Editing Einstein tensor

{{Short description|Tensor used in general relativity
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{{Use American English|date=January 2019}}{{General relativity sidebar}}

In [[differential geometry]], the '''Einstein tensor''' (named after [[Albert Einstein]]; also known as the '''trace-reversed [[Ricci curvature|Ricci tensor]]''') is used to express the [[curvature]] of a [[pseudo-Riemannian manifold]]. In [[general relativity]], it occurs in the [[Einstein field equations]] for [[gravitation]] that describe [[spacetime]] curvature in a manner that is consistent with conservation of energy and momentum.

== Definition ==
The Einstein tensor <math>\boldsymbol{G}</math> is a [[tensor]] of order 2 defined over [[pseudo-Riemannian manifold]]s.  In index-free notation it is defined as
<math display="block">\boldsymbol{G}=\boldsymbol{R}-\frac{1}{2}\boldsymbol{g}R,</math>
where <math>\boldsymbol{R}</math> is the [[Ricci tensor]], <math>\boldsymbol{g}</math> is the [[Metric tensor (general relativity)|metric tensor]] and <math>R</math> is the [[scalar curvature]], which is computed as the [[Trace (linear algebra)|trace]] of the Ricci tensor <math>R_{\mu \nu}</math> by {{tmath|1= R = g^{\mu \nu}R_{\mu \nu } }}. In component form, the previous equation reads as
<math display="block">G_{\mu\nu} = R_{\mu\nu} - {1\over2} g_{\mu\nu}R .</math>

The Einstein tensor is symmetric
<math display="block">G_{\mu\nu} = G_{\nu\mu}</math>
and, like the [[on shell and off shell|on shell]] [[stress–energy tensor]], has zero [[divergence]]:
<math display="block">\nabla_\mu G^{\mu\nu} = 0\,.</math>

== Explicit form ==
The Ricci tensor depends only on the metric tensor, so the Einstein tensor can be defined directly with just the metric tensor. However, this  expression is complex and rarely quoted in textbooks.  The complexity of this expression can be shown using the formula for the Ricci tensor in terms of [[Christoffel symbols]]:
<math display="block">\begin{align}
  G_{\alpha\beta}
    &= R_{\alpha\beta} - \frac{1}{2} g_{\alpha\beta} R \\
    &= R_{\alpha\beta} - \frac{1}{2} g_{\alpha\beta} g^{\gamma\zeta} R_{\gamma\zeta} \\
    &= \left(\delta^\gamma_\alpha \delta^\zeta_\beta - \frac{1}{2} g_{\alpha\beta}g^{\gamma\zeta}\right) R_{\gamma\zeta} \\
    &= \left(\delta^\gamma_\alpha \delta^\zeta_\beta - \frac{1}{2} g_{\alpha\beta}g^{\gamma\zeta}\right)\left(\Gamma^\epsilon{}_{\gamma\zeta,\epsilon} - \Gamma^\epsilon{}_{\gamma\epsilon,\zeta} + \Gamma^\epsilon{}_{\epsilon\sigma} \Gamma^\sigma{}_{\gamma\zeta} - \Gamma^\epsilon{}_{\zeta\sigma} \Gamma^\sigma{}_{\epsilon\gamma}\right), \\[2pt]
  G^{\alpha\beta}
    &= \left(g^{\alpha\gamma} g^{\beta\zeta} - \frac{1}{2} g^{\alpha\beta}g^{\gamma\zeta}\right)\left(\Gamma^\epsilon{}_{\gamma\zeta,\epsilon} - \Gamma^\epsilon{}_{\gamma\epsilon,\zeta} + \Gamma^\epsilon{}_{\epsilon\sigma} \Gamma^\sigma{}_{\gamma\zeta} - \Gamma^\epsilon{}_{\zeta\sigma} \Gamma^\sigma{}_{\epsilon\gamma}\right),
\end{align}</math>
where <math>\delta^\alpha_\beta</math> is the [[Kronecker tensor]] and the Christoffel symbol <math>\Gamma^\alpha{}_{\beta\gamma}</math> is defined as
<math display="block">\Gamma^\alpha{}_{\beta\gamma} = \frac{1}{2} g^{\alpha\epsilon}\left(g_{\beta\epsilon,\gamma} + g_{\gamma\epsilon,\beta} - g_{\beta\gamma,\epsilon}\right).</math>
and terms of the form <math>\Gamma ^\alpha _{\beta \gamma, \mu}</math> or <math>g_{\beta\gamma,\mu}</math> represent partial derivatives in the ''μ''-direction, e.g.:
<math display="block">\Gamma^\alpha{}_{\beta\gamma, \mu} = \partial _\mu \Gamma^\alpha{}_{\beta\gamma} = 
\frac{\partial}{\partial x^\mu}
\Gamma^\alpha{}_{\beta\gamma}</math>

