Understanding Gravity with Embedding Diagrams

[JAM-OTBOC]

A help to understand how gravity works in our universe is called an embedding diagram. It is a three-dimensional representation of what happens to a two-dimensional universe in the presence of mass, i.e. the plane is distorted into a 'gravity' pit.

What happens if we extend this concept to attempt to diagram the same thing for our three-dimensional universe? Does this imply that our 3-D universe is 'embedded' within a higher dimensional space?

The metric tensor used to describe the motion of an object in the presence of mass makes use of a multi-dimensional Reimannian Space. This (as I understand it) is necessary in order to explain the distortion of our three dimensions according to general relativity due to the presence of mass. Into what does our 3D universe distort according to this law? The two dimensional plane needs our 3D universe in order to deform to show gravitation in that realm. What kind of universe is required to explain the distortion of our 3D universe? Could this be some kind of Hyperspace? If so what are it's properties?

 

[bcrowell]

The metric tensor used to describe the motion of an object in the presence of mass makes use of a multi-dimensional Reimannian Space. This (as I understand it) is necessary in order to explain the distortion of our three dimensions according to general relativity due to the presence of mass.

No, the metric tensor does not refer to any additional dimensions. We live in a 3+1-dimensional universe (three spatial dimensions and one time dimension). The equations of general relativity only refer to these four dimensions.

It is possible to *model* a Riemannian space by embedding it in a higher-dimensional space. That doesn't mean the extra dimension is mathematically necessary or physically real. This may be helpful: http://www.lightandmatter.com/html_books/genrel/ch03/ch03.html#Section3.2

 

[JAM-OTBOC]

In a paper by Lu and Suen entitled 'Extrinsic Curvature Embedding Diagrams' the abstract makes the statement: "Embedding diagrams have been used extensively to visualize the properties of curved space in Relativity. We introduce a new kind of embedding diagram based on the extrinsic curvature (instead of the intrinsic curvature). Such an extrinsic curvature embedding diagram, when used together with the usual kind of intrinsic curvature embedding diagram, carries the information of how a surface is embedded in the higher dimensional curved space. Simple examples are given to illustrate the idea." What do these authors mean by higher dimensional curved space?

 

[JAM-OTBOC]

Here is a quote from a topologist:
Embedding refers to how a topological object—a graph, surface, or manifold—is positioned in space. The concept of embedding is central to the idea of an extrinsic view of topology simply because we cannot view something from the outside unless it is somehow situated in some larger, or higher-dimensional, space. Otherwise, from where would we be viewing it? Furthermore, there can be many different ways to embed an object in that larger space. Comments?

 

[ZapperZ]

In a paper by Lu and Suen entitled 'Extrinsic Curvature Embedding Diagrams' the abstract makes the statement: "Embedding diagrams have been used extensively to visualize the properties of curved space in Relativity. We introduce a new kind of embedding diagram based on the extrinsic curvature (instead of the intrinsic curvature). Such an extrinsic curvature embedding diagram, when used together with the usual kind of intrinsic curvature embedding diagram, carries the information of how a surface is embedded in the higher dimensional curved space. Simple examples are given to illustrate the idea." What do these authors mean by higher dimensional curved space?

When you cite a paper, you need to not only cite the name of the first author, but also the journal, volume, page (or article) number, and year. In other words, follow the format of a typical journal.

Here is a quote from a topologist:
Embedding refers to how a topological object—a graph, surface, or manifold—is positioned in space. The concept of embedding is central to the idea of an extrinsic view of topology simply because we cannot view something from the outside unless it is somehow situated in some larger, or higher-dimensional, space. Otherwise, from where would we be viewing it? Furthermore, there can be many different ways to embed an object in that larger space. Comments?

This, typically, is NOT a valid source unless you can make complete citation. After all, no one can tell if you just made this up. If you can't make exact citation to established sources, then be prepared to have your reference to not be considered as valid.

Zz.

 

[JAM-OTBOC]

Was not in a journal - Title: "Extrinsic Curvature Embedding Diagrams" J. L. Lu and W. M. Suen to be submitted to Physical Review D, Feb 7 2008.

 

[George Jones]

Was not in a journal - Title: "Extrinsic Curvature Embedding Diagrams" J. L. Lu and W. M. Suen to be submitted to Physical Review D, Feb 7 2008.

According to the arXiv,

http://arxiv.org/abs/gr-qc/0301012,

the paper was published in Gen.Rel.Grav. 35 (2003) 1175-1189, and, sure enough,

http://www.springerlink.com/content/k6ht568041163425/.

 

[hamster143]

Here is a quote from a topologist:
Embedding refers to how a topological object—a graph, surface, or manifold—is positioned in space. The concept of embedding is central to the idea of an extrinsic view of topology simply because we cannot view something from the outside unless it is somehow situated in some larger, or higher-dimensional, space. Otherwise, from where would we be viewing it? Furthermore, there can be many different ways to embed an object in that larger space. Comments?

Extrinsic view of topology is just one possible view, it is not necessary to do GR.

Humans have this special relationship with flat spaces: our brains are wired to see the universe as flat, and our common sense works particularly well with flat spaces. That's the only justification for embedding. We can get all results by treating our universe as lots of locally flat chunks of space, stitched together in such a way as to create curvature at the macroscopic level. We don't need to worry if, how or where this space is embedded.

 

[JesseM]

Here is a quote from a topologist:
Embedding refers to how a topological object—a graph, surface, or manifold—is positioned in space. The concept of embedding is central to the idea of an extrinsic view of topology simply because we cannot view something from the outside unless it is somehow situated in some larger, or higher-dimensional, space. Otherwise, from where would we be viewing it? Furthermore, there can be many different ways to embed an object in that larger space. Comments?

Assuming the quote is valid, note that the topologist is referring specifically to the "extrinsic view of topology"--I don't think he's denying that you can also deal with curved surfaces just fine using a purely intrinsic perspective. Look at the brief discussion of intrinsic vs. extrinsic in wikipedia's page on differential geometry...general relativity deals with spacetime curvature in a purely intrinsic way, so it does not require any external embedding space (or embedding spacetime) although it may help with intuitions to figure out how curved space or spacetime could be embedded in a higher-dimensional flat space/spacetime in such a way that the curvature would match what's predicted from the intrinsic perspective.

 

[JAM-OTBOC]

Extrinsic view of topology is just one possible view, it is not necessary to do GR.

Humans have this special relationship with flat spaces: our brains are wired to see the universe as flat, and our common sense works particularly well with flat spaces. That's the only justification for embedding. We can get all results by treating our universe as lots of locally flat chunks of space, stitched together in such a way as to create curvature at the macroscopic level. We don't need to worry if, how or where this space is embedded.

Oh but some of us are.