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Solutions of the Einstein field equations are spacetimes solving the Einstein field equations (EFE) of general relativity. They are also named Solutions of General relativity. Those solutions are broadly classed as exact solutions or non-exact solutions, see especially Lorentz metrics. Solving the equations needs at first 20 differential relations as basis. Finally Integrations of the differential form of Einstein's equations results in known solutions.

The Einstein field equations are

${\displaystyle G_{ab}\,=\kappa T_{ab}}$

or more generally

${\displaystyle G_{ab}+\Lambda g_{ab}\,=\kappa T_{ab}}$

where ${\displaystyle \kappa }$ is a constant, and the Einstein tensor on the left side of the equation is equated to the stress-energy tensor representing the energy and momentum present in the spacetime. The Einstein tensor is built up from the metric tensor and its partial derivatives; thus, the EFE are a system of ten partial differential equations to be solved for the metric.

Solving the equations

It is important to realize that the Einstein field equations alone are not enough to determine the evolution of a gravitational system in many cases. They depend on the stress-energy tensor, which in turn depends on the (unknown) metric. If we're only interested in the weak field limit of the theory, we can compute the dynamics of matter using special relativity methods and/or Newtonian laws of gravity and then plug in the resulting stress-energy tensor into the Einstein field equations. But if we want to obtain the exact solution or to deal with strong fields, we need to solve for the evolution of the metric and the stress-energy tensor at the same time.

We start with two sets of equations:

The Einstein field equations (to determine evolution of the metric):

${\displaystyle G_{\alpha \beta }+\Lambda g_{\alpha \beta }\,=\kappa T_{\alpha \beta }}$

The continuity equation (to determine evolution of the stress-energy tensor):

${\displaystyle T_{;\beta }^{\alpha \beta }=0}$

This is clearly not enough, because we only have 14 equations (10 from the Einstein's equations and 4 from the continuity equation) for 20 unknowns (10 components of the metric and 10 components of the stress-energy tensor). We're missing equations of state. It's easy to see that, in the most general case, we need at least 6 more equations, possibly more if there are internal degrees of freedom (such as temperature) which may vary throughout space-time.

In practice, it is usually possible to simplify the problem by replacing the full set of equations of state with a simple approximation. Some common approximations are:

• Vacuum:

${\displaystyle T_{\alpha \beta }\,=0}$

• Perfect fluid:

${\displaystyle T_{\alpha \beta }\,=(\rho +p)u_{\alpha }u_{\beta }+pg_{\alpha \beta }}$ where ${\displaystyle u^{\alpha }u_{\alpha }=-1\!}$

Here ${\displaystyle \rho }$ is the mass-energy density measured in a momentary co-moving frame, ${\displaystyle u_{\alpha }}$ is the fluid's 4-velocity vector field, and ${\displaystyle p}$ is the pressure.

• Non-interacting dust ( a special case of perfect fluid ):

${\displaystyle T_{\alpha \beta }\,=\rho u_{\alpha }u_{\beta }}$

For a perfect fluid, we need to add one more equation of state that relates density ${\displaystyle \rho }$ and pressure ${\displaystyle p}$. This equation will often depend on temperature, so we also need to include a heat transfer equation or to postulate that heat transfer can be neglected.

Next, notice that only 10 of our original 14 equations are independent, because the continuity equation ${\displaystyle T_{;\beta }^{\alpha \beta }=0}$ is a consequence of the Einstein's equations. This reflects the fact that our system is gauge invariant and we need to perform "gauge fixing", i.e. impose 4 constraints on the system, in order to obtain unequivocal results.

A popular choice of gauge is the so-called "De Donder gauge", also known as harmonic gauge or Lorentz gauge.

