# Is space time curvature and absolute concept?

#### kiwiheretic

Seeing we hear gravity warps space and time and now it seems even a rotating disk seems to warp the circumference of the disk relative to the radius and wondering whether an observer rotating with the centre of the disk would see this cirumcumference contraction; So would these distortions/bends/hills/valleys/etc be absolute or is it possible one observer might look at a surface and one person might see a giraffe and another person a hippopotamus?

#### Pmb

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Seeing we hear gravity warps space and time ...
That's not quite right. To begin with, you'll find that the term "curves" rather than "warps" is the term most commonly found in the GR literature. Second, its only correct to say that matter can curve spacetime. Spacetime curvature is the geometric term in GR for tidal gradients in a gravitational field.

...and now it seems even a rotating disk seems to warp the circumference of the disk ..
Not at all. Contraction is not the same thing as spacetime curvature. It's merely a shortening of the circumference and has nothing to do with curvature.

..relative to the radius and wondering whether an observer rotating with the centre of the disk would see this cirumcumference contraction; So would these distortions/bends/hills/valleys/etc be absolute or is it possible one observer might look at a surface and one person might see a giraffe and another person a hippopotamus?
Spacetime curvature is an inherent property of spacetime and no change in coordinates can create it. It's literally impossible for one observer to observe spacetime curvature and not another.

1 person

#### studiot

Is space time curvature and absolute concept?
No it is a local concept on a manifold.

Properties of manifolds come in two varieties.

Global or absolute or extrinsic which are determined by the embedding space and local or intrinsic which can be determined by properties of the points in the manifold alone.

A simple example of an intrinsic property is the curvature which can be determined from the properties (coordinates) of successive points along the line itself, without reference to whether the line exists in 2 space or 3 space or n space.

#### Pmb

PHF Hall of Fame
No it is a local concept on a manifold.
That is incorrect. While its true that curvature is a local property of a manifold, it has a definite curvature at every single point on a manifold and its either zero or non-zero, a property which is independent of the coordinate system. And its for that reason I said that its an absolute property of the manifold itself. And by this I mean how these terms are defined in the mathematical literature and the physics literature, which includes the GR literature.

I believe that you may be confusing local flatness with global flatness. A manifold can have a curvature which is non-zero at a specific point on a manifold. However, if one restricts oneself to observations in a small coordinate system about that point and those properties cannot be measured with the instruments being used then that coordinate system the manifold is said to be locally flat. The "Bible of GR" i.e. Gravitation by Misner, Thorne and Wheeler, has beautiful section on this very point. To determine whether the manifold is curved at that point one merely calculates the Riemann tensor; if it vanishes (e.g. all of the components are zero) at that point then the curvature is zero, otherwise the manifold is curved.

The way to local flatness is defined mathematically in relativity is that its always possible to find a coordinate system such that the interval ds^2 is

ds^2 = dt^2 - dx^2 - dy^2 - dz^2

If you're not familiar with the interval, which is defined by the metric tensor, see:
https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)

You'll need an understanding of tensors to be able to understand it all but you might be able to dig out some understanding now.

Properties of manifolds come in two varieties.

Global or absolute or extrinsic which are determined by the embedding space and local or intrinsic which can be determined by properties of the points in the manifold alone.
That is not a property of a manifold. It's a property of the union of the embedding space and the manifold. And its not called global curvature. The terms are intrinsic and extrinsic. Thus, for example, the surface of a cylinder in 3-space has zero intrinsic curvature but non-zero extrinsic curvature. The surface of the cylinder is a two dimensional manifold and its embedded in the manifold 3-space. But 3-space and the surface of the cylinder are not the same manifolds.

A simple example of an intrinsic property is the curvature which can be determined from the properties (coordinates) of successive points along the line itself, without reference to whether the line exists in 2 space or 3 space or n space.
That's nota very good example because a line is a 1-dimensional manifold and a manifold must have at least or more dimensions in order to be able to define intrinsic curvature.

Kiwi - The term manifold, as its being used here, is just a fancy term used to describe a set of points with certain properties. Relativity utilizes only differentiable manifolds which are defined at

https://en.wikipedia.org/wiki/Differentiable_manifold

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#### kiwiheretic

I guess a differentiable manifold is rougly equivalent to a surface being continuously differentiable in calculus.

#### Pmb

PHF Hall of Fame
I guess a differentiable manifold is rougly equivalent to a surface being continuously differentiable in calculus.
As I recall, that's almost exactly what it means.

#### studiot

PMB
That is incorrect. While its true that curvature is a local property of a manifold, it has a definite curvature at every single point on a manifold and its either zero or non-zero, a property which is independent of the coordinate system. And its for that reason I said that its an absolute property of the manifold itself. And by this I mean how these terms are defined in the mathematical literature and the physics literature, which includes the GR literature.
The only purpose of this paragraph seems to be to contradict me, but instead it only seems to contradict itself (within the first 6 words).

I am sorry you are having personal medical problems, but being grumpy doesn't help communication here.

I have no intention of getting into a futile slanging match with you and I'm sure your Physics is rock solid. However tripping through the minefield of mathematics definitions requires extreme care as almost nothing is as it seems.

A mathematical manifold is nothing more than a set of elements or points which share a common property or collection of properties.
It doesn't have to be all the items in the universe with those properties.
It doesn't have to have any structure, but it may do. (That is there need be no relations between the individual elements and no operations such as addition defined between the elements)
But each and every element much possess or be attributable to the common property of interest.

So if the common property of interest is redness then the set

{Ferrari F1 car, a tomato, Mars} is a manifold.

Not very interesting, mathematically but a set of red tomatoes might be of mathematical interest to a horticulturalist comparing redness statistically.

kiwiheretic
I guess a differentiable manifold is rougly equivalent to a surface being continuously differentiable in calculus.
PMB
As I recall, that's almost exactly what it means.
A good perceptive question and answer, which brings me to topological manifolds, which allow us to do the very desirable operations of calculus on a manifold.

The common property of a topological manifold is that every element of a topological manifolds is homeomorphic to the interior points of a sphere (otherwise called an open ball) in a Euclidian space.

This means that each point has neighbours or that the manifold may be divided into disjoint open subsets. The word open is very important.

Formally a manifold M is an n dimensional topological manifold if there is locally a finite open cover Z that maps each point in M onto an open subset of R(^n).
The pair (M, Z) is a C(^r) manifold if Z is a C(^r) differential structure.

So we have the basis of our calculus.

I think that a one dimensional manifold is an excellent and easy to understand example, compared to the above (simplified) formal definition.

In fact it is so simple that it has been used for hundreds of years by surveyors under the guise of 'Through Chainage'.
We also used it in Kiwiheretic's recent question about Lagrangians and angles.
It is easy to see that there is no global function that will give you the circular curve connecting two roads which meet at angle theta in one dimension. You can determine the curvature from points on the connecting curve, but not from points on the straights.

Turning to two dimensional manifolds,
Naval architects and sheet metal workers have used the concept of ruled surfaces for centuries and more recently civil engineers have used this to construct the iconic cooling towers in power stations.

The curved surface of your cylinder (without the ends) is a ruled surface because it is isomorphic to the plane. The surface of a sphere is not so isomorphic as any cartographer will tell you.
This isomorphism means that you can rule straight lines on a piece of paper or card and then roll it into a cylinder with the lines parallel to the axis remaining straight.

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