Editing Einstein tensor (section)

== 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.