${\displaystyle g^{\mu \nu }\Gamma _{\mu \nu }^{\sigma }=0}$

In numerical relativity, the preferred gauge is the so-called "3+1 decomposition", based on the ADM formalism. In this decomposition, metric is written in the form

${\displaystyle ds^{2}\,=(-N+N^{i}N^{j}\gamma _{ij})dt^{2}+2N^{i}\gamma _{ij}dtdx^{j}+\gamma _{ij}dx^{i}dx^{j}}$, where ${\displaystyle i,j=1\dots 3}$

${\displaystyle N}$ and ${\displaystyle N^{i}}$ can be chosen arbitrarily. The remaining physical degrees of freedom are contained in ${\displaystyle \gamma _{ij}}$, which represents the Riemannian metric on 3-hypersurfaces ${\displaystyle t=const}$.

Once equations of state are chosen and the gauge is fixed, we can try to solve the complete set of equations. Unfortunately, even in the simplest case of gravitational field in the vacuum ( vanishing stress-energy tensor ), the problem turns out too complex to be exactly solvable. To get physical results, we can either turn to numerical methods; try to find exact solutions by imposing symmetries; or try middle-ground approaches such as perturbation methods or linear approximations of the Einstein tensor.

Basis of solutions

Mentioned Tensor relations say that each element of the space must produce an infinitesimal Gauss’ curvature, now seen in 4 dimensions. Differentially seen and resulting by related integrals the solutions differ for vacuum and matter (different fluids, gases, solids) without elastic effects (then named a scalar field) or with possible elastic effects, then named a tensor (a vector-set, differently acting for each direction), each with one vector only. All possible combined solutions are valid.

Boundaries for solutions

The field-equations have multiply infinite number-sets of solutions:

• By above mentioned inner boundaries of elements within a given matter or in vacuum, the “equations of state”, give only 10 equations from Einstein's equations, 4 by “continuity equation” but 6 by given physics of a matter, most generally depending on mentioned elastic properties, to get all 20 priory unknowns (10 components).
• Having got these inner boundaries the Integration of Field equations give additionally an infinite number of sets for geometric solutions by realized or supposed boundaries at the extern limits of integral solutions.
• Each Superposition of single and general solutions – not only of mentioned different material, infinitesimal and integral seen – is again a valid and possible solution.

Integration of the field equations

Such solutions are seen as infinitesimal metrics, but must be shown more clearly by intrinsic curvatures results, shown by an area, a "visible surface" of the space as solution. Each integration – already of a simple scalar-field – gives an arbitrarily chosen constant. It must fulfil real conditions, here only for a beginning and an end. Integration of vector-fields will already give for each direction one solution with by principle unknown functions in all other directions but bounded and limited one to the other. All together must be adapted with and to real physical boundaries at the limits. Rather complicated are integrations for in each direction different vectors of a tensor-field.

Known solutions

Schwarzschild solutions

Einstein was astonished about two promptly given solutions, by Karl Schwarzschild:

• The first Schwarzschild solution is well known as the first solution of Einstein’s General relativity and as his “extern solution”, meaning outside of the Schwarzschild radius. It describes in cylinder coordinates a kind of to its end more and more contracting tunnel-shape. The geometric surface define all possible circles. The solution give all valid extern effects of a black hole.
• The second, so called Inner Schwarzschild solution was presented by Schwarzschild some days later only, mainly found in German original and related texts as "innere Schwarzschild Lösung". Schwarzschild had seen that the inner solution of his black hole is equivalently described as an attractor, a super-massive body within the spere of an homogenous isotropic compressed gas, fluid or solid, valid within boundary of R=Schwarzschild radius.

Friedmann-Lemaître-Robertson-Walker solutions

The first Friedmann equations solution is based on the same mathematical calculus as the second Schwarzschild solution, but explicitly considering molecules of an isotropic homogenous interstellar gas fluid instead of molecules of a super compressed fluid or solid only within its calculated black hole radius.

The second solution described for the first time the Big bang as new solution, assuming that the spatial component of the 4D-metric can be time dependent. The main basis of the actual standard solution at its physical Schwarzschild radius has the same problem: While in black holes the time stops for us, the total beginning of the time must be supposed at its Schwarzschild radius, meanwhile declared sufficiently by mainstream but not yet accepted for its critics.

