# Einstein 1916 — The Foundation of GR (§§15–18) **By A. Einstein** Translated from "Die Grundlage der allgemeinen Relativitätstheorie," *Annalen der Physik*, 49, 1916. This is the foundational paper of General Relativity. We focus on §§15–18 (pp.145–153), which is where Einstein introduces the pseudotensor, claims conservation of energy and momentum, and admits the pseudotensor is not a tensor. This is the section Levi-Civita targeted in his 1917 paper. [[1917_Analytic_Expression_of_the_Gravitational_Tensor]] --- ## Page 37 / p.145 — §15 Opens: The Hamiltonian Function ![[1916_Einstein_Foundation_GR-37.png]] Einstein states that the field equations, combined with the equations of motion (46), give "to a first approximation Newton's law of attraction, and to a second approximation the explanation of the motion of the perihelion of the planet Mercury discovered by Leverrier." He considers these facts "a convincing proof of the correctness of the theory." > [!info] The perihelion is referenced, not derived > The perihelion result (Eq. 75 later in the paper) is only mentioned here as motivation. The actual derivation is not in this paper — Einstein cites his 1915 paper for that. See [[1915_Perihelion_Motion_of_Mercury]] and [[2020_Vankov_Perihelion_Critique]] for the derivation and its critique. ### §15. The Hamiltonian Function for the Gravitational Field. Laws of Momentum and Energy Einstein writes the field equations in Hamiltonian form: $\delta\left\{\int H d\tau = 0\right\}$ $H = g^{\mu\nu}\Gamma^\alpha_{\mu\beta}\Gamma^\beta_{\nu\alpha} \tag{47a}$ $\sqrt{-g} = 1$ > [!info] What is H? > H is the Hamiltonian density for the gravitational field. It is built from g^μν (the inverse metric) and products of Christoffel symbols (Γ). Note: H is composed entirely of the metric and its first derivatives (since Christoffel symbols are first derivatives of the metric). This is significant — H is a first order intrinsic quantity. Everything built from it inherits that character. Taking the variation gives: $\frac{\partial H}{\partial g^{\mu\nu}} = -\Gamma^\alpha_{\mu\beta}\Gamma^\beta_{\nu\alpha}$ $\frac{\partial H}{\partial g^\mu_\sigma} = \Gamma^\sigma_{\mu\nu} \tag{48}$ --- ## Page 38 / p.146 — The Pseudotensor is Born (Eq. 49–50) ![[1916_Einstein_Foundation_GR-38.png]] Carrying out the variation in (47a) gives: $\frac{\partial}{\partial x_\alpha}\left(\frac{\partial H}{\partial g^\mu_\alpha}\right) - \frac{\partial H}{\partial g^{\mu\nu}} = 0 \tag{47b}$ which agrees with the field equations (47), as was to be proved. Einstein then multiplies (47b) by g^μν_σ and manipulates to obtain: $\frac{\partial t^\alpha_\sigma}{\partial x_\alpha} = 0 \tag{49}$ with the definition: $-2\kappa t^\alpha_\sigma = g^{\mu\nu}_\sigma \frac{\partial H}{\partial g^\mu_\alpha} - \delta^\alpha_\sigma H$ which, using (48) and (34), gives: $\kappa t^\alpha_\sigma = \frac{1}{2}\delta^\alpha_\sigma g^{\mu\nu}\Gamma^\lambda_{\mu\beta}\Gamma^\beta_{\nu\lambda} - g^{\mu\nu}\Gamma^\alpha_{\mu\beta}\Gamma^\beta_{\nu\sigma} \tag{50}$ > [!danger] This is the pseudotensor > Eq. 50 defines Einstein's pseudotensor t^α_σ. Look at what's inside: g^μν (inverse metric), Γ (Christoffel symbols), and nothing else. Every ingredient is built from the metric and its first derivatives. This is the object Levi-Civita contracted in 1917 to show its trace reduces to G* (a first order intrinsic quantity), invoking the 1900 non-existence theorem to declare it inadmissible. [[1917_Analytic_Expression_of_the_Gravitational_Tensor#Section 9 The Killing Blow Against Einsteins Pseudotensor]] > > Crothers' Eq. 2.1 in [[2020_Einstein_Pseudotensor_Crothers]] and [[2018_Crothers_GW_Speed_Coordinate_Dependent]] is this same equation. --- ## Page 39 / p.147 — "It is not a tensor" ![[1916_Einstein_Foundation_GR-39.png]] ### Einstein's Admission > **"It is to be noticed that t^α_σ is not a tensor; on the other hand (49) applies to all systems of co-ordinates for which √(-g) = 1. This equation expresses the law of conservation of momentum and of energy for the gravitational field."