# Pseudotensor and Conservation Laws ## The question Einstein's field equations $G_{\mu\nu} = -\kappa T_{\mu\nu}$ are proper tensor equations. Nobody disputes them. The problem is what happens next: to write conservation laws for gravitational energy, Einstein introduced a quantity $t_{\mu\nu}$ called the pseudotensor. This object sits outside the field equations and is used to account for the energy carried by the gravitational field itself. The question is whether this pseudotensor is a valid mathematical object within the framework Einstein used to build his theory. ## Null hypothesis **H₁:** Solutions to Einstein's field equations satisfy conservation laws within the framework of absolute differential calculus. **H₀:** The solutions to Einstein's field equations do not satisfy the conservation laws within the framework of absolute differential calculus. --- ## The framework: absolute differential calculus Founded by Ricci and Levi-Civita (1900). This is the mathematical language Einstein used to build general relativity. Its purpose is to produce results that are independent of coordinate choice. If something depends on your coordinates, it has no physical meaning within the framework. The framework has specific rules about what kinds of mathematical objects are admissible, and those rules were published 15 years before Einstein used them. [[1900_Methodes_de_Calcul_Differentiel_Absolu]] ### The 1900 theorem On page 162 of their foundational paper, Ricci and Levi-Civita proved: > "Les formes de classe superieure ne possedent pas des invariants differentiels du premier ordre." > > "Forms of class greater than zero do not possess first order differential invariants." In plain language: if your space has any curvature at all (class greater than zero), you cannot build a meaningful quantity from only the first derivatives of the metric. You need second derivatives. First order invariants simply do not exist. This is a proven theorem, not a conjecture. For flat space (class zero), there are no intrinsic differential invariants of any order, because flat space has no intrinsic geometry to measure. Either way, first order is ruled out. [[1900_Methodes_de_Calcul_Differentiel_Absolu]] --- ## The hierarchy: what is admissible and what is not | Object | Derivative order | Valid tensor? | Notes | |--------|-----------------|---------------|-------| | $g_{\mu\nu}$ (metric) | 0 (zeroth) | Yes | The foundation | | $\partial g$ (partial of metric) | 1 (first) | Vanishes by coordinate choice | Not invariant | | $\Gamma$ (Christoffel symbol) | 1 (first) | Not even a tensor | Built from $g$ and $\partial g$ | | $G_{\mu\nu}$, $R_{\mu\nu}$ (Einstein/Ricci) | 2 (second) | Yes | Built from $\partial\Gamma = \partial^2 g$ | | $T_{\mu\nu}$ (matter energy) | 2 (second) | Yes | Proper tensor | | $A_{ik}$ (Levi-Civita 1917) | 2 (second) | Yes | Gives zero in vacuum | | **$t_{\mu\nu}$ (Einstein pseudotensor)** | **1 (first)** | **No** | **Inadmissible** | | **$t^{ik}$ (Landau-Lifshitz)** | **1 (first)** | **No** | **Inadmissible** | The field equations themselves ($G_{\mu\nu} = -\kappa T_{\mu\nu}$) are second order and perfectly valid. The pseudotensor is first order and inadmissible by the framework's own rules. --- ## What the pseudotensor is made of Einstein's pseudotensor $t^\alpha_\sigma$ is built from $g^{\mu\nu}$ (the metric) and $\Gamma$ (Christoffel symbols). Christoffel symbols are constructed from $g$ and $\partial g$ (first derivatives of the metric), and they are not tensors. They are connection coefficients that depend on the coordinate system. When you contract the pseudotensor (set upper and lower indices equal and sum), the trace reduces to a quantity Levi-Civita called $G^*$. This quantity contains nothing but metric components and products of Christoffel symbols. First derivatives of the metric all the way down, no second derivatives anywhere. ### The contraction test The identification is visual: there is no 2 next to the derivative symbol. That is all you need to see. You do not have to solve the equation. Just look at the notation. If the derivative is first order ($\partial$) where it needs to be second order ($\partial^2$), it is inadmissible. The 1900 theorem says such an object does not exist as a meaningful invariant. Everything that follows from it is bankrupt. [[2020_Einstein_Pseudotensor_Crothers]] --- ## Einstein knew: November 4, 1915 In the same paper where Einstein demands general covariance and credits Levi-Civita by name, he introduces the pseudotensor (Eq. 20a/20b), admits