Editing Continuity equation (section)

===General relativity===
In [[general relativity]], where spacetime is curved, the continuity equation (in differential form) for energy, charge, or other conserved quantities involves the [[Covariant derivative|''covariant'' divergence]] instead of the ordinary divergence.

For example, the [[stress–energy tensor]] is a second-order [[tensor field]] containing energy–momentum densities, energy–momentum fluxes, and shear stresses, of a mass-energy distribution. The differential form of energy–momentum conservation in general relativity states that the ''covariant'' divergence of the stress-energy tensor is zero:
<math display="block">{T^\mu}_{\nu; \mu} = 0.</math>

This is an important constraint on the form the [[Einstein field equations]] take in [[general relativity]].<ref>{{cite book |title=Relativity DeMystified|author=D. McMahon|publisher=McGraw Hill (USA)|year=2006|isbn=0-07-145545-0}}</ref>

However, the ''ordinary'' [[Tensors in curvilinear coordinates#Second-order tensor field|divergence]] of the stress–energy tensor does ''not'' necessarily vanish:<ref>{{cite book |title=Gravitation |author=C.W. Misner |last2=K.S. Thorne |last3=J.A. Wheeler | publisher=W.H. Freeman & Co |year=1973 |isbn=0-7167-0344-0}}</ref>
<math display="block">\partial_{\mu} T^{\mu\nu} = - \Gamma^{\mu}_{\mu \lambda} T^{\lambda \nu} - \Gamma^{\nu}_{\mu \lambda} T^{\mu \lambda},</math>

The right-hand side strictly vanishes for a flat geometry only.

As a consequence, the ''integral'' form of the continuity equation is difficult to define and not necessarily valid for a region within which spacetime is significantly curved (e.g. around a black hole, or across the whole universe).<ref>{{cite web |url=http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html |title=Is Energy Conserved in General Relativity? |access-date=2014-04-25 |author1=Michael Weiss |author2=John Baez }}</ref>