- Page 1: CRITICISMS OF THE EINSTEIN FIELD EQ
- Page 6 and 7: PREFACE where R ρ σµν is the cu
- Page 8 and 9: PREFACE Craigcefnparc, Wales Myron
- Page 10 and 11: CONTENTS 3.11 Conclusions . . . . .
- Page 12 and 13: CONTENTS 4
- Page 14 and 15: all the Christoffel symbols and all
- Page 16 and 17: geometry such as Levi-Civita and Gr
- Page 18 and 19: 2.1. INTRODUCTION untarily on ECE t
- Page 20 and 21: 2.1. INTRODUCTION Ampère Maxwell l
- Page 22 and 23: 2.2. GEOMETRICAL PRINCIPLES equival
- Page 24 and 25: 2.3. THE FIELD AND WAVE EQUATIONS O
- Page 26 and 27: 2.3. THE FIELD AND WAVE EQUATIONS O
- Page 28 and 29: 2.4. AHARONOV BOHM AND PHASE EFFECT
- Page 30 and 31: 2.4. AHARONOV BOHM AND PHASE EFFECT
- Page 32 and 33: 2.4. AHARONOV BOHM AND PHASE EFFECT
- Page 34 and 35: 2.4. AHARONOV BOHM AND PHASE EFFECT
- Page 36 and 37: 2.5. TENSOR AND VECTOR LAWS OF CLAS
- Page 38 and 39: 2.5. TENSOR AND VECTOR LAWS OF CLAS
- Page 40 and 41: 2.5. TENSOR AND VECTOR LAWS OF CLAS
- Page 42 and 43: 2.5. TENSOR AND VECTOR LAWS OF CLAS
- Page 44 and 45: 2.5. TENSOR AND VECTOR LAWS OF CLAS
- Page 46 and 47: 2.6. SPIN CONNECTION RESONANCE writ
- Page 48 and 49: 2.6. SPIN CONNECTION RESONANCE ∂2
- Page 50 and 51: 2.7. EFFECTS OF GRAVITATION ON OPTI
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2.7. EFFECTS OF GRAVITATION ON OPTI
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2.8. RADIATIVE CORRECTIONS IN ECE T
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2.8. RADIATIVE CORRECTIONS IN ECE T
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2.8. RADIATIVE CORRECTIONS IN ECE T
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2.8. RADIATIVE CORRECTIONS IN ECE T
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2.9. SUMMARY OF ADVANCES MADE BY EC
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2.9. SUMMARY OF ADVANCES MADE BY EC
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2.9. SUMMARY OF ADVANCES MADE BY EC
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2.10. APPENDIX 1: HOMOGENEOUS MAXWE
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2.11. APPENDIX 2: THE INHOMOGENEOUS
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2.12. APPENDIX 3: SOME EXAMPLES OF
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2.13. APPENDIX 4: STANDARD TENSORIA
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2.13. APPENDIX 4: STANDARD TENSORIA
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2.14. APPENDIX 5: ILLUSTRATING THE
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2.14. APPENDIX 5: ILLUSTRATING THE
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2.14. APPENDIX 5: ILLUSTRATING THE
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2.14. APPENDIX 5: ILLUSTRATING THE
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2.14. APPENDIX 5: ILLUSTRATING THE
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2.14. APPENDIX 5: ILLUSTRATING THE
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2.14. APPENDIX 5: ILLUSTRATING THE
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BIBLIOGRAPHY [15] J. B. Marion and
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3.2. SCHWARZSCHILD SPACETIME any di
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3.3. SPHERICAL SYMMETRY be determin
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3.3. SPHERICAL SYMMETRY R c(0) ≤
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3.3. SPHERICAL SYMMETRY and all oth
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3.4. DERIVATION OF SCHWARZSCHILD SP
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3.4. DERIVATION OF SCHWARZSCHILD SP
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3.4. DERIVATION OF SCHWARZSCHILD SP
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3.4. DERIVATION OF SCHWARZSCHILD SP
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3.5. THE PROHIBITION OF POINT-MASS
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3.6. LAPLACE’S ALLEGED BLACK HOLE
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3.7. BLACK HOLE INTERACTIONS AND GR
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3.7. BLACK HOLE INTERACTIONS AND GR
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3.8. FURTHER CONSEQUENCES FOR GRAVI
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3.8. FURTHER CONSEQUENCES FOR GRAVI
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3.8. FURTHER CONSEQUENCES FOR GRAVI
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3.8. FURTHER CONSEQUENCES FOR GRAVI
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3.9. OTHER VIOLATIONS The Standard
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3.9. OTHER VIOLATIONS To further il
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3.10. THREE-DIMENSIONAL SPHERICALLY
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3.10. THREE-DIMENSIONAL SPHERICALLY
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3.11. CONCLUSIONS The Riemann tenso
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BIBLIOGRAPHY [12] Information Leafl
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BIBLIOGRAPHY [43] Kay, D. C. Tensor
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BIBLIOGRAPHY [74] Sintes, A. M.,
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4.1. INTRODUCTION has no physical m