it "has tensorial character only under linear transformations," and derives equations that are coordinate dependent. He demanded coordinate independence, built energy conservation from a coordinate dependent object, then restricted which coordinates are allowed. All in 7 pages. [[1915_Einstein_Field_Equations_Nov4_Nov11]] From the 1912 to 1914 correspondence, Levi-Civita had shown Einstein multiple times that this approach does not work. Einstein's reply was to call a linear transformation "a special case," dismissing the minimum backbone of even his restricted theory. [[2012_Einstein_the_Stubborn_Correspondence]] --- ## Einstein's conservation law: the core problem Einstein needed matter energy ($T$) plus gravitational energy ($t$) to be conserved: $\frac{\partial}{\partial x_\alpha}(T^\alpha_\sigma + t^\alpha_\sigma) = 0$ But $T^\alpha_\sigma$ is a proper tensor and $t^\alpha_\sigma$ is not a tensor. This conservation law uses ordinary divergence (not covariant divergence) and mixes a real tensor with a fake one. Einstein admitted this openly in his 1916 foundational paper, Section 15: "It is to be noticed that $t$ is not a tensor." [[1916_Einstein_Foundation_GR_Sections15_17]] --- ## Levi-Civita (1917): "Einstein's Misunderstanding" In his 1917 paper, Levi-Civita addressed the pseudotensor directly: **Section 8:** He constructs the proper tensor: $A_{ik} = \frac{1}{\kappa}\left(G_{ik} - \frac{1}{2}g_{ik}G\right)$ This IS a tensor. It is built from second order objects (Riemann and Ricci tensors). The field equations become $T_{ik} + A_{ik} = 0$. In vacuum ($T_{ik} = 0$), this gives $A_{ik} = 0$. Zero gravitational energy. No waves. **Section 9:** Titled "EINSTEIN'S MISUNDERSTANDING ABOUT THE GRAVITATIONAL TENSOR." He raises an index on $A_{ik}$ to match the form of Einstein's pseudotensor $t^j_i$, then contracts Einstein's pseudotensor and shows the trace reduces to $G^*$ (Eq. 15). He invokes the 1900 theorem and declares the pseudotensor "not admissible." Einstein never responded. [[1917_Analytic_Expression_of_the_Gravitational_Tensor]] --- ## Schrodinger (1918): vanishes for Schwarzschild Schrodinger showed by direct calculation that with a suitable choice of coordinate system (Cartesian coordinates), all energy components $t^\alpha_\sigma$ of the gravitational field of a sphere (outside of it) vanish. The Schwarzschild solution, the simplest and most important solution in general relativity, has zero gravitational energy according to the pseudotensor when you use Cartesian coordinates. A real gravitational field. Zero energy. By coordinate choice. (*Physikalische Zeitschrift* 19, 4-7, 1918) --- ## Bauer (1918): nonzero energy for nothing Bauer showed the complementary absurdity. Take flat empty spacetime (Minkowski space) and write it in spherical coordinates. The pseudotensor gives nonzero energy. No gravitational field anywhere. Infinite total energy. By coordinate choice. A quantity that vanishes for a real gravitational field (Schrodinger) and appears for no gravitational field (Bauer) is not a physical quantity. --- ## Einstein's response (1918) Einstein's response to Schrodinger acknowledges three problems with his own pseudotensor: 1. $t^\alpha_\sigma$ is not a tensor 2. $t_{\sigma\tau}$ is not symmetric 3. Lorentz and Levi-Civita both carry reservations His defense: "It can very well be that there exist gravitational fields without stresses and without energy density." He then adds that he cannot find a "more suitable definition" and dismisses "further formal demands." [[1918_Einstein_Response_to_Schrodinger]] > [!critical] Einstein's defense undermines gravitational waves > If gravitational fields can have no energy, they cannot carry energy to detectors. The man who created GR admitted in 1918 that his own energy framework allows gravitational fields without energy density. This concession is fatal to gravitational wave detection claims. --- ## Noether (1918): conservation laws are identities Emmy Noether's second theorem proves that in generally covariant theories, the conservation laws are not dynamical statements. They are mathematical identities that hold regardless of whether the field equations are satisfied. The "conservation" of $T + t$ is not a physical law constraining the system. It is a tautology that follows from the symmetry group of the theory. This means the pseudotensor approach is not merely mathematically inadmissible. It is conceptually misconceived. In a generally covariant theory, you cannot have nontrivial conservation laws in the standard sense. The