# De Sitter Vacuum: Is it the Only Positive CC Solution?

• wabbit
In summary, the conversation discusses the implications of a spacetime with zero Weyl curvature and an Einstein tensor proportional to the metric. It is determined that this implies constant scalar curvature and maximally symmetric space. This leads to the conclusion that such a space is conformally equivalent to one of the model constant curvature spaces, such as de Sitter. However, the issue of conformal transformations and isometry is also discussed, with a suggestion for further reading on the topic.
wabbit
Gold Member
Assuming a spacetime with zero Weyl curvature and an Einstein tensor proportional to the metric, is it true that in a finite neighborhood of any point, that spacetime must be isometric to a de Sitter vacuum, or are there other possible solutions, and if so how are they classified?

Thanks

Einstein tensor proportional to the metric implies constant scalar curvature. Then vanishing of the Weyl curvature implies that the space is maximally symmetric ##R_{mnpq} = k ( g_{mp}g_{nq} - g_{mq}g_{np})## and therefore of constant sectional curvature. Therefore such a space is, by a conformal rescaling of the metric, equivalent to one of the model constant curvature spaces. For signature ##(1,d-1)## and positive cosmological constant, this is indeed de Sitter.

wabbit
Thanks - you say "by a conformal rescaling", so it isn't isometric, only conformally equivalent ? I must say conformal transformations isn't something I am really familiar with.

wabbit said:
Thanks - you say "by a conformal rescaling", so it isn't isometric, only conformally equivalent ? I must say conformal transformations isn't something I am really familiar with.

The issue is that the denominator of the formula for sectional curvature involves precisely the same contractions that correspond to the Riemann tensor of a maximally symmetric manifold. So we can rescale ##g' = e^{2\sigma(x)} g## without changing the sectional curvature. With this relation we say that ##g'## is pointwise conformal to ##g##. If there is a diffeomorphism that pulls ##g'## back to ##g##, then we say that the metrics are conformally equivalent and there is a genuine isometry. I think this is a stronger condition than the assumptions warrant.

It is probably overkill and yet might not even answer all questions that you might have, but the most specific reference I know of is Besse, Einstein Manifolds. Some results are discussed in the first few pages of this lecture.

wabbit
Ah, the situation seems more complex than I thought - will check these, thanks for the references.

## 1. What is the De Sitter Vacuum?

The De Sitter Vacuum is a solution to Einstein's equations of general relativity that describes the universe as a flat, empty space with a positive cosmological constant (CC). This means that the vacuum energy density is constant throughout space and time.

## 2. Is the De Sitter Vacuum the only solution to the positive CC problem?

No, there are other proposed solutions to the positive CC problem, such as the quintessence model and the holographic principle. However, the De Sitter Vacuum is considered to be the simplest and most well-supported solution.

## 3. How does the De Sitter Vacuum explain the observed acceleration of the universe?

The De Sitter Vacuum predicts that the expansion of the universe will accelerate due to the repulsive force of the positive CC. This is consistent with observations of the accelerating expansion of the universe.

## 4. Are there any observations or experiments that support the existence of the De Sitter Vacuum?

Yes, the De Sitter Vacuum is supported by many observational data including the cosmic microwave background radiation, the large-scale structure of the universe, and the observed expansion rate of the universe. Additionally, experiments such as the Hubble Space Telescope and the Planck satellite have also provided evidence for the existence of the De Sitter Vacuum.

## 5. Are there any challenges or limitations to the De Sitter Vacuum theory?

There are some challenges and limitations to the De Sitter Vacuum theory, such as the cosmological constant problem which attempts to explain the extremely small value of the CC that is observed. There are also ongoing debates and research regarding the validity of the De Sitter Vacuum and its implications for our understanding of the universe.

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