# Spacetime metric?

1. May 8, 2007

### delta001

I'm a layman here, so please put any answers in terms that a layman can understand. You can use calculus though :)

What is the spacetime metric, and what are the equations describing it?

2. May 8, 2007

### MeJennifer

A metric is simply a definition on how distances are measured in a space.

3. May 8, 2007

### delta001

I know that, like the pythagorean theorem in 2D space, but what is the theory behing the metric of the spacetime, like quantum foam and whatnot.

4. May 8, 2007

### delta001

Also, what is a tensor?

5. May 8, 2007

### delta001

And a manifold, what's that?

So many questions!

6. May 8, 2007

### neutrino

Well, as to post #4, there are a couple of threads in this very forum (probably with the same or similar words as its title).

7. May 8, 2007

### cristo

Staff Emeritus
The answers to these questions you pose will not mean anything to you if you don't know any higher maths. Sorry to be blunt but its true. If there are any more specific questions feel free to ask, but these are far too broad to give you a meaningful answer.

8. May 8, 2007

### delta001

I know calculus, like I said...?

9. May 8, 2007

### cristo

Staff Emeritus
Are you reading something in particular and trying to understand it? I can give you a definition of a manifold, but I doubt it'll be of much use to you:

10. May 8, 2007

### yenchin

Intuitively (over-simplified of course), a manifold is a generalization of a surface, i.e. a higher dimensional "surface", but it need not be sitting inside some even higher dimensional R^n.

How much of Calculus do you know?

11. May 8, 2007

### pmb_phy

The metric is a function which maps two vectors to a scalar. If the two vectors are a displacement (then the vectors are identical) and the scalar has the value of the spacetime interval. In Euclidean geometry this would be called the "distance" between two points. The distance between two points on a manifold is given by

$ds^2 = \eta_{\alpha\beta} dx^{\alpha} dx^{\beta}$

The $\eta_{\alpha\beta}$ are the components of the metric. In geometrical notaton this is given by

$ds^2 = \eta(dV,dV)$ where dV is a displacement vector.

http://www.geocities.com/physics_world/gr_math/geo_tensor.htm
http://www.geocities.com/physics_world/ma/intro_tensor.htm

The spacetime metric also defines the scalar product between two vectors A and B as

A*B = $\eta(A,B)$

The general expression is

A*B = g(A,B)

which holds in all coordinate systems, not just in Lorentzian coordinates which the $\eta$ denotes

Best wishes

Pete

Last edited: May 9, 2007
12. May 8, 2007

### pmb_phy

This is my own personal opinion so please feel free to ignore it: In my opinion the term distance shouldn't be used to describe what the metric measures. If people use this definition then they are likely to use notions like "moving through spacetime". The idea of motion may give the reader the wrong idea of something down the road.

Best wishes

Pete

13. May 8, 2007

### Hurkyl

Staff Emeritus
For your information, there are two senses in which the word "metric" is used. MeJennifer described one usage, and pmb_phy described the other. (Which, technically, is a "metric tensor", although it's typical to simply call it a "metric")

Minkowski space doesn't have a metric; just a metric tensor.