very thing Einstein tried to build (energy conservation via $T + t$) is the one thing general covariance forbids as a physical constraint. --- ## Crothers (2020): same defect in the "improved" version Landau and Lifshitz built a symmetric pseudotensor to fix angular momentum conservation. More complex (4 lines of Christoffel products vs 2), but same ingredients: $g_{\mu\nu}$ and $\Gamma$ throughout. Same contraction test, same verdict. Making the pseudotensor symmetric did not fix the structural defect. [[2020_Einstein_Pseudotensor_Crothers]] [[2020_Landau_Lifshitz_Pseudotensor_Crothers]] Crothers' one-page proof: contract the pseudotensor, observe first order invariant, invoke the 1900 theorem, declare inadmissible. The same argument that Levi-Civita made in 1917, applied to the replacement that was supposed to fix the original. --- ## Infeld and Plebanski (1960): radiation annihilated by coordinate choice Leopold Infeld was Einstein's collaborator and co-author of the EIH equations of motion (1938). In *Motion and Relativity* (1960), he and Plebanski demonstrated: > "We can **annihilate the radiation** by a proper choice of the laws of motion." (p.166) > "We can speak only about radiation in the case of single jumps. However, **its existence or non-existence or its value will depend upon the choice of arbitrary harmonic functions.**" (p.201) > Einstein often remarked: "**We do not have any satisfactory classical theory of radiation.** Ritz understood this fact." (p.201) The existence of gravitational radiation is not a physical invariant. It depends on the choice of arbitrary mathematical functions in the approximation procedure. A physical observable should not depend on arbitrary choices in the coordinate representation. Einstein's own collaborator, the co-author of the equations of motion, declared the radiation coordinate dependent. [[1960_Infeld_Plebanski_Motion_and_Relativity]] Peters and Mathews (1963), the authors of the quadrupole radiation formula used for the Hulse-Taylor binary, cite this book. Their acknowledgment: "The question has been raised, however, whether the energy so calculated has any physical meaning." They then choose to "not concern ourselves with this question" and proceed anyway. [[1963_Peters_Mathews_Gravitational_Radiation]] --- ## Crothers (2018): wave speed is coordinate dependent The gravitational wave equation is obtained by linearizing Einstein's field equations and imposing the harmonic gauge condition (de Donder gauge). The Ricci tensor splits into two parts. The first contains the wave operator. The second is a mess of cross terms. To get a clean wave equation, you need the second part to vanish. That requires choosing specific coordinates. The wave equation is not a consequence of GR. It is a consequence of GR plus a coordinate choice. A different choice of coordinates does not satisfy the harmonic condition, and the cross terms do not vanish, giving a different wave equation with a different propagation speed. The speed $c$ is put in by choosing coordinates that produce it. Different coordinates, different speeds. [[2018_Crothers_GW_Speed_Coordinate_Dependent]] > [!critical] *Petitio principii* > Assume coordinates that give wave equation with speed $c$. Observe that the wave equation has speed $c$. Conclude gravitational waves propagate at $c$. This is a textbook circular argument. --- ## What a valid tensor would need A valid gravitational energy tensor requires second derivatives. The Riemann tensor contains $\partial\Gamma$ (which equals $\partial^2 g$) plus $\Gamma \times \Gamma$. It IS a proper tensor. The Ricci scalar $R$ is a proper second order invariant, exactly the kind the 1900 theorem says CAN exist. But the only valid gravitational energy tensor built from these (Levi-Civita's $A_{ik}$) gives zero in vacuum. The choice: inadmissible pseudotensors (nonzero energy, mathematically invalid) vs valid tensor (zero energy, mathematically sound). 108 years, no third option. --- ## The unimodular formulation In unimodular coordinates ($\sqrt{-g} = 1$), which is how Einstein first presented his completed theory, both the matter-absent and matter-present field equations can be written explicitly in terms of the pseudotensor. If the pseudotensor is invalid, the unimodular formulation collapses. This is not a peripheral technicality. It strikes at the formulation Einstein himself used on November 4, 1915. [[Crothers_for_Dummies]] [[Crothers_Sky_Scholar]] --- ## Conclusion Failed to falsify H₀. The conservation laws of general relativity are not satisfied within the framework of absolute differential calculus. The evidence: 1. **The 1900 theorem** (Ricci and Levi-Civita): first order intrinsic differential invariants do not exist. The pseudotensor is first order. It is inadmissible. 