De Sitter solutions

De Sitter universe is another solution to Albert Einstein's field equations of GR by Willem de Sitter. It is an exponential solution. The exponentially expanding universe of the FLRW form has the scale factor:

${\displaystyle a(t)=e^{Ht}\,}$, (H = Hubble constant)

describing the expansion of physical spatial distances.

• A de Sitter universe is one with no ordinary matter content (supported e.g. by dark energy) but with a positive cosmological constant which sets the expansion rate, H with the effect that a larger cosmological constant leads to a larger expansion rate.

A mathematically equivalent negative exponent would produce implosion. A complex exponent shows an oscillation:

Geoffrey Burbidge’s solution

According to Geoffrey Burbidge and Margaret_Burbidge, the universe is oscillatory and as such expands and contracts periodically over infinite time between the following periodic extremes:

• Einstein had initially preferred a static solution of the GR. Because of its instability he had revised it and named it by himself as his biggest stupor.
• Such an initially maximal huge universe is still affected by a rest of gravity. At thereby initially minimum gas-tension the space must thereby move more and more move to a center inside, finally imploding by gravity.
• At the end of implosion with its minimum space, seen as a maximal compressed interstellar gas will stop to move. Then begins the other extreme, an increasing movement outwards finally exploding.
• The resulting Burbidge’s theory was called the B²FH theory after the participants.

This theory, due to its controversial nature, has brought fame and infamy to Burbidge.

Halton Arp's solution

Intrinsic redshift meant by principle only that some stellar objects anyhow not obey standard laws (e.g. Hubble's law) and that something like this has to be declared. Arp et. al. discovered highly discontinuous redshifts within filamentary superclusters, voids and related quasars, until now meaning that a too rare statistic probability cannot completely overrule intrinsic effects by generally different considerations of the standard model. It seems that redshift of several "child galaxies" are significantly higher than its "parent galaxies". For Quasars it could be based on gravitational redshift (Einstein’s clocks run differently at different gravity-centres by Einstein effects: a same kind of emitted photon starts at different places at different frequencies affected by different gravities). A problem arised because Intrinsic redshift became then a hypothesis from various non-standard cosmologies that a significant portion of the observed redshift of extragalactic objects (e.g. quasars and galaxies) may be caused by a phenomenon other than known redshift mechanisms (cosmological redshift, Doppler redshift, gravitational redshift). A proposed Redshift quantization could not be confirmed until now. Therefore also this controversial theory is widely not accepted.

Fritz Zwicky’s solution

Einstein himself wrote - until today confusing physics - that photons have a "zero rest mass (remark: considered in standard cosmology) but non-zero relativistic mass". Zwicky related to the relativistic mass and the Einstein effects because Einstein confirmed Plancks view that photons are particles with a relativistic mass. Gravitational redshifts was utilized by him simply by another view about photons, meaning now:

• Gravity influences photons in direction of its related radius or simply always centripetally in direction of its centre, meanwhile partly supported by some newer graviton’s theories.
• By this non-conform "relativistic mass view" about physics an increasing gravitational potential “slows” the energy of a photon by redshift by loosing its momentum.

Not accccepted by the mainstream, increasing gravitational potential was recently used to declare the Pioneer anomaly.

Exact solutions

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Exact solutions are Lorentz metrics that are conformable to a physically realistic stress-energy tensor and which are obtained by solving the EFE exactly in closed form.

Non-exact solutions

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Those solutions that are not exact are called non-exact solutions. Such solutions mainly arise due to the difficulty of solving the EFE in closed form and often take the form of approximations to ideal systems. Many non-exact solutions may be devoid of physical content, but serve as useful counterexamples to theoretical conjectures.

Applications

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There are practical as well as theoretical reasons for studying solutions of the Einstein field equations.

From a purely mathematical viewpoint, it is interesting to know the set of solutions of the Einstein field equations. Some of these solutions are parametrised by one or more parameters.

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