** > [!danger] The admission and the contradiction > In the same paragraph, Einstein: > 1. Admits t^α_σ is not a tensor > 2. Claims it expresses the law of conservation of momentum and energy > > But GR's entire framework demands that physically meaningful quantities be tensors (coordinate independent). An object that is not a tensor has no coordinate independent physical meaning. Einstein is using a mathematically inadmissible object to express the most fundamental laws of physics. > > He also restricts Eq. 49 to coordinate systems where √(-g) = 1. A proper tensor equation holds in ALL coordinate systems. The restriction to special coordinates is another sign that this is not a proper tensorial conservation law. ### The "Energy Components" > "The quantities t^α_σ we call the 'energy components' of the gravitational field." Einstein then rewrites the field equations in "mixed" form (Eq. 51): $\frac{\partial}{\partial x_\alpha}(g^{\sigma\beta}\Gamma^\alpha_{\mu\beta}) = -\kappa(t^\sigma_\mu - \frac{1}{2}\delta^\sigma_\mu t) \tag{51}$ with √(-g) = 1. > [!info] Eq. 51 compared to Levi-Civita's Eq. 2.8 > Compare this with Levi-Civita's proper tensor version: G^μ_ν/κ + T^μ_ν = 0 (Eq. 2.8 in [[2018_Crothers_GW_Speed_Coordinate_Dependent]]). Both attempt to express gravitational energy. Einstein's version uses the pseudotensor (inadmissible). Levi-Civita's uses the Einstein tensor G^μ_ν (proper tensor, valid). They give different answers: Einstein's gives nonzero gravitational energy, Levi-Civita's gives zero in vacuum. --- ## Page 40 / p.148 — §16. The General Form of the Field Equations ![[1916_Einstein_Foundation_GR-40.png]] Einstein compares his matter free field equations (from §15) with Newton's ∇²ϕ = 0, and says we need the equation corresponding to Poisson's equation ∇²ϕ = 4πκρ, where ρ is the density of matter. He introduces the matter energy tensor T^α_σ (a proper, symmetric tensor) and states that we must introduce it "which, like the energy-components t_σ [equations (49) and (50)] of the gravitational field, will have mixed character." ### Eq. 52: The Combined Conservation Law $\frac{\partial}{\partial x_\alpha}(g^{\sigma\beta}T^\alpha_{\mu\beta}) = -\kappa[(t^\sigma_\mu + T^\sigma_\mu) - \frac{1}{2}\delta^\sigma_\mu(t + T)] \tag{52}$ > [!info] Real tensor + fake tensor = "conservation" > Eq. 52 combines the proper matter tensor T^σ_μ with the fake pseudotensor t^σ_μ. The total "conservation" law (Eq. 56 on the next page) passes through both objects. The real physics (matter energy, T) is coupled to a mathematically invalid object (gravitational energy, t). The entire conservation scheme depends on the invalid piece. --- ## Page 41 / p.149 — §17. Einstein Admits the Energy Tensor is Not Justified ![[1916_Einstein_Foundation_GR-41.png]] Einstein writes the field equations in mixed form (Eq. 53): $\frac{\partial}{\partial x_\alpha}\Gamma^\alpha_{\mu\nu} + \Gamma^\alpha_{\mu\beta}\Gamma^\beta_{\nu\alpha} = -\kappa(T_{\mu\nu} - \frac{1}{2}g_{\mu\nu}T) \tag{53}$ with √(-g) = 1. Then this remarkable admission: > **"It must be admitted that this introduction of the energy-tensor of matter is not justified by the relativity postulate alone. For this reason we have here deduced it from the requirement that the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy."** > [!danger] Einstein admits the energy tensor is not justified by GR > The introduction of T_μν is motivated by the *requirement* that gravitational energy should act like other energy. This is an assumption imposed on the theory, not a consequence of it. GR by itself does not tell you how to represent matter energy. Einstein deduces it from a physical requirement (that gravity gravitates), not from the mathematical structure of the theory. ### §17. The Laws of Conservation in the General Case From (55) and (52a), it follows that: $\frac{\partial(t^\sigma_\mu + T^\sigma_\mu)}{\partial x_\sigma} = 0 \tag{56}$ > **"Thus it results from our field equations of gravitation that the laws of conservation of momentum and energy are satisfied."** > [!danger] The conservation law that passes through a non-tensor > Eq. 56 is the total conservation law. It uses an ordinary divergence (∂, not the covariant derivative ∇). It combines the proper tensor T^σ_μ with the inadmissible pseudotensor t^σ_μ. This is the equation Crothers and Levi-Civita identify as the core problem: the conservation of energy and momentum in GR depends on an object the framework says cannot have physical meaning. --- ## Page 42 / p.150–151 — §17 (continued) and §18 ![[1916_Einstein_Foundation_GR-42.png]] ### Eq. 57: Conservation of Matter Energy $\frac{\partial T^\alpha_\sigma}{\partial x_\alpha} + \frac{1}{2}\frac{\partial g^{\mu\nu}}{\partial x_\sigma}T_{\mu\nu} = 0 \tag{57}$ Einstein notes: "this equation predicates nothing more or less than the vanishing of divergence of the material energy-tensor." But the second term on the left shows that **laws of conservation of momentum and energy do not apply in the strict sense for matter alone**. They apply "only when the g_μν are constant, i.e. when the field intensities of