Before cancellations, this formula results in <math>2 \times (6 + 6 + 9 + 9) = 60</math> individual terms. Cancellations bring this number down somewhat.<!-- exactly how much? -->

In the special case of a locally [[inertial reference frame]] near a point, the first derivatives of the metric tensor vanish and the component form of the Einstein tensor is considerably simplified:

<math display="block">\begin{align}
  G_{\alpha\beta}
    & = g^{\gamma\mu}\left[ g_{\gamma[\beta,\mu]\alpha} + g_{\alpha[\mu,\beta]\gamma} - \frac{1}{2} g_{\alpha\beta} g^{\epsilon\sigma} \left(g_{\epsilon[\mu,\sigma]\gamma} + g_{\gamma[\sigma,\mu]\epsilon}\right)\right] \\
    & = g^{\gamma\mu} \left(\delta^\epsilon_\alpha \delta^\sigma_\beta - \frac{1}{2} g^{\epsilon\sigma}g_{\alpha\beta}\right)\left(g_{\epsilon[\mu,\sigma]\gamma} + g_{\gamma[\sigma,\mu]\epsilon}\right),
\end{align}</math>
where square brackets conventionally denote [[Antisymmetric tensor|antisymmetrization]] over bracketed indices, i.e.
<math display="block">g_{\alpha[\beta,\gamma]\epsilon} \, = \frac{1}{2} \left(g_{\alpha\beta,\gamma\epsilon} - g_{\alpha\gamma,\beta\epsilon}\right).</math>

== Trace ==
The [[trace (linear algebra)|trace]] of the Einstein tensor can be computed by [[Tensor contraction|contract]]ing the equation in the [[#Definition|definition]] with the [[metric tensor]] {{tmath|1= g^{\mu\nu} }}. In <math>n</math> dimensions (of arbitrary signature):
<math display="block">\begin{align}
g^{\mu\nu}G_{\mu\nu} &= g^{\mu\nu}R_{\mu\nu} - {1\over2} g^{\mu\nu}g_{\mu\nu}R \\
G &= R - {1\over2} (nR) = {{2-n}\over2}R
\end{align}</math>

Therefore, in the special case of {{tmath|1= n = 4 }} dimensions, {{tmath|1= G = -R }}. That is, the trace of the Einstein tensor is the negative of the [[Ricci tensor]]'s trace. Thus, another name for the Einstein tensor is the ''trace-reversed Ricci tensor''. This <math>n=4</math> case is especially relevant in the [[theory of general relativity]].

== Use in general relativity ==

The Einstein tensor allows the [[Einstein field equations]] to be written in the concise form:
<math display="block">G_{\mu\nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu\nu} ,</math>
where <math>\Lambda</math> is the [[cosmological constant]] and <math>\kappa</math> is the [[Einstein gravitational constant]].

From the [[#Explicit form|explicit form of the Einstein tensor]], the Einstein tensor is a [[nonlinear]] function of the metric tensor, but is linear in the second [[partial derivative]]s of the metric. As a symmetric order-2 tensor, the Einstein tensor has 10 independent components in a 4-dimensional space. It follows that the Einstein field equations are a set of 10 [[Types of differential equations|quasilinear]] second-order partial differential equations for the metric tensor.

