Schwarzchild metric spherically symmetric space or s-t?

In summary: Therefore, any symmetry we have in spacetime is always spatial, not temporal.In summary, the conversation discusses the question of whether the Schwarzschild metric is spherically symmetric with respect to space or space-time. It is determined that the metric is only symmetric with respect to space because there is no useful notion of three-dimensional rotation in four-dimensional spacetime. The conversation also touches on the concept of maximally symmetric spacetimes and clarifies that symmetry in spacetime is always spatial, not temporal.
  • #1
binbagsss
1,254
11
This is probably a stupid question, but, is the Schwarzschild metric spherically symmetric just with respect to space or space-time?

Looking at the derivation, my thoughts are that it is just wrt space because the derivation is use of 3 space-like Killing vectors , these describe 2-spheres, and ##S^{2}## spheres foliate ##ℝ^{3}##...

Thanks
 
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  • #2
Timelike and spacelike separation are always different, and lightlike motion is different from a "diagonal line" in space. I don't see how anything could have a spherical symmetry in spacetime. The Schwarzschild metric is just symmetric in space.
 
  • #3
Yes, it's just spatial. There is no useful notion of spherical symmetry in the sense of 3-spheres, because there is no useful notion of four-dimensional rotation. The closest thing we have to a rotation mixing space and time is actually a boost, and boosts are not the same as rotations.
 
  • #4
Thanks. And why the 3 spatial dimensions, why not 2 spatial and 1 time?
 
  • #5
binbagsss said:
Thanks. And why the 3 spatial dimensions, why not 2 spatial and 1 time?

Could you clarify what the question is?
 
  • #6
bcrowell said:
Could you clarify what the question is?
Sorry, so we have a notion of 2-spheres that foliate 3-d space, but not 3-spheres that foliate 4-d space; why is the spherical symmetry of the Schwarzschild metric spatial , not, say 2 space dimentions and 1 time dimension?
 
  • #7
The time dimension is different from the spatial dimensions.
Did you ever see anything symmetric in time?
 
  • #8
mfb said:
The time dimension is different from the spatial dimensions.
Did you ever see anything symmetric in time?
Sorry I don't understand the question, Aren't flat space-time and de-sitter space-time maximally symmetric?
 
  • #9
binbagsss said:
we have a notion of 2-spheres that foliate 3-d space, but not 3-spheres that foliate 4-d space

Sure we do; there are 4-d spacetimes (such as closed FRW spacetime) that are foliated by spacelike 3-spheres.
binbagsss said:
why is the spherical symmetry of the Schwarzschild metric spatial , not, say 2 space dimentions and 1 time dimension?

Because "spherical symmetry" means "there is a set of Killing vector fields closed under commutation and with closed integral curves". A spacetime that was "spherically symmetric in time" would have to have a set of such Killing vector fields with one of them being timelike. I'm not aware of any such spacetime, but even if there is one, it certainly is not Schwarzschild spacetime; Schwarzschild spacetime does have a timelike Killing vector field (at least, it does outside the horizon), but that Killing vector field does not have closed integral curves and its commutator with the spherical symmetry Killing vector fields (which are spacelike) is zero.

binbagsss said:
Aren't flat space-time and de-sitter space-time maximally symmetric?

"Maximally symmetric" does not mean "all the dimensions are the same". It means that those spacetimes have the maximum possible number of Killing vector fields (in 4-d spacetime, that number is 10). It does not mean that all of those Killing vector fields are the same as the ones that define spherical symmetry.

If you'll notice, I've thrown a bunch of jargon at you; that was deliberate. If you don't understand the terms I used above, I strongly suggest looking them up and taking the time to understand them. They are critical to a proper understanding of what "symmetry" in general and "spherical symmetry" in particular mean, and I think that understanding will help to answer your questions.
 
  • #10
binbagsss said:
Sorry I don't understand the question, Aren't flat space-time and de-sitter space-time maximally symmetric?
Okay, to be more precise: Did you ever see anything symmetric, but not constant in time?
Or even worse, something that is symmetric in a time/space plane. How would such a concept even look like? Motion through time is always different from motion through space, so you always have a way to break the symmetry.
 

1. What is the Schwarzchild metric?

The Schwarzchild metric is a mathematical formula that describes the gravitational field around a spherically symmetric mass, such as a star or a planet. It was first derived by Karl Schwarzchild in 1916 as part of Albert Einstein's theory of general relativity.

2. What is a spherically symmetric space?

A spherically symmetric space is a three-dimensional space that has the same properties in all directions from a central point. In other words, if you rotated the space around its center, it would look the same from any angle. This is often used to describe the shape of objects in space, such as planets, stars, and galaxies.

3. What does the "s-t" in Schwarzchild metric spherically symmetric space stand for?

The "s-t" in Schwarzchild metric spherically symmetric space stands for Schwarzschild time, which is a coordinate used in the metric to describe the passage of time in a gravitational field. This is important because time can behave differently in the presence of a strong gravitational field, as predicted by Einstein's theory of general relativity.

4. How is the Schwarzchild metric used in astrophysics?

The Schwarzchild metric is used in astrophysics to model the gravitational field around massive objects, such as stars and black holes. It is also used to study the motion of objects in these gravitational fields, and to make predictions about their behavior. The metric has been used to explain phenomena such as gravitational lensing and the precession of Mercury's orbit.

5. Is the Schwarzchild metric spherically symmetric space an accurate representation of reality?

While the Schwarzchild metric has been widely accepted and used in astrophysics, it is not a perfect representation of reality. This is because it assumes a static, non-rotating mass, which is not always the case in the universe. Additionally, the metric does not take into account other factors such as the presence of other masses or the expansion of the universe. However, it is still a useful tool for understanding and studying the effects of gravity on a large scale.

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