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4.3. RESULTS AND DISCUSSION line el
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4.3. RESULTS AND DISCUSSION in whic
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4.3. RESULTS AND DISCUSSION where
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4.4. EXACT SOLUTIONS OF THE EINSTEI
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Metric Compatibility ———— o
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Christoffel Connection Γ 0 01 =
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R 2 323 = 2 sin2 ϑ G M c 2 r R 2 3
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Metric gµν = ⎛ −A ⎜ ⎝ √
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R 2 020 = −R 2 002 R 2 112 = 4 C
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i µj Current Density Class 2 (-R
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Riemann Tensor R 0 101 = |r0 − r|
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Einstein Tensor 4.4. EXACT SOLUTION
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Current Density J 2 , J 3 0 -0.2 -0
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Charge Density ρ 4 3.5 3 2.5 2 1.5
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Metric 4.4. EXACT SOLUTIONS OF THE
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R 2 323 = sin 2 „ ϑ r 2 |r| ` |r
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i µj Current Density Class 3 (-R
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R 0 „ α d 313 = −e−2 d t β
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Ricci Scalar −2 β−2 α Rsc = 2
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Current Density J 1 4 3.5 3 2.5 2 1
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Γ 2 21 = Γ 2 12 Γ 2 33 = − cos
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Ricci Tensor Ric 00 = a 2 r G M +
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Contravariant Metric g µν ⎛ −
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R 2 112 = d d r µ r − µ 2 r2 (r
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i µj Current Density Class 2 (-R
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Γ 3 03 = Γ 3 13 = d d t D 2 D d d
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R 1 331 = −R1 313 4.4. EXACT SOLU
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Ricci Scalar 4 A C Rsc = − 2 D
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Metric gµν = ⎛ − ⎜ ⎝ Q2 r
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Q 2 − r M R 2 112 = − r2 (Q2
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Charge Density ρ 5 4 3 2 1 0 4.4.
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Charge Density ρ Current Density J
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Contravariant Metric g µν ⎛ −
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R 2 3 C + κ r B 112 = − 2 + r2 A
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Charge Density ρ 40 20 0 -20 -40 4
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Charge Density ρ 150 100 50 0 -50
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Γ 0 02 16 cos ϑ sin ϑ J = − 2
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R 0 220 = −R0202 R 0 223 = − 6
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R 2 301 = 12 cos ϑ sin ϑ J ` 2 r
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Ricci Scalar 4.4. EXACT SOLUTIONS O
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Current Density J 1 Current Density
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Current Density J 1 Current Density
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Coordinates Metric 0 B x = @ t r ϑ
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R 0 210 = −R0 201 4.4. EXACT SOLU
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R 3 021 = −R3012 R 3 030 = −R30
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G 21 = G 12 4.4. EXACT SOLUTIONS OF
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Γ 2 12 = 1 r Γ 2 21 = Γ 2 12 Γ
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Ricci Tensor Ric 00 = Q2 Q 2 − 2
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Metric gµν = ⎛ − ⎜ ⎝ 1 0
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R 1 313 = R 1 331 = −R 1 313 R 3
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Charge Density ρ 0.1 0.08 0.06 0.0
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Coordinates x = Metric ⎛ ⎜ ⎝
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R 1 221 = −R 1 212 R 1 313 = r2 s
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Charge Density ρ 40 30 20 10 0 -10
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4.4.17 FLRW metric 4.4. EXACT SOLUT
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Riemann Tensor R 0 2 d a d t 101 =
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Bianchi identity (Ricci cyclic equa
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Current Density J 1 Current Density
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Current Density J 2 , J 3 0 -20 -40
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Christoffel Connection Γ 0 11 = a
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R 2 112 = − R 2 121 = −R 2 112
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Charge Density ρ 0 -5 -10 -15 4.4.
