We have been discussion the law of conservation of energy in the context of classical general relativity. So far I have not been able to convince anyone here that the maths shows that energy is conserved. Lubos Motl and Matti Pitkanen have posted some contrary arguments on their blogs to add to the old one by Sean Carroll. We have also been trading points and counterpoint in the comments with Ervin Goldfain joining in, also in disagreement with me. To avoid going over the same arguments repeatedly we have agreed to disagree, for now.
If you think such a discussion about Energy in physics seems off the wall, think again. This subject and related issues concerning gravitational waves have occupied physicists for years. Some well-known names in the world of science have exchanged some heated words and still not everyone agrees on the outcome.
But it is too soon to end our debate. There are still a few more points I want to make. It was said that my claim in favour of energy conservation means that I am “convinced that all relativists are wrong”. This is not the case. Historically many relativists have been on my side. This is actually a debate that began as soon as general relativity was formulated by Einstein. Einstein in fact developed the first complete formulation of energy conservation in GR, but Hilbert objected. The argument has raged ever since with as many different views on the subject as there have been relativists and cosmologists. Amongst those who have accepted the law of energy conservation and produced their own formulations are Dirac, Landau, Wald, Weinberg and of course Einstein himself, so to say I am contradicting all relativists is far from true.
It has also been said that all the textbooks show that energy is not conserved in general relativity, except in special cases. This is also not true. Most GR textbooks do not tackle the general formulation of energy conservation in GR. They just deal with special cases such as a static background gravitational field with a killing vector. This does not mean that energy conservation only works in special cases as some people claim. The textbooks just don’t cover the general case. Some textbooks do cover it but by using pseudotensor methods (e.g. Dirac, Weinberg, Landau & Lifshitz) A few textbooks do suggest that energy is not conserved, e.g. Peebles, but these are the minority.
I am going to recount some of the history of the debate. To keep it orderly I’ll give it as a timeline of events with my own contribution immodestly tacked on the end. We start in 1915 with conservation of energy a well established concept recently unified with the conservation of mass by Einstein. The World is at war and Einstein is about to publish his general theory.
July 1915: Einstein lectures on his incomplete theory of general relativity to Hilbert, Klein and possible Noether at Göttingen, convincing them that his ideas are important.
October 1915: Albert Einstein publishes a tentative equation for general relativity with being the Ricci curvature tensor and being the covariant generalization of the energy-momentum tensor.
November 1915: Einstein realises that his previous equation cannot be right because the divergence of the energy momentum tensor is zero as required by local energy conservation. To correct it he writes the new equation . These are the Einstein Field equations which work because the left hand side has zero divergence due to the Bianchi identities.
November 1915: David Hilbert publishes a calculation showing how the Einstein Field Equations can be derived from a least action principle. In fact his work is dated prior to Einstein’s but they had been in communication and it is reasonable to give the priority for the equations to Einstein and for the action formulation to Hilbert.
1916: Hilbert publishes a note with an equation for a conserved energy vector in general relativity.
1916: Einstein publishes a full formulation of energy conservation in general relativity in which a pseudotensor quantity is added to the energy-momentum tensor and another superpotential term to give a conserved energy current.
1916: Einstein predicts the existence of gravitational waves which will carry away energy and momentum from orbiting stars. He derives the quadrupole radiation formula to quantify the rate at which energy is dispersed.
July 1917: Oscar Klein points out (with help from Noether) that conservation of Hilbert’s energy vector is an identity that does not require the field equation.
1917: In response to Klein, Hilbert publishes an article questioning the validity of energy conservation in general relativity. He says that the energy equations do not exist at all and this is a general characteristic of the theory.
1917: Writing to Klein, Hilbert says that general relativity has only improper energy theorems. By this he means that the pseudotensor methods are not covariant.
1917: Klein writes to Einstein making the claim that energy conservation in general relativity is an identity. This is based on Hilbert’s energy vector.
1917: To construct a static cosmological model Einstein introduced the cosmological constant as an extra term in his field equations.
March 1918: Einstein writes back to Klein explaining that in his formulation of energy conservation the divergence of the current is not an identity because it requires the field equations.
July 1918: Emmy Noether publishes two theorems on symmetry in physics. The first showed that symmetry in any theory derived from an action principle implies a conservation law. In particular, energy conservation is implied by time invariance. The second shows that in the case of gauge theories with local symmetry such as general relativity, there are divergence identities such as the Bianchi Identities.