The [[contracted Bianchi identities]] can also be easily expressed with the aid of the Einstein tensor:
<math display="block">\nabla_{\mu} G^{\mu\nu} = 0.</math>

The (contracted) Bianchi identities automatically ensure the covariant conservation of the [[stress–energy tensor]] in curved spacetimes:
<math display="block">\nabla_{\mu} T^{\mu\nu} = 0.</math>

The physical significance of the Einstein tensor is highlighted by this identity. In terms of the densitized stress tensor contracted on a [[Killing vector]] {{tmath|1= \xi^\mu }}, an ordinary conservation law holds:
<math display="block">\partial_{\mu}\left(\sqrt{-g}\ T^\mu{}_\nu \xi^\nu\right) = 0.</math>

== Uniqueness ==
{{see also|Lovelock's theorem}}
[[David Lovelock]] has shown that, in a four-dimensional [[differentiable manifold]], the Einstein tensor is the only [[tensor]]ial and [[divergence]]-free function of the <math>g_{\mu\nu}</math> and at most their first and second partial derivatives.<ref>
{{cite journal
 |last    = Lovelock
 |first   = D.
 |title   = The Einstein Tensor and Its Generalizations
 |journal = Journal of Mathematical Physics
 |year    = 1971
 |volume  = 12
 |issue   = 3
 |pages   = 498–502
 |bibcode = 1971JMP....12..498L
 |doi     = 10.1063/1.1665613
 |doi-access= free
}}</ref><ref>
{{cite journal
 |last=Lovelock |first=D.
 |title=The Four‐Dimensionality of Space and the Einstein Tensor
 |journal=Journal of Mathematical Physics
 |year=1972 |volume=13 |issue=6 |pages=874–876
 |url = https://aip.scitation.org/doi/10.1063/1.1666069
 |doi = 10.1063/1.1666069
 |bibcode =  1972JMP....13..874L
|url-access=subscription
 }}</ref><ref>
{{cite journal
 |last=Lovelock |first=D.
 |title=The uniqueness of the Einstein field equations in a four-dimensional space
 |journal=Archive for Rational Mechanics and Analysis
 |year=1969 |volume=33 |issue=1 |pages=54–70
 |bibcode = 1969ArRMA..33...54L |doi = 10.1007/BF00248156 |s2cid=119985583
}}</ref><ref>
{{cite journal
 |last=Farhoudi |first=M. 
 |title=Lovelock Tensor as Generalized Einstein Tensor
 |journal=General Relativity and Gravitation
 |year=2009 |volume=41 |issue=1 |pages=17–29
 |arxiv=gr-qc/9510060
 |doi=10.1007/s10714-008-0658-9
 |bibcode = 2009GReGr..41..117F |s2cid=119159537 
}}</ref><ref>
{{cite book
 | author=Rindler, Wolfgang
 | author-link=Wolfgang Rindler
 | title=Relativity: Special, General, and Cosmological
 | publisher=[[Oxford University Press]]
 | year=2001 
 | isbn=978-0-19-850836-6
 | page = 299
}}</ref>

However, the [[Einstein field equation]] is not the only equation which satisfies the three conditions:<ref>
{{cite book
 | author=Schutz, Bernard
 | author-link=Bernard Schutz
 | title=A First Course in General Relativity
 | publisher=[[Cambridge University Press]]
 | edition=2
 | date=May 31, 2009
 | isbn=978-0-521-88705-2
 | page=[https://archive.org/details/firstcourseingen00bern_0/page/185 185]
 | url-access=registration
 | url=https://archive.org/details/firstcourseingen00bern_0/page/185
}}</ref> 
# Resemble but generalize [[Gauss's law for gravity|Newton–Poisson gravitational equation]]
# Apply to all coordinate systems, and
# Guarantee local covariant conservation of energy–momentum for any metric tensor.

Many alternative theories have been proposed, such as the [[Einstein–Cartan theory]], that also satisfy the above conditions.

== See also ==
{{Portal|Physics}}
* [[Contracted Bianchi identities]]
* [[Vermeil's theorem]]
* [[Mathematics of general relativity]]
* [[General relativity resources]]

== Notes ==
{{reflist}}

== References ==
* {{cite book
 | last = Ohanian | first = Hans C.
 | author2 = Remo Ruffini
 | title = Gravitation and Spacetime
 | edition = Second
 | publisher = [[W. W. Norton & Company]]
 | year = 1994
 | isbn = 978-0-393-96501-8
}}
* {{cite book
 | last = Martin | first = John Legat
 | title = General Relativity: A First Course for Physicists
 | edition = Revised
 | series = Prentice Hall International Series in Physics and Applied Physics
 | year = 1995
 | publisher = [[Prentice Hall]]
 | isbn = 978-0-13-291196-2
}}

{{tensors}}

[[Category:Tensors in general relativity]]
[[Category:Albert Einstein|Tensor]]