# Einstein's field equations

Field equation on a wall in Leiden

Within the general theory of relativity , the Einstein field equations (according to Albert Einstein , also gravitational equations ) formulate the physical phenomenon of gravitation mathematically using methods of differential geometry .

The basic idea is to link an energy-momentum distribution with the geometry of space-time . According to the special theory of relativity, energy and momentum are combined to form a four -tensor , the energy-momentum tensor , while a metric tensor represents the geometry of space-time.

## Basic assumptions and requirements

In order to set up the field equations, physical considerations are first necessary, since the form of the equations must be postulated. The physical starting point of Einstein's considerations is the principle of equivalence : mass and energy are equivalent and every form of energy induces heavy mass .

Just as the mass causes the gravitational field in Newton's theory of gravitation , the most natural approach for its generalization is that the gravitational field is mathematically dependent on the shape of the energy-momentum tensor . Now is not an arbitrary symmetric tensor , since it has to satisfy. That is, the divergence of the energy-momentum tensor has locally, at a fixed spatial and time coordinate, disappear so that the energy and momentum conservation law is maintained. The equivalence principle is taken into account in the contribution of the energy-momentum tensor. In addition to the mass energy density (mass or energy per volume), the energy-momentum tensor also contains other contributions, for example the pressure that a radiation field can exert. ${\ displaystyle \ T _ {\ mu \ nu}}$${\ displaystyle \ T _ {\ mu \ nu}}$${\ displaystyle \ nabla _ {\ nu} T ^ {\ mu \ nu} = 0}$

According to the equivalence principle, the effect of gravity should be represented as the curvature of space-time . The energy-momentum tensor as the source of the field should accordingly be opposed to a tensor of the same shape on the other side of the equation, which describes the geometric properties (curvature) of space-time, the Einstein tensor , built up from the basic metric tensor and the curvature covariants derived from it and invariants (see below). So the field equations take the form: ${\ displaystyle G _ {\ mu \ nu}}$

${\ displaystyle \ G _ {\ mu \ nu} = \ kappa T _ {\ mu \ nu}}$

The constant is called Einstein's gravitational constant or simply Einstein's constant and is assumed to be the constant of proportionality ( is the constant of gravitation ). ${\ displaystyle \ kappa = 8 \ pi G / c ^ {4}}$${\ displaystyle G}$

From the considerations so far, these requirements result in summary:

1. ${\ displaystyle \ G _ {\ mu \ nu} = 0}$    for a flat spacetime, d. H. in the absence of gravity.
2. ${\ displaystyle \ \ nabla _ {\ nu} T ^ {\ mu \ nu} = 0}$   for the conservation of energy momentum.
3. ${\ displaystyle \ \ nabla _ {\ nu} G ^ {\ mu \ nu} = 0}$  due to the above requirement for .${\ displaystyle T _ {\ mu \ nu}}$
4. ${\ displaystyle \ T _ {\ mu \ nu}}$  is a symmetric second order tensor, so this must also hold for.${\ displaystyle G _ {\ mu \ nu}}$
5. ${\ displaystyle \ G _ {\ mu \ nu}}$  is accordingly a combination of the basic geometric covariants, which are second order (symmetric) tensors, the curvature tensor and the metric tensor .${\ displaystyle R _ {\ mu \ nu}}$${\ displaystyle g _ {\ mu \ nu}}$

## The field equations

The field equations result from these requirements:

${\ displaystyle R _ {\ mu \ nu} - {\ frac {1} {2}} g _ {\ mu \ nu} R = \ kappa T _ {\ mu \ nu} = {\ frac {8 \ pi G} { c ^ {4}}} T _ {\ mu \ nu}}$.

Here is the gravitational constant , the speed of light , the Ricci tensor , the curvature scalar and the metric tensor . ${\ displaystyle G}$${\ displaystyle c}$${\ displaystyle R _ {\ mu \ nu}}$${\ displaystyle R}$${\ displaystyle g _ {\ mu \ nu}}$

The field equations can also be defined with the opposite sign in front of the Einstein constant

${\ displaystyle R _ {\ mu \ nu} - {\ frac {1} {2}} g _ {\ mu \ nu} R = - \ kappa T _ {\ mu \ nu} = - {\ frac {8 \ pi G } {c ^ {4}}} T _ {\ mu \ nu}}$.

This sign is purely dependent on the convention used and is not physically significant; both conventions are widely used.

The field equations can also be transformed and represented as

${\ displaystyle R _ {\ mu \ nu} = \ kappa (T _ {\ mu \ nu} - {\ tfrac {1} {2}} g _ {\ mu \ nu} T)}$.

Here is the Laue scalar . ${\ displaystyle T = T _ {\ mu} ^ {\ mu} = g ^ {\ mu \ nu} T _ {\ mu \ nu}}$

The above requirements also allow a term proportional to the metric tensor on the left, which leads to field equations with a cosmological constant (see below).

In addition to the field equations, there is also the equation of motion for test particles moving on a geodesic , the so-called geodesic equation (see general theory of relativity ). Overall, the field and motion equations express a dynamic mutual influence of energy-momentum distribution and the geometry of space-time.