1918: Felix Klein uses Noether’s theorems to derive a third boundary theorem to show why the conservation law of energy in general relativity must take a particular form that he considers to make it an identity.
1918: Einstein comments on the power and generality of Noether’s theorems but does not accept the conclusion that energy conservation is an identity.
1919: Arthur Eddington measures the deflection of starlight by the Sun during a solar eclipse. The observation confirms the prediction of general relativity and provides massive press publicity for the theory.
1922: Arthur Eddington expresses skepticism about the existence of gravitational waves saying that they “travel at the speed of thought”.
1922: Friedman finds cosmological solutions of general relativity that describe an expanding universe.
1927: Lemaitre predicts an expanding universe
1929: Edwin Hubble observes the expanding universe in galactic redshifts. This led to Einstein dropping his cosmological constant.
1936: After working on exact solutions for gravitational waves with Rosen, Einstein concludes that gravitational waves can not exist, reversing his 1916 prediction. This sparked a vigorous twenty year debate over the reality of gravitational waves.
1936: After working with Robertson, Einstein eventually concedes that gravitational waves do exist. Rosen who had departed for the Soviet Union did not accept this concession. He never changed his mind even as late as 1970.
1951: Landau and Lifshitz publish “The Classical Theory of Fields” as part of a series of textbooks on theoretical physics. It deals with energy and momentum in general relativity using a symmetric pseudotensor. The symmetry means that they can also show conservation of angular momentum using the same structure.
1955: Rosen computes the energy in exact gravitational wave solutions using pseudotensors and finds the result ot be zero. He presents this as evidence that gravitational waves are not real.
1957: Herman Bondi introduced a formalism now known as Bondi Energy to study energy in general relativity and gravitational waves in particular. This work was very influential and formed a turning point in the understanding of gravitational waves and energy in general realtivity.
1957: Weber and Wheeler find a gravitational wave solution that does transmit energy.
1957: Richard Feynman describes the sticky-bead thought experiment to show that gravitational waves are real. The idea was popularised by Herman Bondi and finally led to the general acceptance of the reality of gravitational waves.
1959: Andrzej Trautman gave the formulation of energy conservation for the special case where a static background is given by the existence of a killing vector field.
1959: Komar defined a superpotential for general cases whose divergence vanishes as an identity. The superpotential uses an auxiliary vector field similar to the killing vector field in Trautmans theory, but the Komar field does not need to satisfy any special conditions so the solution is more general. The Komar potential has the advantage over pseudotensor methods that it is expressed in a covariant form. However, the zero divergence of the superpotential is an identity.
1961: Arnowitt, Deser and Misner formulate the ADM mass/energy for systems in asymptotically flat spacetimes
1961: In his book “Geometrodynamics” Archibald Wheeler says that energy conservation in a closed universe reduces to a trivial 0 = 0 equation.
1964: Weber begins experiments to try to detect gravitational waves.
1972: Steven Weinberg uses a pseudotensor method to show energy conservation in his textbook “Gravitation and Cosmology”
1974: The discovery of the Hulse-Taylor binary pulsar shows that gravitational energy is radiated as originally predicted by Einstein.
1975: In his concise introductory text to general relativity Dirac derives a pseudotensor using Noether’s theorem to prove energy and momentum conservation.
1979: Schoen and Yau prove the positive energy theorem for ADM energy. (A simpler proof was given by Witten in 1981)
1993: Phillip Peebles in his book “Principles of Physical Cosmology” claims that energy conservation is violated for the cosmic microwave background
1997: Philip Gibbs shows that if Noether’s theorem is generalised to include second derivatives of the fields and is applied to the symmetries generated by a vector field, a conserved current with a covariant current can be derived. The current which has an explicit dependence on the vector field is equal to a term that is zero when the Einstein Field Equations are satisfied, plus the Komar superpotential.
1998: Observational evidence (Riess, Pulmutter) leads to the reintroduction of the cosmological constant, now called dark energy
For further historical details and references on the Klein-Hilbert-Einstein-Noether debate see “A note on General Relativity, Energy Conservation and Noether’s Theorems” by Katherine Brading in “The Universe of General Relativity” ed A.J. Fox, J. Eisenstaedt.
A good read on the history of gravitational waves is “Traveling at the speed of thought” by Daniel Kennefick