Into what dimension is Spacetime Curved?

[PhanthomJay]

I think I may have asked this question a few years ago, but I forget the responses.

We know that gravity is the curvature of spacetime in the presence of mass and energy.. The curvature of spacetime was proved by experiment during a solar eclipse, whereby light from a star behind the sun was nevertheless observable due to its bending (curving) caused by the sun's mass.

Question: Assuming the above statements are true, was the curvature still into the 3rd spatial dimension, or was the curvature into a higher order spatial dimension? What I mean is that for a 2D surface of a sphere, the path followed by a particle or photon along its surface is curved into the 3rd dimension. Why would not 4D spacetime curvature be into the 5th dimension? Balloon analogy, anyone?

 

[alexg]

The curvature is in the three dimensions, making space/time non-Euclidean.

 

[PhanthomJay]

The curvature is in the three dimensions, making space/time non-Euclidean.

Oh, thanks...suppose the star was known to be behind the sun located near the tangent of its surface at its right most edge..would the curvature be to the right of the sun, above the sun, or below the sun?

 

[alexg]

Oh, thanks...suppose the star was known to be behind the sun located near the tangent of its surface at its right most edge..would the curvature be to the right of the sun, above the sun, or below the sun?

The left.

 

[PhanthomJay]

The left.

Why? If the curvature was relatively small, it would still be hidden, no?

 

[Nabeshin]

What I mean is that for a 2D surface of a sphere, the path followed by a particle or photon along its surface is curved into the 3rd dimension. Why would not 4D spacetime curvature be into the 5th dimension? Balloon analogy, anyone?

This is a limitation of our ability to visualize things, nothing more. When we imagine a 2D surface of a sphere, we automatically embed it into a 3-dimensional universe. The path taken by an object on that surface is still completely two dimensional. Indeed, to describe it, we need to make absolutely NO reference to a third dimension at all. We say that the surface has an intrinsic property, known as curvature, meaning that it is simply something about the surface which is known without reference to outside entities (in other dimensions).

So too is it the case with our 4D universe. To describe the curvature, we do NOT need to appeal to a 5th dimension through which to curve it.

 

[PhanthomJay]

This is a limitation of our ability to visualize things, nothing more.

Well, yes, but seeing is sometimes believing. When the light from that star was observed, surely it appeared on one side or the other.. supposing a star was located light years away directly behind the center of the sun when observed from Earth. If spacetime curvature was large enough, would its pinpoint of light be observed to the right, left, top, or bottom of the sun? Why would one geodesic path be favored over the other?

 

[Cosmo Novice]

Well, yes, but seeing is sometimes believing. When the light from that star was observed, surely it appeared on one side or the other.. supposing a star was located light years away directly behind the center of the sun when observed from Earth. If spacetime curvature was large enough, would its pinpoint of light be observed to the right, left, top, or bottom of the sun? Why would one geodesic path be favored over the other?

Essentially if it was behind the center you would have gravitational lensing except the Sun is probably not massive enough to affect photons in such a massive way and is far too bright. The mechanism would be the same as a black hole gravitationally lensing a distant galaxy.

 

[Drakkith]

Well, yes, but seeing is sometimes believing. When the light from that star was observed, surely it appeared on one side or the other.. supposing a star was located light years away directly behind the center of the sun when observed from Earth. If spacetime curvature was large enough, would its pinpoint of light be observed to the right, left, top, or bottom of the sun? Why would one geodesic path be favored over the other?

If the Sun were much more massive and the star was "exactly" in the right spot, the light would appear as a ring around the sun.

 

[alexg]

Why?

Because you gave me three choices so I went with the fourth.

 

[PhanthomJay]

If the Sun were much more massive and the star was "exactly" in the right spot, the light would appear as a ring around the sun.

That makes perfect sense, thank you for the response!

 

[DaveC426913]

That makes perfect sense, thank you for the response!

Search Einstein Rings.

 

[alexg]

As it is, the sun will deflect a light beam passing close by 1.75 arc seconds.

 

[Imax]

Hi PhanthomJay:

We know that gravity is the curvature of spacetime in the presence of mass and energy. The curvature of spacetime was proved by experiment during a solar eclipse, whereby light from a star behind the sun was nevertheless observable due to its bending (curving) caused by the sun's mass.

Question: Assuming the above statements are true, was the curvature still into the 3rd spatial dimension, or was the curvature into a higher order spatial dimension?

Spacetime can have a curvature near massive objects like our sun. It's not necessary to embed 4D spacetime in higher dimensions, but sometimes it's conceptually easier to imagine what could happen to 2D space when looking at it as a 3D sphere (i.e. the balloon analogy). Trying to embed a 4D spacetime in a higher dimension would mean that you need 5D, but I have enough problems imagining what could happen in 3D ?

Einstein looked at spacetime as a fabric, and that fabric could bend around large masses like our sun. Such masses can induce curvatures in 4D spacetime, including curvatures in 3D space and curvatures in time.

 

[twofish-quant]

Curvature is a metaphor.

I draw a triangle and measure it's area. It turns out to be 1/2*base*height. Then one morning I get up and I draw a triangle, and measure it's area, and it turns out that it is more or less than 1/2*base*height.

