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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.
The Einstein tensor allows the Einstein field equations to be written in the concise form: + =, where is the cosmological constant and is the Einstein gravitational constant. From the explicit form of the Einstein tensor , the Einstein tensor is a nonlinear function of the metric tensor, but is linear in the second partial derivatives of the ...
Einstein's field equations: = where the Ricci curvature tensor = and the scalar curvature = relate the metric (and the associated curvature tensors) to the stress–energy tensor. This tensor equation is a complicated set of nonlinear partial differential equations for the metric components.
The metric tensor is a central object in general relativity that describes the local geometry of spacetime (as a result of solving the Einstein field equations). Using the weak-field approximation , the metric tensor can also be thought of as representing the 'gravitational potential'.
where is the Einstein tensor, is the cosmological constant (sometimes taken to be zero for simplicity), is the metric tensor, is a constant, and is the stress–energy tensor. The Einstein field equations relate the Einstein tensor to the stress–energy tensor, which represents the distribution of energy, momentum and stress in the spacetime ...
These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy Maxwell's equations).Following a standard recipe which is widely used in mathematical physics, these tensor fields should also give rise to specific contributions to the stress–energy tensor. [1]
In general relativity, the stress–energy tensor is studied in the context of the Einstein field equations which are often written as + =, where is the Ricci tensor, is the Ricci scalar (the tensor contraction of the Ricci tensor), is the metric tensor, Λ is the cosmological constant (negligible at the scale of a galaxy or smaller), and ...
The Einstein field equation (EFE) describing the geometry of spacetime is given as = where is the Ricci tensor, is the Ricci scalar, is the energy–momentum tensor, = / is the Einstein gravitational constant, and is the spacetime metric tensor that represents the solutions of the equation.