View source for Metric Tensor in General Relativity

== Introduction ==

In the field of [[General Relativity|general relativity]], the metric tensor (also known as the fundamental tensor) is a key concept that describes the geometric structure of spacetime. It is a symmetric two-tensor that provides a way to compute the distance between any two nearby points in spacetime.

== Definition ==

The metric tensor is denoted by the symbol g and its components are usually represented in a matrix form. In a four-dimensional spacetime, the metric tensor is a 4x4 matrix with 10 independent components due to its symmetry. The metric tensor is defined by the line element, which is a differential quantity that describes the infinitesimal distance between two neighboring points in spacetime. The line element is given by:

ds^2 = g_{μν} dx^μ dx^ν

Where:
- ds^2 is the square of the infinitesimal distance,
- g_{μν} are the components of the metric tensor,
- dx^μ and dx^ν are the infinitesimal coordinates differences.

[[Image:Detail-74125.jpg|thumb|center|A representation of a curved spacetime grid.|class=only_on_mobile]]
[[Image:Detail-74126.jpg|thumb|center|A representation of a curved spacetime grid.|class=only_on_desktop]]

== Properties ==

The metric tensor has several important properties:

1. It is symmetric: g_{μν} = g_{νμ}. This means that the order of the indices does not matter.

2. It is non-degenerate: its determinant is not zero. This ensures that the metric tensor defines a proper distance.

3. It is used to raise and lower indices of tensors, changing their variance.

4. It defines the curvature of spacetime through the [[Riemann Curvature Tensor|Riemann curvature tensor]].

== Role in General Relativity ==

In general relativity, the metric tensor plays a crucial role. It is used in the [[Einstein Field Equations|Einstein field equations]], which describe how matter and energy influence the curvature of spacetime. The metric tensor is a solution of these equations and describes the geometry of the spacetime in the presence of matter and energy.

== Types of Metric Tensors ==

There are several types of metric tensors that are solutions to the Einstein field equations. These include:

1. [[Minkowski Metric|Minkowski metric]]: This is the metric tensor of flat spacetime, which is the spacetime of special relativity.

2. [[Schwarzschild Metric|Schwarzschild metric]]: This is the metric tensor of a spacetime outside a spherical, non-rotating mass.

3. [[Kerr Metric|Kerr metric]]: This is the metric tensor of a spacetime outside a rotating mass.

4. [[Friedmann-Lemaître-Robertson-Walker Metric|Friedmann-Lemaître-Robertson-Walker (FLRW) metric]]: This is the metric tensor of a homogeneous and isotropic spacetime, which is used to model the universe in cosmology.

== Conclusion ==

The metric tensor is a fundamental concept in general relativity. It describes the geometry of spacetime and plays a crucial role in the theory. Different types of metric tensors represent different spacetime geometries, which correspond to different physical situations.

== See Also ==

- [[Riemann Curvature Tensor]]
- [[Einstein Field Equations]]
- [[Minkowski Metric]]
- [[Schwarzschild Metric]]
- [[Kerr Metric]]
- [[Friedmann-Lemaître-Robertson-Walker Metric]]

[[Category:General Relativity]]
[[Category:Physics]]
[[Category:Mathematical Physics]]