By drawing some triangles, I figure out that they behave "as if" a triangle on a sphere would behave so I call them "curved." The weird part is that the triangles don't actually have to be on a sphere. It could be that I'm somewhere that triangles are just weird.

So "curvature" is a metaphor. It's like saying that someone has a "cold personality" or that "sales of toasters are hot."

 

[alt]

Curvature is a metaphor.

I draw a triangle and measure it's area. It turns out to be 1/2*base*height. Then one morning I get up and I draw a triangle, and measure it's area, and it turns out that it is more or less than 1/2*base*height.

By drawing some triangles, I figure out that they behave "as if" a triangle on a sphere would behave so I call them "curved." The weird part is that the triangles don't actually have to be on a sphere. It could be that I'm somewhere that triangles are just weird.

So "curvature" is a metaphor. It's like saying that someone has a "cold personality" or that "sales of toasters are hot."

Wow! Thanks for putting it so succinctly. And, well .. so honestly!

 

[Imax]

Curvature is not necessarily a metaphor. See for example http://en.wikipedia.org/wiki/Riemannian_geometry or
http://en.wikipedia.org/wiki/Riemannian_manifold

 

[twofish-quant]

Curvature is not necessarily a metaphor. See for example http://en.wikipedia.org/wiki/Riemannian_geometry or
http://en.wikipedia.org/wiki/Riemannian_manifold

There are several different ways of looking at it. The way that mathematicians look at it "curvature" is this abstract thing with a precise mathematical definition which includes "curvature" on a objects in flat space.

However, this precise definition is much more general than when layman talk about "curvature".

The way that you tell if something is "curved" is that you point in a direction, you walk in a circle and then see what direction you are pointing in. If you are pointing in the same direction then the region of space that you are walking in is "flat". If not, they you are walking in "curved" space.

Now *why* space is curved is another issue. It could be that space is "curved" because you are stuck on a surface that is embedded in a "flat" space. Or to could be that you just happen to be in a region of space in which distances and angles just behave weird.

 

[Imax]

The way that you tell if something is "curved" is that you point in a direction, you walk in a circle and then see what direction you are pointing in. If you are pointing in the same direction then the region of space that you are walking in is "flat". If not, they you are walking in "curved" space.

If you take a vector and you move that vector around a loop and you end up with that vector pointing in the same direction at the point where you started from, then you’re in a space that is orientable (i.e. Minkowski space). This may not be an experiment that can explain spacetime curvature.

One experiment that can give some insight into space-time curvature is to look at a pulsar as it moves behind our Sun. For a regular pulsar, the interval between pulses should vary with time, if time has a curvature.

 

[alt]

If you take a vector and you move that vector around a loop and you end up with that vector pointing in the same direction at the point where you started from, then you’re in a space that is orientable (i.e. Minkowski space). This may not be an experiment that can explain spacetime curvature.

One experiment that can give some insight into space-time curvature is to look at a pulsar as it moves behind our Sun. For a regular pulsar, the interval between pulses should vary with time, if time has a curvature.

That seems to bring it back then, to the original question. Into what dimension is spacetime curved ?

And now that I mention spacetime, I seem to recall a thread here some months ago, from a mentor no less, to the effect that 'Spacetime is a fantasy' (or a fairytale or something like that). I will try to find it, though I fear I probably won't understand it.

 

[Imax]

One way of looking at the Universe is to think of it like a Compact Laurentian Manifold. Within this framework, there is no need for expansion into a higher dimension. The Universe doesn’t need to be static, but it can expand or contract. A Compact Laurentian Manifold can allow closed space like and closed time like curves, which may be what is happening near black holes. The Universe doesn’t need to be simply connected.

 

[alt]

One way of looking at the Universe is to think of it like a Compact Laurentian Manifold. Within this framework, there is no need for expansion into a higher dimension. The Universe doesn’t need to be static, but it can expand or contract. A Compact Laurentian Manifold can allow closed space like and closed time like curves, which may be what is happening near black holes. The Universe doesn’t need to be simply connected.

I googled Compact Laurentian Manifold but got nothing. So I am none the wiser, though I appreciate your effort.

 

[Imax]

Oops, my bad.

It’s “Compact Lorentzian Manifold.”

 

[Nabeshin]

I googled Compact Laurentian Manifold but got nothing. So I am none the wiser, though I appreciate your effort.

The essential point in what Imax is trying to say is the same as I expressed in my earlier post in this thread. Spacetime, i.e. our universe, can be described mathematically as a four-dimensional surface. With only these four dimensions (t,x,y,z), we can also describe fully the phenomenon of 'curvature', which, as has also been expressed in this thread, you can think of simply as angles in triangles not adding to 180 degrees, or the area of circles not being what you might think, etc. The punch line is that curvature is a property INTRINSIC to the surface, meaning you do not need to reference anything outside the surface (another dimension) in order to describe this. So the answer to the question "Into which dimension does spacetime curve?" is that "Spacetime does not curve INTO any dimension, but is itself simply intrinsically possesses the quality we call curvature."