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Current Density J 2 , J 3 0 -50 -10
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Current Density J 2 , J 3 0 -50 -10
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Riemann Tensor R 0 101 = cosh2 3
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Ricci Scalar Rsc = 6 2 cosh 2 3 t
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Current Density J 1 , J 2 , J 3 0 -
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Riemann Tensor R 0 2 p1 −2 101 =
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i µj Current Density Class 1 (-R
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Current Density J 2 , J 3 3 2 1 0 -
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R 1 202 = − (n − m) 2 (n + m +
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Einstein Tensor G 00 = 2 (2 m n −
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Γ 1 G M (2 G M − r) 00 = − r3
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R 2 323 = 2 sin2 ϑ G M r R 2 332 =
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Metric 0 B gµν = B @ − 32 G3 M
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R 0 212 = r 4.4. EXACT SOLUTIONS OF
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Ricci Tensor 4.4. EXACT SOLUTIONS O
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4.4. EXACT SOLUTIONS OF THE EINSTEI
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R 1 212 = − m u2 + 2 m R 1 221 =
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i µj Current Density Class 3 (-R
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R 0 303 = − R 0 330 = −R 0 303
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0 i0 Scalar Charge Density (-R i )
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Current Density J 2 , J 3 10 8 6 4
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R 0 330 = −R0303 4.4. EXACT SOLUT
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i µj Current Density Class 1 (-R
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Riemann Tensor R 0 2 101 = − “
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i µj Current Density Class 1 (-R
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Metric Compatibility ———— o
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Einstein Tensor G 00 = − G 11 = u
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Current Density J 1 0.07 0.06 0.05
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Riemann Tensor ———— all ele
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Contravariant Metric 4.4. EXACT SOL
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i µj Current Density Class 2 (-R
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Metric Compatibility ———— o
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4.4. EXACT SOLUTIONS OF THE EINSTEI
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4.4.33 Hayward-Kim-Lee wormhole typ
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R 1 001 = R 1 010 = −R 1 001 4 m
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i µj Current Density Class 1 (-R
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Metric Compatibility ———— o
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Einstein Tensor G 00 = − 2 24 r
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Γ 2 33 = − cos ϑ sin ϑ Γ 3 13
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Current Density J 1 2 1.5 1 0.5 4.4
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Metric Compatibility ———— o
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Ricci Scalar 2 r b (r) Rsc = 4.4. E
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Contravariant Metric g µν 0 1 0 0
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R 2 020 = −R 2 002 R 2 112 = 2 B
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Einstein Tensor A G00 = C G22 = 4.4
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Christoffel Connection Γ 0 01 = a
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R 2 “ `3 ´ 2 a a r2 + 1 3 − b
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Hodge Dual of Bianchi Identity —
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Current Density J 2 , J 3 1000 800
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R 2 112 = −1 R 2 121 = −R 2 112
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Current Density J 1 Current Density
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Γ 1 33 = µ3 (τ − ρ) sin 2 ϑ
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R 3 003 = − 2 9 (τ − ρ) 2 R 3
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Current Density J 2 , J 3 0 -0.2 -0
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Γ 0 03 = “ ” ` d d f d x f 3 d
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4.4. EXACT SOLUTIONS OF THE EINSTEI
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R 0 301 = d d x 3 f 4.4. EXACT SOLU
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4.4. EXACT SOLUTIONS OF THE EINSTEI
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4.4. EXACT SOLUTIONS OF THE EINSTEI
- Page 386 and 387:
R 3 101 = R 3 021 = −R3012 R 3 03
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Ric 33 = − Ric 30 = Ric 03 Ric 31
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i µj Current Density Class 1 (-R
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Current Density J 1 2 1.5 1 0.5 0 -
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Riemann Tensor R 0 303 = − d 2 d
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i µj Current Density Class 2 (-R
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Metric Compatibility ———— o
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Ricci Scalar 4.4. EXACT SOLUTIONS O
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Christoffel Connection Γ 1 00 = si
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i µj Current Density Class 1 (-R
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Metric Compatibility ———— o
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R 2 112 = 1 R 2 121 = −R2112 R 2
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Einstein Tensor G 00 = G 03 = G 11
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Current Density J 1 Current Density
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Christoffel Connection Γ 0 01 = co
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i µj Current Density Class 1 (-R
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Γ 0 20 = Γ 0 02 Γ 0 23 = − 2
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R 0 220 = −R 0 202 R 0 4 y (6 y
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R 2 320 = −R 2 302 R 2 323 = x2
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Ricci Scalar Rsc = − 6 48 y 4
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Charge Density ρ 8 7 6 5 4 3 2 1 4
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Metric gµν = ⎛ −1 0 0 0 ⎜ 1
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Ricci Scalar Rsc = 6 a 2 Bianchi id
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Current Density J 2 , J 3 0 -2 -4 -
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R 0 101 = 2 a 4 − 8 R 0 110 = −
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Hodge Dual of Bianchi Identity —
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Current Density J 3 20 15 10 5 0 4.
