Does the Curvature of 4-D Space-Time Extend into a Fifth Dimension?

[engstudent321]

I have ben doing some reading on general relativity but I can not find an answer to my question.

If a curved 2-D surface (like the surface of a sphere) must curve into a third dimension, does the curvature of 4-D space-time 'extend' into some fifth dimension? Maybe I am missing something simple but if anyone could clarify this, it would be great.

Thanks.

 

[StatusX]

You're thinking of a surface that's embedded into flat three dimensional space. While this is a perfectly good way to talk about surfaces, and was indeed the way they were first studied, it's possible to talk about surfaces abstractly, without having any such embedding in mind, by simply giving the distance between any two points. Although a little hard to visualize, this is enough to define things like curvature, and in a sense, is more natural.

(If you think about it, the only reason we can visualize flat space so easily is because our brains are built that way (because we evolved in space that is flat to a very good approximation, etc, etc). Mathematically, we shouldn't expect flat space to be any more "real" than curved space, and so both should be defined independently.)

In any case, GR is most easily formulated without reference to some ambient flat spacetime. Although we could think of these curved spacetimes as being embedded in some higher dimensional flat spacetime, this would lead both to more complicated equations and a redundancy of description, since there are many ways to embed the same space. Incidentally, there is a theorem that it takes up to 2n-dimensional flat space to embed a general curved n-dimensional space, so 5 wouldn't be enough.

 

[shoehorn]

Hi engstudent, welcome to PF.

If a curved 2-D surface (like the surface of a sphere) must curve into a third dimension...

There's your problem right there. A surface does not need to be embedded in a higher-dimensional space in order to have curvature. In geometry there are two broad types of "curvature." The first is called "extrinsic" curvature and arises as a result of a geometric object existing in some higher-dimensional space. As an example, take a piece of string and hold a point in the middle of the string between your thumb and forefinger. The endpoints of the string will be lower than the point at which you're holding the string, and it's pretty obvious that the string "curves" around the point you're holding. This is called extrinsic curvature.

The second type of curvature is called "intrinsic," and is a type of curvature that a geometric object has regardless of whether it's embedded in a higher-dimensional space. To go back to our example of a string, a surprising fact is that while the string can have an extrinsic curvature if it exists in two or more dimensions, it cannot have an intrinsic curvature; this is a general property of one-dimensional objects, of which a piece of string is a good example.

To the broader question of whether our spacetime must exist in a higher-dimensional space in order for it to be curved, the answer is no. Standard general relativity asserts that our universe is a four-dimensional spacetime and that nothing exists outside this spacetime. In particular, there isn't (according to empirically tested theories such as GR) a "higher dimensional space" in which our universe exists. Thus, from the point of view of GR, spacetime cannot have extrinsic curvature. It can, however, have intrinsic curvature, a property usually described in terms of a mathematical object called a Riemann tensor.

If you're interested, a recent thread asked a related question. There's also a slightly older thread that discusses cosmological implications of this. A word of warning though: in the first thread you should ignore anything said by MeJennifer and A.T. since their claims are wrong. There are lots of decent contributions to the second thread, although you'd be well advised to ignore anything said there by Robert100 (and anyone who agrees with him) since, once again, they're wrong in what they claim.

 

[engstudent321]

Thanks a lot shoehorn. That answered my question.

 

[logboi]

what do u think about time and space?

 

[gravity guru]

instead of flat space wouldn"t it be better to look at space as being homogenous and mass is strechting it to a point. gravity guru.

 

[gravity guru]

ok. i answer myself at home too, i see gravity as an active force not a passive one, it seems to me the best way to gather is to make a physical action with the object to be gathered and the action must be invisible ,flexible and originate from a physical object to fit the observation of gravity. when it"s between an electron and a proton at some distance from each other there is one thing that fills the bill fine and that"s a single open end magnetic line that travels 186,000 mps. north line seeking any south pole. gravity guru.