# Shape of de Sitter Universe: Is Hyperboloid Misleading?

• I
• gerald V
In summary, the de Sitter universe can be described as a submanifold of a flat space with a line element defined by a constant ##k## which can be positive or negative. While it is often depicted as a hyperboloid, this representation may be misleading and it may be more appropriate to depict it as a 4-sphere. Additionally, the de Sitter Robertson-Walker spacetime is the vacuum solution of the Friedman equations with positive cosmological constant and can have different curvature values. These are three different coordinate charts on the same spacetime, not three different spacetime geometries.
gerald V
TL;DR Summary
How is the de Sitter universe best depicted?
I am confused about the shape of the de Sitter universe. The Misner-Thorne-Wheeler says it can be regarded as the submanifold given by ##-x_1^2 + x_2^2 + x_3^2 +x_4^2 + x_5^2 = k## of a flat space with lineelement ##\mbox{d}s^2 = -\mbox{d}x_1^2 + \mbox{d}x_2^2 + \mbox{d}x_3^2 +\mbox{d}x_4^2 + \mbox{d}x_5^2## (I am aware that there are generalizations with more dimensions as well as with more mixed signs, but this is not my point). ##k## is a constant which can be positive or negative. One oftenly sees figures depicting this de Sitter universe (two dimensions supressed) as a nice hyperboloid. In the following I only regard two degrees of freedom. The line element shall read ##\mbox{d}s^2 = -\mbox{d}x^2 + \mbox{d}y^2##, and the equation ##-x^2 + y^2 = k## shall define a 1-dimensional submanifold. If depicted on a sheet of paper, this equation yields a hyperbola. But a sheet of paper has euclidean symmetry, not pseudoeuclidean. If one wants to take into account the pseudoeuclidean metric of the embedding space, then one has to do the tricks familiar from sketches for Special Relativity, with length contraction and so on. But if one does so, the „hyperbola“ looks like a circle on a sheet of paper, right? What else could it look like?Is my conclusion right? So istn’t the depiction of the de Sitter universe as a hyperboloid completely misleading? Wouldn’t it be more appropriate to depict it as 4-sphere? I am aware that somebody might argue that this question has no answer, because actually there be no embedding space for our universe - what sounds to me a bit like an evasion.

Thank you very much in advance.

Usually the de Sitter Robertson-Walker spacetime is the "vacuum" solution of the Friedman equations with positive cosmological constant. It can have all three curvature values ##k=\pm 1##, ##k=0## of the FLRW metric.

vanhees71 said:
It can have all three curvature values ##k=\pm 1##, ##k=0## of the FLRW metric.
To be clear, these are three different coordinate charts on the same spacetime; they are not three different spacetime geometries that all share the same name ("de Sitter"). It's also worth noting that not all of those charts cover the entire spacetime.

cianfa72 and vanhees71

## 1. What is the shape of the de Sitter Universe?

The de Sitter Universe is a 4-dimensional spacetime with a constant positive curvature, meaning it has a spherical shape.

## 2. Why is the term "hyperboloid" often used to describe the de Sitter Universe?

The term "hyperboloid" is often used because the de Sitter Universe can also be represented as a hyperboloid in 5-dimensional space. This is a mathematical tool used to visualize the universe and does not accurately represent its actual shape.

## 3. How does the shape of the de Sitter Universe differ from the shape of the observable universe?

The shape of the de Sitter Universe is a 4-dimensional spacetime, while the observable universe is the 3-dimensional space that we can observe. Therefore, the shape of the de Sitter Universe is not directly observable.

## 4. What implications does the shape of the de Sitter Universe have on our understanding of the universe?

The shape of the de Sitter Universe has implications on the expansion and future fate of the universe. Its positive curvature suggests that the universe will continue to expand forever and eventually reach a state of maximum entropy.

## 5. Can we ever know for sure the exact shape of the de Sitter Universe?

No, we can never know for sure the exact shape of the de Sitter Universe as it is beyond our observable universe. However, through mathematical models and observations, we can continue to refine our understanding of its shape and properties.

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