gravitation vanish." > [!important] Conservation only works when there's no gravity > Einstein admits that energy and momentum conservation for matter alone only holds when gravity is absent (g_μν constant). The moment gravity is present, you need the pseudotensor t^σ_μ to restore conservation. But the pseudotensor is inadmissible. So GR can only conserve energy by using a mathematically invalid object. Without it, conservation fails in the presence of gravity — which is the only situation where GR is supposed to apply. Eq. 57a rewrites this as: $\frac{\partial T^\alpha_\sigma}{\partial x_\alpha} = -\Gamma^\beta_{\alpha\sigma}T^\alpha_\beta \tag{57a}$ "The right side expresses the energetic effect of the gravitational field on matter." --- ## Page 43 / p.152 — §18 and §19 ![[1916_Einstein_Foundation_GR-43.png]] ### §18. The Laws of Momentum and Energy for Matter, as a Consequence of the Field Equations Einstein notes that "it is not necessary to introduce definite assumptions as to the physical nature of matter (in the narrower sense)." The general principle of relativity "is unable on principle to tell us anything about this." > [!info] GR cannot determine the nature of matter > Einstein admits the theory cannot tell you what matter is. It remains an open question whether electromagnetism plus gravity is sufficient for a theory of matter, or whether additional physics is needed. The theory takes T_μν as given from outside, it does not derive it. ### §19. Euler's Equations for a Frictionless Adiabatic Fluid The contravariant energy tensor of a perfect fluid: $T^{\alpha\beta} = -g^{\alpha\beta}p + \rho\frac{dx_\alpha}{ds}\frac{dx_\beta}{ds} \tag{58}$ where p is pressure and ρ is density. > [!info] Where Big Bang cosmology comes from > This is the perfect fluid energy tensor that all Big Bang models use. Every cosmological model (Friedmann, FLRW, etc.) treats the entire universe as a single indivisible fluid of uniform density and pressure. This is why Crothers notes that Big Bang models contain "arbitrarily many masses" treated as one mass — they use Eq. 58 with uniform ρ and p everywhere. [[2018_Crothers_GW_Speed_Coordinate_Dependent#Page 7 p7 LIGO and the Two Body Problem]] --- ## Page 44–45 / pp.153 ![[1916_Einstein_Foundation_GR-44.png]] ![[1916_Einstein_Foundation_GR-45.png]] Einstein notes the equations are "over-defined" (eleven equations for ten functions g_μν) but this is resolved because equations (57a) are already contained in (53), leaving "only seven independent equations." The remaining freedom is in the choice of coordinates: "three of the functions of space may be chosen at will." Then §20 begins: Maxwell's Electromagnetic Field Equations for Free Space. --- ## Summary: What Levi-Civita Attacked | Item | Einstein's Claim | The Problem | |---|---|---| | **Eq. 50** (the pseudotensor) | Represents gravitational energy | Built entirely from metric + first derivatives. Not a tensor. Contracting it gives a first order intrinsic invariant that cannot exist (1900 theorem). | | **Eq. 49** (∂t^α_σ/∂x_α = 0) | Conservation of gravitational energy | Uses ordinary divergence (∂), not covariant divergence (∇). Only valid for coordinates where √(-g) = 1. Not a proper tensor equation. | | **Eq. 56** (∂(t+T)/∂x = 0) | Total conservation of energy | Combines a valid tensor (T) with an invalid pseudotensor (t). The conservation law depends on the invalid piece. | | **Eq. 57** | Conservation for matter alone | Only holds when gravity vanishes (g_μν constant). In the presence of gravity, conservation requires the pseudotensor. | | **"It is not a tensor"** (p.147) | Einstein's own admission | He admits it, then uses it anyway. The entire gravitational energy framework rests on an object the author acknowledges is not a tensor. | --- ## Citation Einstein, A. "The Foundation of the General Theory of Relativity." *Annalen der Physik*, 49, 1916. Translated from "Die Grundlage der allgemeinen Relativitätstheorie." Zotero: `VSABH3DA` --- ## See Also - [[1917_Analytic_Expression_of_the_Gravitational_Tensor]] — Levi-Civita's direct response to §15 of this paper - [[2020_Einstein_Pseudotensor_Crothers]] — Crothers' single page argument against Eq. 50 - [[2020_Landau_Lifshitz_Pseudotensor_Crothers]] — Same argument extended to the Landau-Lifshitz version - [[2018_Crothers_GW_Speed_Coordinate_Dependent]] — Full critique including the conservation law problem - [[1900_Methodes_de_Calcul_Differentiel_Absolu]] — The 1900 theorem that kills the pseudotensor's contraction (p.162) - [[00_Gravitational_Waves_Index]] — Master index and presentation outline