High School Spacetime Curve: Mass Effects & 5th Dimension

[diazdaiz]

i am new at relativity, it said mass can curve spacetime, does this mean spacetime will curve to a new 5th dimension (1-3 for space dimension, 4 for time dimension)?

 

[martinbn]

i am new at relativity, it said mass can curve spacetime, does this mean spacetime will curve to a new 5th dimension (1-3 for space dimension, 4 for time dimension)?

No, it doesn't mean that.

 

[Ibix]

There are two types of curvature, intrinsic and extrinsic curvature.

Intrinsic curvature can be measured without reference to an external space - for example, the intrinsic curvature of the surface of the Earth can be detected by drawing a large triangle and noting that the angles don't sum to 180°.

On the other hand, extrinsic curvature requires a space to be embedded in a higher-dimensional space. For example if you take the surface of a cylinder, this is curved in the sense that straight lines in 3d space that touch the surface of the cylinder don't necessarily stay touching the cylinder.

A surface can have extrinsic curvature but not intrinsic curvature (for example, the surface of the cylinder has no intrinsic curvature - triangles drawn on it have angles that sum to 180°). I suppose it might have intrinsic curvature but not extrinsic (can't think of an example, though). Or it might have both.

In relativity, we only care about intrinsic curvature. As far as we know, the universe is not embedded in a higher dimensional space, so it doesn't make sense to talk about extrinsic curvature. So, as martinbn says, the answer to your question is no. We have no need to propose more than four dimensions to talk about the kind of curvature that is used in relativity.

 

[jbriggs444]

I suppose it might have intrinsic curvature but not extrinsic

Bit of a contrived example, but consider the surface of a hemisphere. Project this surface vertically onto its equatorial plane. Inherit the distance metric from the original hemisphere to judge "straight lines" in the resulting space. It now has intrinsic but not extrinsic curvature.

 

[Ibix]

Bit of a contrived example, but consider the surface of a hemisphere. Project this surface vertically onto its equatorial plane. Inherit the distance metric from the original hemisphere to judge "straight lines" in the resulting space. It now has intrinsic but not extrinsic curvature.

Or the other way around, I guess. Embed the manifold in a higher dimensional manifold whose metric is contrived to match that of the embedded manifold where appropriate.