DOC. 25 FIELD EQUATIONS OF GRAVITATION 117
Doc. 25
[p. 844] Session of the physical-mathematical class on November 25, 1915
The Field Equations of Gravitation
by A. Einstein
In two recently published papers1 I have shown how to obtain field equations of
gravitation that comply with the postulate of general relativity, i.e., which in their
general formulation are covariant under arbitrary substitutions of space-time variables.
Historically they evolved in the following sequence. First, I found equations that
contain the Newtonian theory as an approximation and are also covariant under
arbitrary substitutions of determinant 1. Then I found that these equations are
equivalent to generally-covariant ones if the scalar of the energy tensor of "matter"
vanishes. The coordinate system could then be specialized by the simple rule that
v/-g must equal 1, which leads to an immense simplification of the equations of the
theory. It has to be mentioned, however, that this requires the introduction of the
hypothesis that the scalar of the energy tensor of matter vanishes.
I now quite recently found that one can get away without this hypothesis about
the energy tensor of matter merely by inserting it into the field equations in a slightly
different way. The field equations for vacuum, onto which I based the explanation
of the Mercury perihelion, remain unaffected by this modification. In order not to [2]
force the reader constantly to consult the previous publications, I repeat here the
considerations in their entirety.
One derives from the well-known Riemann-covariant of rank four the following
covariant of rank two:
Gim = Rim + Sim (1)
Rim = -El d{iml}/dxl + Elp{ilp}{mpl} (1a)
Sim = -El d{ill}/dxm - Elp{imp}{pll} (1b)
1Sitzungsber. 44 p. 778 and 46, p. 799 (1915). [1]

- Publisher's Foreword Page ix
- Translator's Preface Page xi
### Selected Documents Page 1

- Selected Documents Page 1
- Doc. 1 On the Principle of Relativity Page 3
- Doc. 2 Covariance Properties of the Field Equations of the Theory of Gravitation Based on the General Theory of Relativity Page 6
- Doc. 3 Inaugural Lectures and Responses Page 16
- Doc. 4 Remarks on P. Harzer's Paper: "On the Dragging of Light in Glass and on Aberration" Page 19
- Doc. 5 Contributions to Quantum Theory Page 20
- Doc. 6 Response to Paul Harzer's Reply Page 27
- Doc. 7 Lecture Notes for Course on Relativity at the University of Berlin, Winter Semester 1914-1915 Page 27
- Doc. 8 Manifesto to the Europeans Page 28
- Doc. 9 The Formal Foundation of the General Theory of Relativity Page 30
- Doc. 10 Review of Alexander Brill, The Principle of Relativity: An Introduction to the Theory Page 85
- Doc. 11 Review of H. A. Lorentz, The Principle of Relativity: Three Lectures ... Page 85
- Doc. 12 Expert Opinion on Legal Dispute between Anschütz & Co. and Sperry Gyroscope Company Page 85
- Doc. 13 Experimental Proof of Ampere's Molecular Currents Page 85
- Doc. 14 Experimental Proof of the Existence of Ampere's Molecular Currents Page 86
- Doc. 15 Experimental Proof of Ampere's Molecular Currents Page 87
- Doc. 16 Correction of My Joint Paper with J. W. de Haas: "Experimental Proof of Ampere's Molecular Currents" Page 87
- Doc. 17 Comment on the Essay Submitted by Knapp: "The Shearing of the Light-Ether ..." Page 87
- Doc. 18 Response to a Paper by M. von Laue: "A Theorem in Probability Calculus and Its Application to Radiation Theory" Page 88
- Doc. 19 Supplementary Expert Opinion Page 95
- Doc. 20 My Opinion on the War Page 96
- Doc. 21 On the General Theory of Relativity Page 98
- Doc. 22 On the General Theory of Relativity (Addendum) Page 108
- Doc. 23 Comment on Our Paper: "Experimental Proof of Ampere's Molecular Currents" Page 111
- Doc. 24 Explanation of the Perihelion Motion of Mercury from the General Theory of Relativity Page 112
- Doc. 25 The Field Equations of Gravitation Page 117
- Doc. 26 On the Theory of Tetrode and Sackur for the Entropy Constant Page 121
- Doc. 27 A New Formal Interpretation of Maxwell's Field Equations of Electrodynamics Page 132
- Doc. 28 A Simple Experiment to Demonstrate Ampere's Molecular Currents Page 138
- Doc. 29 Ernst Mach Page 141
- Doc. 30 The Foundation of the General Theory of Relativity Page 146
- Doc. 31 Appendix. Formulation of the Theory on the Basis of a Variational Principle Page 200
- Doc. 32 Approximative Integration of the Field Equations of Gravitation Page 201
- Doc. 33 Einstein's Memorial Lecture on Karl Schwarzschild Page 211
- Doc. 34 Emission and Absorption of Radiation in Quantum Theory Page 212
- Doc. 35 "Preface" to Erwin Freundlich, The Foundations of Einstein's Theory of Gravitation Page 217
- Doc. 36 Review of H. A. Lorentz, Statistical Theories in Thermodynamics: Five Lectures ... Page 218
- Doc. 37 Author's Summary of The Foundation of the General Theory of Relativity Page 219
- Doc. 38 On the Quantum Theory of Radiation Page 220
- Doc. 39 Elementary Theory of Water Waves and of Flight Page 234
- Doc. 40 5. On Friedlich Kottler's Paper: "On Einstein's Equivalence Hypothesis and Gravitation" Page 237
- Doc. 41 Hamilton's Principle and the General Theory of Relativity Page 240
- Doc. 42 On the Special and General Theory of Relativity (A Popular Account) Page 247
- Doc. 43 Cosmological Considerations in the General Theory of Relativity Page 421
- Doc. 44 Reply to the Plaintiff's Written Statement of 27 December 1916 Page 433
- Doc. 45 On the Quantum Theorem of Sommerfeld and Epstein Page 434
- Doc. 46 Review of Hermann von Helmholtz, Two Lectures on Goethe Page 444
- Doc. 47 A Derivation of Jacobi's Theorem Page 445
- Doc. 48 Marian von Smoluchowski Page 448
- Doc. 49 The Nightmare Page 449