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diazdaiz

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In summary, the answer to your question is no, we don't need to propose more than four dimensions to talk about the kind of curvature that is used in relativity.

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diazdaiz

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martinbn

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No, it doesn't mean that.diazdaiz said:

- #3

Ibix

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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.

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jbriggs444

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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 said:I suppose it might have intrinsic curvature but not extrinsic

- #5

Ibix

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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.jbriggs444 said: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.

A spacetime curve is a mathematical representation of the curvature of space and time caused by the presence of massive objects. It is a fundamental concept in Einstein's theory of general relativity.

Mass is the main factor that determines the amount of curvature in spacetime. The more massive an object is, the greater its gravitational pull, and the more it will bend the fabric of spacetime around it.

The 5th dimension is a hypothetical dimension that is often used in theories of physics, such as string theory, to explain the behavior of particles and the structure of the universe. In the context of spacetime curvature, the 5th dimension is thought to play a role in how gravity behaves and how mass affects spacetime.

Spacetime curvature and time dilation are closely related. As an object moves through curved spacetime, its path will be affected by the gravitational pull of massive objects, causing it to experience time at a different rate compared to an observer in a different location. This is known as time dilation.

Yes, spacetime curvature can be observed and measured through various experiments and observations, such as the bending of light around massive objects, the orbit of planets around the sun, and the gravitational waves detected by LIGO. These observations provide evidence for the existence of spacetime curvature and the effects of mass on it.

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