I Shape of de Sitter Universe: Is Hyperboloid Misleading?

  • I
  • Thread starter Thread starter gerald V
  • Start date Start date
  • Tags Tags
    Shape Universe
gerald V
Messages
66
Reaction score
3
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.
 
Physics news on Phys.org
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.
 
  • Like
Likes cianfa72 and vanhees71
In this video I can see a person walking around lines of curvature on a sphere with an arrow strapped to his waist. His task is to keep the arrow pointed in the same direction How does he do this ? Does he use a reference point like the stars? (that only move very slowly) If that is how he keeps the arrow pointing in the same direction, is that equivalent to saying that he orients the arrow wrt the 3d space that the sphere is embedded in? So ,although one refers to intrinsic curvature...
ASSUMPTIONS 1. Two identical clocks A and B in the same inertial frame are stationary relative to each other a fixed distance L apart. Time passes at the same rate for both. 2. Both clocks are able to send/receive light signals and to write/read the send/receive times into signals. 3. The speed of light is anisotropic. METHOD 1. At time t[A1] and time t[B1], clock A sends a light signal to clock B. The clock B time is unknown to A. 2. Clock B receives the signal from A at time t[B2] and...
So, to calculate a proper time of a worldline in SR using an inertial frame is quite easy. But I struggled a bit using a "rotating frame metric" and now I'm not sure whether I'll do it right. Couls someone point me in the right direction? "What have you tried?" Well, trying to help truly absolute layppl with some variation of a "Circular Twin Paradox" not using an inertial frame of reference for whatevere reason. I thought it would be a bit of a challenge so I made a derivation or...

Similar threads

Replies
3
Views
1K
Replies
35
Views
4K
Replies
1
Views
2K
Replies
1
Views
1K
Replies
42
Views
6K
Back
Top