Then odd things were observed and magnetic fields proposed to explain what was going in. We didn't know about gravity till it was realized that it was needed to explain the observations of the motions of planets. We didn't always think electron, quarks etc etc existed until the data suggested that they must.

This is what dark energy is. It is the prediction of some new physics required in order to match theory with what is observed. Not all new predictions turn out to be true since other things may be proposed that turn out to fit the observations better and provide a coherent theory. Dark energy may one day go the way of spontaneous generation or the fluid theory of heat, if that is what observations and theory tell us.

Thankyou for reading this far! One last point I would like to make is that I find it odd when people point out they are being 'open minded' by refusing to believe what the data is telling us. Of course we need to consider all possible explanations of the data, whether the explanation contains dark energy or not, no one would argue with this. However the data overwhelmingly points to the existence of dark energy. Despite this there are folks whose theoretical prejudice against its existence leads them to scoff at the very idea that it could exist and insist that models explaining observations that work far less well are more reasonable. This is not being open minded, this is in fact being closed minded to the possibility of the new physics implied by dark energy.

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However the key point about the above was that regardless of the value you think is appropriate for the $$\Lambda$$ term to have, it was always known to be a perfectly valid term from a theoretical point of view. The 'blunder' related to fitting to data, but the existence of the $$\Lambda$$ term is not only allowed but suggested by the form of the Einstein Field Equations.
Suggested?
Ok, very well, in the interest of pleasant discussions and courtesy, let's just say then that I lack the intelligence to see that.

Regards,
MJ

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Well 'suggested' may have been a little strong a phrase, the point is that it dosn't change any of the postulates of relativity or whether the Einstein tensor satisfies the formal requirements such as the Bianchi identities. In general the $$\Lambda$$ term is permitted, whether or not it is zero requires looking at the data to answer, you can't predict it a priori.

must be wrong

Since General Relativity has dark mater with galaxies and further dark recipe for big bang, it must be wrong. Where it is wrong? Simple, the Einstein field equation. Since the sole origin of the equation is curved spacetime assumption, the assumption must be wrong!!

a case in point for closed minded theoretical prejudice :zzz:

sorry to see all that work go to waste. it, of course, had no effect on me, but thats because i dont understand it.

whatever idea you were trying to convey/prove, grats to you!

marcus
Gold Member
Dearly Missed
sorry to see all that work go to waste. It doesnt go to waste. It goes into the bank. We have a bunch of information resources here, that quite a few people have worked on and contributed to.
Ultimately it is your only chance at a decent quality forum.

I started an astronomy sticky, or a reference library that one of the mods made into a sticky, that has links. A lot of people contributed. Some of that information is probably still useful, although it hasnt been kept up to date.

Chronos
Gold Member
An elegant explanation by Wallace - If I didn't happen to know he grew up on a sugar beet farm, I would swear he took a few mail order courses in cosmology. The convergence of disparate observations insist upon something that looks very much like a cosmological constant [aka dark energy]. Scientists are not temple priests trying to explain to the 'emperor' why they failed to predict the last solar eclipse. Evidence favoring the 'standard' model bombards them from all directions [e.g., WMAP]. The odds against that being pure coincidence are, using borrowed 'statistical' methodologies, about a google to one.

Garth
Gold Member
Despite this there are folks whose theoretical prejudice against its existence leads them to scoff at the very idea that it could exist and insist that models explaining observations that work far less well are more reasonable. This is not being open minded, this is in fact being closed minded to the possibility of the new physics implied by dark energy.

Thank you Wallace for a very erudite post.

May I make an observation?

It is not just DE that presents a problem for hard science, but also exotic non-baryonic DM and Inflation.

The existence of the Higgs Boson/Inflaton, non-baryonic DM particle and the nature of DE have not yet been verified by laboratory physics even after several decades of intense investigation. Now they may well be discovered next year (always mañana!), perhaps by the LHC, but not until that event and when their properties have been measured and found to be concordant with cosmological constraints, shall we really know what we are talking about.

I agree there have been several cross checks on their existence from cosmological observations: SNe Ia, CMB, large scale structure formation, etc. however these are all dependent on interpretation of the cosmological data and that interpretation is itself model dependent.

Until these entities are discovered in the laboratory it is well to keep an open mind, degeneracies exist in the interpretation of data and we might be mistaken. For example, what will a QG theory bring?

Garth

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Of course Garth, no one would disagree with any of what you have said. However if I may be see bold as to say so, the way you point them out time and time again seems to imply that they are novel insights that 'mainstream' cosmologist do not take seriously or concern themselves with. Nothing could be further from the truth!

Pedagogically, for people coming to forums like these to learn some (in this case) cosmology, it would seem more useful to focus on the successes of the standard model and the particulars of the strong evidence supporting evidence for it. Of course they should also be exposed to competing ideas and the inadequacies and shortfalls of the standard model.

I strongly believe that people are best engaged by first being exposed to the marvelous success of a field of science before they can then understand the nuances of where it is incomplete and what the future may hold. People often come here to these forums questioning something about say dark energy or dark matter that is based on a simple misconception about the current theory, rather than an understanding of what some of the true issues with the standard model are. It is useful for their understanding to explain the misconception and point out the reasons we have arrived at such a strange looking theory. If however they are encouraged to 'keep an open mind' and always 'question' the current paradigm from the get go, I believe it is far less likely they will engage with the field and be more likely to walk away, thinking that they de-bunked cosmology in five minutes and there is nothing else of interest to them in the field.

Of course people should always question any assertion or theory in science, but the point is to empower people to be able to ask good questions, by engaging and informing them about what has been achieved, rather than encouraging bad questions based on misinformation and ignorance.

I'm sure I'm over-reacting and don't mean to offend, but I think this is an important issue. You've been around here a lot longer than I and I'm sure you have helped many more people than I gain a greater understanding through your obvious knowledge and expertise. I just feel that there are better ways to achieve what we are both trying to do than what I sometimes see done.

Garth
Gold Member
Thank you Wallace, be assured I have not taken offence!

Of course, I agree that the standard model should be the one that is taught well to novices coming to the subject, and these Forums, with a genuine desire to learn and understand. That is why I complemented you on your OP.

However, my experience has been that students find the subject more engaging and impressive if presented within the spirit of healthy scientific scepticism.

All the best,

Garth

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i personally found wallace's post most informative with adequate reference to current technological limits to theory verification. i am very very novice but very interested and found this thread most helpful in understanding a bit more about dark energy and einsteins development of certain theories.

Haelfix
Theres a few other ways to think about the cosmological constant.

One way is to look at the Einstein Hilbert Lagrangian and think like a particle physicist.

The inclusion of the cosmological constant term is the most general (leading order) lagrangian consistent with the symmetries of the system, so it satisfies the GellMan principle. Even if you want to make it zero at one scale, upon renormalization group flow, it will tend to be generated at another.

The only way to make it zero and to keep it zero is if you have an additional symmetry that protects it. At this time, we know of no such mechanism.

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Excellent post Wallace, thank you for it.

A question arises... Do you process any of your sugar beets into alcohol fuel?