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0 i0 Scalar Charge Density (-R i )
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Metric gµν = ⎛ −ε 0 0 0 ⎜
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R 0 331 = −R 0 313 R 1 001 = d 2
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Ricci Tensor Ric 22 = Ric 33 = 2 d
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4.4.52 Electrovacuum metric 4.4. EX
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Einstein Tensor G 00 = − 3 2 2 G
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Γ 1 22 = −Σ Γ 1 33 = −Σ Γ
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R 2 121 = −R 2 112 R 2 323 = −
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i µj Current Density Class 2 (-R
- Page 458 and 459:
Riemann Tensor R 0 101 = A R 0 110
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Bianchi identity (Ricci cyclic equa
- Page 462 and 463:
Christoffel Connection Γ 0 01 =
- Page 464 and 465:
R 3 003 = e−2 a x n 2 e 2 a x
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Charge Density ρ Current Density J
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Coordinates x = Metric ⎛ ⎜ ⎝
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R 1 221 = −R 1 212 R 1 313 = −
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0 i0 Scalar Charge Density (-R i )
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Metric Compatibility ———— o
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0 i0 Scalar Charge Density (-R i )
- Page 478 and 479:
Metric Compatibility ———— o
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0 i0 Scalar Charge Density (-R i )
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Γ 2 32 = Γ 2 23 Γ 3 00 = e−2 t
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Ricci Tensor Ric00 = − 2 (a − 1
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Contravariant Metric g µν = ⎛
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R 2 323 = sin 2 ϑ e −2 λ e λ
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i µj Current Density Class 3 (-R
- Page 492 and 493:
Metric Compatibility ———— o
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R 3 103 d d r = − y ` d d t λ´
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i µj Current Density Class 3 (-R
- Page 498 and 499:
R 0 313 = − b2 cos 2 (b v + a u)
- Page 500 and 501:
Hodge Dual of Bianchi Identity —
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4.4. EXACT SOLUTIONS OF THE EINSTEI
- Page 504 and 505:
BIBLIOGRAPHY 496
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5.2. EINSTEIN’S EARLY WORK AND HO
- Page 508 and 509:
5.2. EINSTEIN’S EARLY WORK AND HO
- Page 510 and 511:
5.2. EINSTEIN’S EARLY WORK AND HO
- Page 512 and 513:
5.2. EINSTEIN’S EARLY WORK AND HO
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5.2. EINSTEIN’S EARLY WORK AND HO
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5.2. EINSTEIN’S EARLY WORK AND HO
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5.2. EINSTEIN’S EARLY WORK AND HO
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5.2. EINSTEIN’S EARLY WORK AND HO
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5.3. EINSTEIN AND GENERAL RELATIVIT
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5.3. EINSTEIN AND GENERAL RELATIVIT
- Page 526 and 527:
5.4. TESTING RELATIVITY, BY OBSERVI
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5.4. TESTING RELATIVITY, BY OBSERVI
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5.5. BLACK HOLES, SINGULARITIES AND
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5.5. BLACK HOLES, SINGULARITIES AND
- Page 534 and 535:
5.5. BLACK HOLES, SINGULARITIES AND
- Page 536 and 537:
5.6. NEW COSMOLOGIES place in the f
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5.6. NEW COSMOLOGIES out, as spinni
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5.6. NEW COSMOLOGIES their orbit wi
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5.6. NEW COSMOLOGIES and the force
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5.6. NEW COSMOLOGIES as the univers
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5.6. NEW COSMOLOGIES are thought to
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5.7. DARK MATTER IN FOCUS theory is