Metric Tensor (general Relativity)

In General Relativity , the metric tensor (or simply the '''metric''') is the fundamental object of study. It may loosely be thought of as a generalization of the Gravitational Field familiar from Newtonian Gravitation . The metric captures all the geometric and causal structure of Spacetime . Using the metric one can define such notions as distance, volume, angle, future, past, and curvature.

:''Notation and conventions'': Throughout this article we work with a Metric Signature that is mostly positive (-+++); see Sign Convention . As is customary in relativity, Units are used where the Speed Of Light ''c'' = 1. The Gravitation Constant ''G'' will be kept explicit. The Summation Convention , where repeated indices are automatically summed over, will be employed.

DEFINITION

Mathematically, spacetime is represented by a 4-dimensional Differentiable Manifold ''M'' and the metric is given as a Covariant , second-rank, Symmetric Tensor on ''M'', conventionally denoted by ''g''. Moreover the metric is required to be Nondegenerate with Signature (-+++). A manifold ''M'' equipped with such a metric is called a Lorentzian Manifold .

Explicitly, the metric is a Symmetric Bilinear Form on each Tangent Space of ''M'' which varies in a smooth (or differentiable) manner from point to point. Given two tangent vectors ''u'' and ''v'' and a point ''x'' in ''M'', the metric can be evaluated on ''u'' and ''v'' to give a real number:
:

g_x(u,v) = g_x(v,u) \in \mathbb{R}.

This can be thought of as a generalization of the Dot Product in ordinary Euclidean Space . This analogy is not exact, however. Rather, than Euclidean space — where the dot product is Positive Definite — the metric gives each tangent space the structure of Minkowski Space .

LOCAL COORDINATES AND MATRIX REPRESENTATIONS

Physicists usually work in Local Coordinates (i.e. coordinates defined on some Local Patch of ''M''). In local coordinates

x^\mu

(where

\mu

is an index which runs from 0 to 3) the metric can be written in the form
:

g = g_{\mu
u} dx^\mu \otimes dx^
u.

The factors

dx^\mu

are One-form Gradient s of the scalar coordinate fields

x^\mu

. The metric is thus a linear combination of Tensor Product s of one-form gradients of coordinates. The coefficients

g_{\mu
u}

are a set of 16 real-valued functions (since the tensor ''g'' is actually a ''tensor field'' defined at all points of a Spacetime manifold). In order for the metric to be symmetric we must have
:

g_{\mu
u} = g_{
u\mu}\,

giving 10 independent coefficients. If we denote the symmetric Tensor Product by juxtaposition (so that

dx^\mu dx^
u = dx^
u dx^\mu

) we can write the metric in the form
:

g = g_{\mu
u}dx^\mu dx^
u.\,

If the local coordinates are specified, or understood from context, the metric can be written as a 4×4 Symmetric Matrix with entries

g_{\mu
u}

. The nondegeneracy of

g_{\mu 
u}

means that this matrix is Non-singular (i.e. has non-vanishing determinant), while the Lorentzian signature of ''g'' implies that the matrix has one negative and three positive Eigenvalues . Note that physicists often refer to this matrix or the coordinates

g_{\mu
u}

themselves as the metric (see, however, Abstract Index Notation ).

With the quantity

dx^\mu

being an infinitesimal coordinate displacement, the metric acts as an infinitesimal invariant interval squared or Line Element . For this reason one often sees the notation

ds^2

for the metric:
:

ds^2 = g_{\mu
u}dx^\mu dx^
u.\,

In general relativity, the terms ''metric'' and ''line element'' are often used interchangeably.

The line element

ds^2

imparts information about the causal structure of the spacetime. When

ds^2 < 0

, the interval is Timelike and the square root of the absolute value of ''ds²'' is an incremental Proper Time . Only timelike intervals can be physically traversed by a massive object. When

ds^2=0

, the interval is Lightlike , and can only be traversed by light. When

ds^2 > 0

, the interval is spacelike and the square root of ''ds²'' acts as an incremental Proper Length . Spacelike intervals cannot be traversed, since they connect events that are out of each other's Light Cone s. Event s can be causally related only if they are within each other's light cones.

The metric components obviously depend on the chosen local coordinate system. Under a change of coordinates

x^\mu 	o x^{\bar \mu}

the metric components transform as
:

g_{\bar \mu \bar 
u} = rac{\partial x^
ho}{\partial x^{\bar \mu}}rac{\partial x^\sigma}{\partial x^{\bar 
u}} g_{
ho\sigma} = \Lambda^
ho {}_{\bar \mu} \, \Lambda^\sigma {}_{\bar 
u} \, g_{
ho \sigma} .

EXAMPLES

Flat spacetime

The simplest example of a Lorentzian manifold is Flat Spacetime which can be given as R⁴ with coordinates

(t,x,y,z)

and the metric
:

ds^2 = -dt^2 + dx^2 + dy^2 + dz^2.\,

Note that these coordinates actually cover all of R⁴. The flat space metric (or Minkowski metric) is often denoted by the symbol η and is the metric used in Special Relativity . In the above coordinates, the matrix representation of η is
:

\eta = \begin{pmatrix}-1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&1\end{pmatrix}

In Spherical Coordinates

(t,r,	heta,\phi)

, the flat space metric takes the form
:

ds^2 = -dt^2 + dr^2 + r^2d\Omega^2\,

where
:

d\Omega^2 = d	heta^2 + \sin^2	heta\,d\phi^2

is the standard metric on the 2-sphere .

Schwarzschild metric

Besides the flat space metric the most important metric in general relativity is the Schwarzschild Metric which can be given in one set of local coordinates by
:

ds^{2} = -\left(1-rac{2GM}{r} 
ight)dt^2 + \left(1-rac{2GM}{r}
ight)^{-1}dr^2+ r^2 d\Omega^2

where, again,

d\Omega^2

is the standard metric on the 2-sphere . Here ''G'' is the Gravitation Constant and ''M'' is a constant with the dimensions of Mass .

VOLUME

The metric ''g'' defines a natural Volume Form , which can be used to integrate over spacetimes. In local coordinates

x^\mu

of a manifold, the volume form can be written

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Metric Tensor (general Relativity)