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(named after ‘albert einstein’)
also known as the trace-reversed Ricci tensor)
In differential geometry, the Einstein 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
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Definition
The Einstein tensor \mathbf {G} is a tensor of order 2 defined over pseudo-Riemannian manifolds.
In index-free notation it is defined as
{\displaystyle \mathbf {G} =\mathbf {R} -{\frac {1}{2}}\mathbf {g} R,}
where \mathbf {R} is the Ricci tensor, \mathbf {g} is the metric tensor and R is the scalar curvature, which is computed as the trace of the Ricci Tensor R_{\mu \nu} by {\displaystyle R=g^{\mu \nu }R_{\mu \nu }=R_{\mu }^{\mu }}. In component form, the previous equation reads as
{\displaystyle G_{\mu \nu }=R_{\mu \nu }-{1 \over 2}g_{\mu \nu }R.}
The Einstein tensor is symmetric
{\displaystyle G_{\mu \nu }=G_{\nu \mu }}
and, like the on shell stress–energy tensor, and has zero divergence:
{\displaystyle \nabla _{\mu }G^{\mu \nu }=0\,.}
Explicit form[edit]
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:
{\displaystyle {\begin{aligned}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 {\alpha }^{\gamma }\delta {\beta }^{\zeta }-{\frac {1}{2}}g_{\alpha \beta }g^{\gamma \zeta }\right)R_{\gamma \zeta }\&=\left(\delta {\alpha }^{\gamma }\delta {\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),\[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{aligned}}}
where \delta {\beta }^{\alpha } is the Kronecker tensor and the Christoffel symbol {\displaystyle \Gamma ^{\alpha }{}{\beta \gamma }} is defined as
{\displaystyle \Gamma ^{\alpha }{}{\beta \gamma }={\frac {1}{2}}g^{\alpha \epsilon }\left(g{\beta \epsilon ,\gamma }+g_{\gamma \epsilon ,\beta }-g_{\beta \gamma ,\epsilon }\right).}
and terms of the form {\displaystyle \Gamma _{\beta \gamma ,\mu }^{\alpha }} represent its partial derivative in the μ-direction, i.e.:
{\displaystyle \Gamma ^{\alpha }{}{\beta \gamma ,\mu }=\partial {\mu }\Gamma ^{\alpha }{}{\beta \gamma }={\frac {\partial }{\partial x^{\mu }}}\Gamma ^{\alpha }{}{\beta \gamma }}
Before cancellations, this formula results in {\displaystyle 2\times (6+6+9+9)=60} individual terms. Cancellations bring this number down somewhat.
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:
{\displaystyle {\begin{aligned}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 {\alpha }^{\epsilon }\delta {\beta }^{\sigma }-{\frac {1}{2}}g^{\epsilon \sigma }g_{\alpha \beta }\right)\left(g_{\epsilon [\mu ,\sigma ]\gamma }+g_{\gamma [\sigma ,\mu ]\epsilon }\right),\end{aligned}}}
where square brackets conventionally denote antisymmetrization over bracketed indices, i.e.
{\displaystyle g_{\alpha [\beta ,\gamma ]\epsilon }\,={\frac {1}{2}}\left(g_{\alpha \beta ,\gamma \epsilon }-g_{\alpha \gamma ,\beta \epsilon }\right).}
Trace[edit]
The trace of the Einstein tensor can be computed by contracting the equation in the definition with the metric tensor g^{\mu \nu }. In n dimensions (of arbitrary signature):
{\displaystyle {\begin{aligned}g^{\mu \nu }G_{\mu \nu }&=g^{\mu \nu }R_{\mu \nu }-{1 \over 2}g^{\mu \nu }g_{\mu \nu }R\G&=R-{1 \over 2}(nR)={{2-n} \over 2}R\end{aligned}}}
Therefore, in the special case of n = 4 dimensions, {\displaystyle 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 n=4 case is especially relevant in the theory of general relativity.
Use in general relativity[edit]
The Einstein tensor allows the Einstein field equations to be written in the concise form:
{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu },}
where \Lambda is the cosmological constant and \kappa is the Einstein gravitational constant.
From the explicit form of the Einstein tensor, the Einstein tensor is a nonlinear function of the metric tensor, but is linear in the second partial derivatives 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 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:
{\displaystyle \nabla _{\mu }G^{\mu \nu }=0.}
The (contracted) Bianchi identities automatically ensure the covariant conservation of the stress–energy tensor in curved spacetimes:
{\displaystyle \nabla _{\mu }T^{\mu \nu }=0.}
The physical significance of the Einstein tensor is highlighted by this identity. In terms of the densitized stress tensor contracted on a Killing vector \xi ^{\mu }, an ordinary conservation law holds:
{\displaystyle \partial {\mu }\left({\sqrt {-g}}T^{\mu }{}{\nu }\xi ^{\nu }\right)=0.}
Uniqueness[edit]
David Lovelock has shown that, in a four-dimensional differentiable manifold, the Einstein tensor is the only tensorial and divergence-free function of the g_{\mu \nu } and at most their first and second partial derivatives.[1][2][3][4][5]
However, the Einstein field equation is not the only equation which satisfies the three conditions:[6]
Resemble but generalize 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[edit]
Contracted Bianchi identities
Vermeil’s theorem
Mathematics of general relativity
General relativity resources
Notes[edit]
^ Lovelock, D. (1971). “The Einstein Tensor and Its Generalizations”. Journal of Mathematical Physics. 12 (3): 498–502. Bibcode:1971JMP….12..498L. doi:10.1063/1.1665613. Archived from the original on 2013-02-24.
^ Lovelock, D. (1972). “The Four‐Dimensionality of Space and the Einstein Tensor”. Journal of Mathematical Physics. 13 (6): 874–876. Bibcode:1972JMP….13..874L. doi:10.1063/1.1666069.
^ Lovelock, D. (1969). “The uniqueness of the Einstein field equations in a four-dimensional space”. Archive for Rational Mechanics and Analysis. 33 (1): 54–70. Bibcode:1969ArRMA..33…54L. doi:10.1007/BF00248156.
^ Farhoudi, M. (2009). “Lovelock Tensor as Generalized Einstein Tensor”. General Relativity and Gravitation. 41 (1): 17–29. arXiv:gr-qc/9510060. Bibcode:2009GReGr..41..117F. doi:10.1007/s10714-008-0658-9.
^ Rindler, Wolfgang (2001). Relativity: Special, General, and Cosmological. Oxford University Press. p. 299. ISBN 978-0-19-850836-6.
^ Schutz, Bernard (May 31, 2009). A First Course in General Relativity (2 ed.). Cambridge University Press. p. 185. ISBN 978-0-521-88705-2.
References[edit]
Ohanian, Hans C.; Remo Ruffini (1994). Gravitation and Spacetime (Second ed.). W. W. Norton & Company. ISBN 978-0-393-96501-8.
Martin, John Legat (1995). General Relativity: A First Course for Physicists. Prentice Hall International Series in Physics and Applied Physics (Revised ed.). Prentice Hall.
ISBN 978-0-13-291196-2
en.wikipedia.org /wiki/Einstein_tensor
Einstein tensor
Contributors to Wikimedia projects6-7 minutes 10/11/2004
DOI: 10.1063/1.1665613, Show Details
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