definition 44.2 Einstein tensor

open in the book · parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:136 · p. 1264

ground object -- no derivation owed

Rests on

depends_on definition 13.153 Contractions ¶
- depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
  - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.47 Differentiable structure ¶
      - depends_on definition 13.44 Atlas ¶
  - depends_on definition 13.84 Tensor ¶
    - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.81 Vector on a manifold ¶ ↺
- depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
  - depends_on definition 13.145 Affine connection ¶
    - depends_on definition 13.90 Mixed tensor ¶
      - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.88 General coordinate transformation ¶
    - depends_on definition 13.82 Vector field ¶
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
  - depends_on definition 13.147 Torsion ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
  - depends_on proposition 7.105 Clairaut–Schwarz ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-64 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶

Supports

depends_on postulate 44.3 Einstein field equations ¶
- depends_on corollary 44.24 Conservation is an integrability condition ¶
- depends_on phenomenon 49.18 Orbital decay of a binary pulsar ¶
- depends_on phenomenon 46.1 phen:gravitational-waves ¶
  - depends_on phenomenon 49.18 Orbital decay of a binary pulsar ¶ ↺
  - tests experiment GW150914 ¶
    - cites (method) derived bib Advanced LIGO ¶
    - cites (method) derived bib Advanced LIGO ¶ ↺
    - cites (evidence) derived bib Observation of Gravitational Waves from a Binary Black Hole Merger ¶
    - cites (method) derived bib Observation of Gravitational Waves from a Binary Black Hole Merger ¶ ↺
    - cites (evidence) derived bib Observation of Gravitational Waves from a Binary Black Hole Merger ¶ ↺
    - cites (evidence) derived bib Observation of Gravitational Waves from a Binary Black Hole Merger ¶ ↺
    - cites (evidence) derived bib Observation of Gravitational Waves from a Binary Black Hole Merger ¶ ↺
    - cites (evidence) derived bib Observation of Gravitational Waves from a Binary Black Hole Merger ¶ ↺
  - tests experiment The Hulse–Taylor Binary Pulsar (PSR B1913+16) ¶
    - cites (evidence) derived bib Discovery of a pulsar in a binary system ¶
    - cites (evidence) derived bib Discovery of a pulsar in a binary system ¶ ↺
    - cites (data) derived bib Discovery of a pulsar in a binary system ¶ ↺
    - cites (evidence) derived bib Discovery of a pulsar in a binary system ¶ ↺
    - cites (evidence) derived bib A new test of general relativity: Gravitational radiation and the binary pulsar PSR 1913+16 ¶
    - cites (evidence) derived bib A new test of general relativity: Gravitational radiation and the binary pulsar PSR 1913+16 ¶ ↺
    - cites (evidence) derived bib Relativistic Measurements from Timing the Binary Pulsar PSR B1913+16 ¶
    - cites (data) derived bib Relativistic Measurements from Timing the Binary Pulsar PSR B1913+16 ¶ ↺
    - cites (data) derived bib Relativistic Measurements from Timing the Binary Pulsar PSR B1913+16 ¶ ↺
    - cites (evidence) derived bib Relativistic Measurements from Timing the Binary Pulsar PSR B1913+16 ¶ ↺
    - cites (evidence) derived bib Relativistic Measurements from Timing the Binary Pulsar PSR B1913+16 ¶ ↺
    - cites (evidence) derived bib Relativistic Measurements from Timing the Binary Pulsar PSR B1913+16 ¶ ↺
- depends_on proposition 44.29 Linearized field equations ¶
- depends_on proposition 44.4 Trace-reversed form ¶
  - depends_on proposition 44.39 SEC is the attraction condition ¶
  - depends_on theorem 44.27 Newtonian correspondence of the field equations ¶
    - depends_on corollary 44.35 The field equations are essentially unique ¶
    - depends_on phenomenon 49.6 A maximum mass for a cold degenerate star ¶
    - depends_on phenomenon 49.16 Pulsars: clocks at nuclear density ¶
    - depends_on phenomenon 49.14 A neutron star is observed being born ¶
    - depends_on proposition 44.29 Linearized field equations ¶ ↺
- depends_on theorem 49.9 Tolman–Oppenheimer–Volkoff equations ¶
  - depends_on example 49.11 The uniform-density star ¶
    - depends_on proposition 49.12 Buchdahl bound ¶
      - depends_on phenomenon 49.19 Neutron stars reach two solar masses ¶
  - depends_on phenomenon 49.19 Neutron stars reach two solar masses ¶ ↺
- depends_on theorem 48.5 The Friedmann equations ¶
  - depends_on phenomenon 48.17 The expansion is accelerating ¶
  - depends_on phenomenon 48.10 A primordial floor of light elements ¶
  - depends_on phenomenon 51.6 Acoustic peaks, and a spatially flat universe ¶
    - depends_on phenomenon 51.7 E-mode polarization, and no primordial B mode ¶
  - depends_on theorem 48.14 Linear growth of density perturbations ¶
    - depends_on phenomenon 48.16 Flat rotation curves and the missing mass ¶
- depends_on theorem 44.42 The constraint equations ¶
  - depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
    - depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶
  - depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶ ↺
depends_on theorem 44.23 Contracted Bianchi identity ¶
- depends_on corollary 44.24 Conservation is an integrability condition ¶ ↺
- depends_on proposition 44.41 Four of the ten equations are constraints ¶
- depends_on proposition 44.30 The cosmological term ¶
  - depends_on proposition 44.31 de Sitter geometry ¶
- depends_on theorem 44.33 Lovelock's uniqueness theorem; imported ¶
  - depends_on corollary 44.35 The field equations are essentially unique ¶ ↺
depends_on theorem 44.33 Lovelock's uniqueness theorem; imported ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Contractions	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:145
`depends_on`	←	Einstein field equations	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:158
`depends_on`	←	Contracted Bianchi identity	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:994
`depends_on`	←	Lovelock's uniqueness theorem; imported	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1427