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A continuity equation or transport equation is an equation that describes ... This is an important constraint on the form the Einstein field equations take in general ...
The Einstein field equations (EFE) may be written in the form: [5] [1] + = EFE on a wall in Leiden, Netherlands. where is the Einstein tensor, is the metric tensor, is the stress–energy tensor, is the cosmological constant and is the Einstein gravitational constant.
Next, notice that only 10 of the original 14 equations are independent, because the continuity equation ; = is a consequence of Einstein's equations. This reflects the fact that the system is gauge invariant (in general, absent some symmetry, any choice of a curvilinear coordinate net on the same system would correspond to a numerically ...
The second is: ¨ = (+) + which is derived from the first together with the trace of Einstein's field equations (the dimension of the two equations is time −2). a is the scale factor , G , Λ , and c are universal constants ( G is the Newtonian constant of gravitation , Λ is the cosmological constant with dimension length −2 , and c is the ...
The equations of motion are contained in the continuity equation of the stress–energy tensor: =, where is the covariant derivative. [5] For a perfect fluid, = (+) +. Here is the total mass-energy density (including both rest mass and internal energy density) of the fluid, is the fluid pressure, is the four-velocity of the fluid, and is the metric tensor. [2]
The Einstein field equations are a system of coupled, nonlinear partial differential equations. In general, this makes them hard to solve. In general, this makes them hard to solve. Nonetheless, several effective techniques for obtaining exact solutions have been established.
Whereas the "wave equation", the "diffusion equation", and the "continuity equation" all have standard forms (and various special cases or generalizations), there is no single, special equation referred to as "the field equation". The topic broadly splits into equations of classical field theory and quantum field theory.
The cosmological constant was originally introduced in Einstein's 1917 paper entitled “The cosmological considerations in the General Theory of Reality”. [2] Einstein included the cosmological constant as a term in his field equations for general relativity because he was dissatisfied that otherwise his equations did not allow for a static universe: gravity would cause a universe that was ...