The field equations form a system of 16 coupled partial differential equations , which are reduced to 10 by symmetries. There are also the four Bianchi identities that result from the conservation of energy and momentum and further reduce the system. A number of exact solutions are known, which mostly meet certain additional symmetry requirements. In material-free space, the field equations have a hyperbolic character, i.e. the solutions correspond to wave equations (with the speed of light as the maximum speed of propagation). In general, they can only be solved numerically, for which there are sophisticated techniques and a separate specialty (numerical relativity). There are also some exact mathematical results such as the well-being of the Cauchy problem ( Yvonne Choquet-Bruhat ), the singularity theorems of Roger Penrose and Stephen Hawking, or results of Demetrios Christodoulou on the stability of the Minkowski space (with Sergiu Klainerman ) and the instability of the naked Singularities .

In the limiting case of weak gravitational fields and low velocities, the usual Newtonian gravitational equations of a mass distribution result (and the equations that result here are of the elliptical type as partial differential equations ). For small fields, the post-Newton approximation was also developed in order to be able to compare the general theory of relativity with alternative theories of gravity on the basis of observations, for example.

## The vacuum field equations

For example, if one looks at the outer space of stars, where there is no matter as an approximation, then it is set. One then calls ${\ displaystyle T _ {\ mu \ nu} = 0}$

${\ displaystyle R _ {\ mu \ nu} - {\ frac {1} {2}} g _ {\ mu \ nu} R = 0}$

the vacuum field equations and their solutions vacuum solutions . By multiplying with and with the help of and the conclusion that in vacuum and thus ${\ displaystyle g ^ {\ mu \ nu}}$${\ displaystyle R = g ^ {\ mu \ nu} R _ {\ mu \ nu}}$${\ displaystyle g ^ {\ mu \ nu} g _ {\ mu \ nu} = 4}$${\ displaystyle R = 0}$

${\ displaystyle R _ {\ mu \ nu} = 0}$.

For the surroundings of a non-rotating and electrically neutral ball of the mass , for example, the outer Schwarzschild solution is obtained in spherical coordinates , the line element of which is the shape ${\ displaystyle M}$

${\ displaystyle \ mathrm {d} s ^ {2} = g _ {\ mu \ nu} \, \ mathrm {d} x ^ {\ mu} \ mathrm {d} x ^ {\ nu} = - c ^ { 2} {\ Bigl (} 1 - {\ frac {2GM} {c ^ {2} r}} {\ Bigr)} \, \ mathrm {d} t ^ {2} + {\ Bigl (} 1- { \ frac {2GM} {c ^ {2} r}} {\ Bigr)} ^ {- 1} \, \ mathrm {d} r ^ {2} + r ^ {2} \, \ mathrm {d} \ theta ^ {2} + r ^ {2} \ sin ^ {2} \ theta \, \ mathrm {d} \ phi ^ {2}}$

owns.

The invariant of the theory, generalizes the special-relativistic concept of proper time , among other things by taking into account the gravitation of the celestial body under consideration. Special features arise when the radius falls below a critical value , namely for (see black hole ). ${\ displaystyle \ mathrm {d} s,}$${\ displaystyle r}$${\ displaystyle r <2GM / c ^ {2}}$

## Einstein-Maxwell equations

Used for the electromagnetic energy momentum tensor${\ displaystyle T _ {\ mu \ nu}}$

${\ displaystyle T _ {\ mu \ nu} = {\ frac {1} {\ mu _ {0}}} \ left (F _ {\ mu} {} ^ {\ alpha} F _ {\ alpha \ nu} + { \ frac {1} {4}} F _ {\ alpha \ beta} F ^ {\ alpha \ beta} g _ {\ mu \ nu} \ right)}$

inserted into the field equations

${\ displaystyle R _ {\ mu \ nu} - {\ frac {1} {2}} g _ {\ mu \ nu} R = {\ frac {8 \ pi G} {c ^ {4} \ mu _ {0 }}} \, \ left (F _ {\ mu} {} ^ {\ alpha} F _ {\ alpha \ nu} + {\ frac {1} {4}} F _ {\ alpha \ beta} F ^ {\ alpha \ beta} g _ {\ mu \ nu} \ right)}$

so one speaks of the Einstein-Maxwell equations .

## The cosmological constant

Based on the basic assumptions given above, another additive term can be added to the Einstein tensor , which consists of a constant and the metric tensor. Thus the requirement of freedom from divergence is still met and so the field equations take the form ${\ displaystyle \ Lambda}$

${\ displaystyle R _ {\ mu \ nu} - {\ frac {1} {2}} g _ {\ mu \ nu} R + \ Lambda g _ {\ mu \ nu} = {\ frac {8 \ pi G} {c ^ {4}}} T _ {\ mu \ nu}}$

on. Here is the cosmological constant that Einstein built into the field equations and chosen so that the universe becomes static; this was the most sensible view at the time. However, it turned out that the universe so described by the theory is unstable. When Edwin Hubble finally demonstrated that the universe was expanding, Einstein rejected his constant. Today, however, it plays a role again through the results of observational cosmology from the 1990s. ${\ displaystyle \ Lambda}$

## literature

• Albert Einstein: The field equations of gravitation. Meeting reports of the Prussian Academy of Sciences in Berlin, pp. 844–847, November 25, 1915.
• Yvonne Choquet-Bruhat: General relativity and the Einstein equations. Oxford Univ. Press, Oxford 2009, ISBN 978-0-19-923072-3 .
• Hans Stephani : Exact solutions of Einstein's field equations. Cambridge Univ. Press, Cambridge 2003, ISBN 0-521-46136-7 .
• Bernd G. Schmidt: Einstein's field equations and their physical implications. Springer, Berlin 2000, ISBN 3-540-67073-4 .