2. **Levi-Civita (1917)**: contracted the pseudotensor, invoked the theorem, declared it "not admissible." Titled his section "Einstein's Misunderstanding." Einstein never responded. 3. **Schrodinger (1918)**: the pseudotensor vanishes entirely for the Schwarzschild solution in Cartesian coordinates. A real gravitational field with zero energy. 4. **Bauer (1918)**: the pseudotensor gives nonzero energy for flat empty spacetime in spherical coordinates. No field, infinite energy. 5. **Einstein (1918)**: admitted it is not a tensor, not symmetric, and that Lorentz and Levi-Civita both object. Conceded "gravitational fields without energy density" can exist. 6. **Noether (1918)**: in generally covariant theories, conservation laws are identities, not dynamical constraints. The pseudotensor approach is conceptually misconceived. 7. **Infeld and Plebanski (1960)**: gravitational radiation can be annihilated by coordinate choice. Its existence depends on "arbitrary harmonic functions." Einstein himself admitted there is "no satisfactory classical theory of radiation." 8. **Crothers (2020)**: the Landau-Lifshitz "improvement" has the same structural defect. Still first order, still inadmissible. 9. **Crothers (2018)**: the wave speed $c$ is a gauge choice, not a prediction. The wave equation requires the harmonic condition, which is a coordinate restriction, not a physical law. The only valid gravitational energy tensor, Levi-Civita's $A_{ik}$ (built from second order derivatives), gives zero in vacuum. No energy, no waves. The man who co-invented absolute differential calculus declared the pseudotensor inadmissible in 1917. His verdict has never been overturned. Nobody has produced a valid (second order) gravitational energy tensor that is nonzero in vacuum. H₁ is not supported. --- ## Sources Ricci, M.M.G. and Levi-Civita, T. (1900). "Methodes de calcul differentiel absolu et leurs applications." *Mathematische Annalen*, 54, 125-201. [[1900_Methodes_de_Calcul_Differentiel_Absolu]] Einstein, A. (1915). "Zur allgemeinen Relativitatstheorie." *Sitzungsberichte der Preussischen Akademie der Wissenschaften*, November 4 and 11, 1915. [[1915_Einstein_Field_Equations_Nov4_Nov11]] Einstein, A. (1916). "Die Grundlage der allgemeinen Relativitatstheorie." *Annalen der Physik*, 49, 769-822. [[1916_Einstein_Foundation_GR_Sections15_17]] Levi-Civita, T. (1917). "Sulla espressione analitica spettante al tensore gravitazionale nella teoria di Einstein." *Rendiconti della Reale Accademia dei Lincei*, 26, 381-391. [[1917_Analytic_Expression_of_the_Gravitational_Tensor]] Einstein, A. (1918). "Notiz zu E. Schrodingers Arbeit." *Physikalische Zeitschrift*, 19, 115-116. [[1918_Einstein_Response_to_Schrodinger]] Infeld, L. and Plebanski, J. (1960). *Motion and Relativity*. Pergamon Press. [[1960_Infeld_Plebanski_Motion_and_Relativity]] Crothers, S.J. (2020). "On the Pseudo-Riemannian Manifold and Einstein's Pseudotensor." [[2020_Einstein_Pseudotensor_Crothers]] Crothers, S.J. (2020). "On the Landau-Lifshitz Pseudotensor." [[2020_Landau_Lifshitz_Pseudotensor_Crothers]] Crothers, S.J. (2018). "Gravitational Waves: Propagation Speed is Co-ordinate Dependent." APS April 2018. [[2018_Crothers_GW_Speed_Coordinate_Dependent]] Peters, P.C. and Mathews, J. (1963). "Gravitational Radiation from Point Masses in a Keplerian Orbit." *Physical Review*, 131(1), 435-440. [[1963_Peters_Mathews_Gravitational_Radiation]] Earman, J. and Glymour, C. (1978). "Lost in the Tensors." *Studies in History and Philosophy of Science*, 9(4), 251-278. [[1978_Earman_Glymour_Lost_in_the_Tensors]] Iorio, R. and Levi-Civita, T. (correspondence, 1912-1914). [[2012_Einstein_the_Stubborn_Correspondence]] --- ## See Also - [[Gravitational_Waves_Null]] — Companion null: do gravitational waves exist as physical phenomena? - [[Perihelion_Precession_Null]] — The perihelion formula is kinematics, not dynamics - [[Cosmological_Dynamics_Null]] — GM carries no information beyond kinematics - [[Crothers_for_Dummies]] — Plain language pseudotensor breakdown - [[Gravitational_Waves_Crothers]] — Full technical analysis - [[Crothers_Sky_Scholar]] — Video walkthrough of Einstein's 1915 paper - [[00_Gravitational_Waves_Index]] — Original combined presentation - [[00_Null_Hypothesis_Index]] — Master null hypothesis index