Spacetime metric?

In summary: 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 byds^2 = \eta_{\alpha\beta} dx^{\alpha} dx^{\beta}The \eta_{\alpha\beta} are the components of the metric. In geometrical notaton this is given bydS^2 = \eta(dV,dV) where dV is a displacement vector.For more information
  • #1
delta001
20
0
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?
 
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  • #2
A metric is simply a definition on how distances are measured in a space.
 
  • #3
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.
 
  • #5
And a manifold, what's that?

So many questions!
 
  • #6
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
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
I know calculus, like I said...?
 
  • #9
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:
A set of points M is defined to be a manifold if each point in M has an open neighbourhood which has a continuous one-to-one map onto an open set of Rn for some n
 
  • #10
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
delta001 said:
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?
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

[itex]ds^2 = \eta_{\alpha\beta} dx^{\alpha} dx^{\beta}[/itex]

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

[itex] ds^2 = \eta(dV,dV)[/itex] where dV is a displacement vector.

For more information please see my website at
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 = [itex]\eta(A,B)[/itex]

The general expression is

A*B = g(A,B)

which holds in all coordinate systems, not just in Lorentzian coordinates which the [itex]\eta[/itex] denotes

Best wishes

Pete
 
Last edited:
  • #12
MeJennifer said:
A metric is simply a definition on how distances are measured in a space.
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
delta001 said:
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?
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.
 

1. What is a spacetime metric?

A spacetime metric is a mathematical representation of the geometry of spacetime. It defines the structure of spacetime, including the distances between points and the curvature of spacetime caused by the presence of mass and energy.

2. How is the spacetime metric related to Einstein's theory of general relativity?

The spacetime metric is a key concept in Einstein's theory of general relativity. It describes how mass and energy warp the fabric of spacetime, and how this curvature affects the motion of objects in the universe.

3. What is the difference between Minkowski and Riemannian spacetime metrics?

Minkowski spacetime metrics are used in special relativity to describe a flat, non-curved spacetime. Riemannian spacetime metrics, on the other hand, are used in general relativity to describe a curved, gravitational spacetime.

4. How is the spacetime metric measured or calculated?

The spacetime metric is typically calculated using mathematical equations and tools such as tensors and differential geometry. It can also be measured through observations of the behavior of objects in spacetime, such as the bending of light around massive objects.

5. What implications does the spacetime metric have for our understanding of the universe?

The spacetime metric is crucial in our understanding of the universe, as it provides the framework for Einstein's theory of general relativity and helps explain the behavior of objects in the cosmos. It also plays a key role in cosmology, the study of the origin and evolution of the universe.

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