In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor.[1][2] It is therefore a tensor field of order three. It vanishes for the case of Riemannian geometry and can be used to study non-Riemannian spacetimes.[3]
Definition
By components, it is defined as follows.[1]
\( {\displaystyle Q_{\mu \alpha \beta }=\nabla _{\mu }g_{\alpha \beta }} \)
It measures the rate of change of the components of the metric tensor along the flow of a given vector field, since
\( {\displaystyle \nabla _{\mu }\equiv \nabla _{\partial _{\mu }}} \)
where \( {\displaystyle \{\partial _{\mu }\}_{\mu =0,1,2,3}} \) is the coordinate basis of vector fields of the tangent bundle, in the case of having a 4-dimensional manifold.
Relation to connection
We say that a connection \( \Gamma \) is compatible with the metric when its associated covariant derivative of the metric tensor (call it \( {\displaystyle \nabla ^{\Gamma }} \), for example) is zero, i.e.
\( {\displaystyle \nabla _{\mu }^{\Gamma }g_{\alpha \beta }=0.}
If the connection is also torsion-free (i.e. totally symmetric) then it is known as the Levi-Civita connection, which is the only one without torsion and compatible with the metric tensor. If we see it from a geometrical point of view, a non-vanishing nonmetricity tensor for a metric tensor g implies that the modulus of a vector defined on the tangent bundle to a certain point } p of the manifold, changes when it is evaluated along the direction (flow) of another arbitrary vector.
References
Hehl, Friedrich W.; McCrea, J. Dermott; Mielke, Eckehard W.; Ne'eman, Yuval (July 1995). "Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance". Physics Reports. 258 (1–2): 1–171. arXiv:gr-qc/9402012. doi:10.1016/0370-1573(94)00111-F.
Kopeikin, Sergei; Efroimsky, Michael; Kaplan, George (2011), Relativistic Celestial Mechanics of the Solar System, John Wiley & Sons, p. 242, ISBN 9783527408566.
Puntigam, Roland A.; Lämmerzahl, Claus; Hehl, Friedrich W. (May 1997). "Maxwell's theory on a post-Riemannian spacetime and the equivalence principle". Classical and Quantum Gravity. 14 (5): 1347–1356. arXiv:gr-qc/9607023. doi:10.1088/0264-9381/14/5/033.
vte
Tensors
Glossary of tensor theory
Scope
Mathematics
coordinate system multilinear algebra Euclidean geometry tensor algebra dyadic algebra differential geometry exterior calculus tensor calculus
PhysicsEngineering
continuum mechanics electromagnetism transport phenomena general relativity computer vision
Notation
index notation multi-index notation Einstein notation Ricci calculus Penrose graphical notation Voigt notation abstract index notation tetrad (index notation) Van der Waerden notation
Tensor
definitions
tensor (intrinsic definition) tensor field tensor density tensors in curvilinear coordinates mixed tensor antisymmetric tensor symmetric tensor tensor operator tensor bundle two-point tensor
Operations
tensor product exterior product tensor contraction transpose (2nd-order tensors) raising and lowering indices Hodge star operator covariant derivative exterior derivative exterior covariant derivative Lie derivative
Related
abstractions
dimension basis vector, vector space multivector covariance and contravariance of vectors linear transformation matrix spinor Cartan formalism (physics) differential form exterior form connection form geodesic manifold fiber bundle Levi-Civita connection affine connection
Notable tensors
Mathematics
Kronecker delta Levi-Civita symbol metric tensor nonmetricity tensor Christoffel symbols Ricci curvature Riemann curvature tensor Weyl tensor torsion tensor
Physics
moment of inertia angular momentum tensor spin tensor Cauchy stress tensor stress–energy tensor EM tensor gluon field strength tensor Einstein tensor metric tensor (GR)
Mathematicians
Leonhard Euler Carl Friedrich Gauss Augustin-Louis Cauchy Hermann Grassmann Gregorio Ricci-Curbastro Tullio Levi-Civita Jan Arnoldus Schouten Bernhard Riemann Elwin Bruno Christoffel Woldemar Voigt Élie Cartan Hermann Weyl Albert Einstein
External links
Iosifidis, Damianos; Petkou, Anastasios C.; Tsagas, Christos G. (May 2019). "Torsion/nonmetricity duality in f(R) gravity". General Relativity and Gravitation. 51 (5): 66. arXiv:1810.06602. doi:10.1007/s10714-019-2539-9. ISSN 0001-7701.
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