As Einstein's equations of General relativity being solved, by whom and how was it solved. Explain it in detail along with mathematical expressions and equations.
Albert Einstein formulated the theory of General Relativity, which describes the force of gravity as the curvature of spacetime caused by mass and energy. However, solving the equations of General Relativity is a complex task and requires mathematical techniques beyond the scope of this response. Nonetheless, I can provide a general overview of how these equations are solved and highlight some key contributions.
The Einstein field equations, the core components of General Relativity, describe the relationship between the curvature of spacetime (represented by the metric tensor) and the distribution of matter and energy (represented by the stress-energy tensor). These equations can be expressed as:
Rμν - 1/2 R gμν = 8πG/c^4 Tμν
where Rμν represents the curvature tensor, R is the scalar curvature, gμν is the metric tensor, G is the gravitational constant, c is the speed of light, and Tμν is the stress-energy tensor.
One of the fundamental challenges in solving these equations is the high level of mathematical complexity involved, which requires advanced differential geometry and tensor calculus techniques. Consequently, the solutions to the field equations are not always easy to obtain, and only a few exact solutions are known.
One of the most famous solutions is the Schwarzschild solution, obtained by Karl Schwarzschild in 1916. This solution describes the spacetime around a non-rotating, spherically symmetric mass. The metric tensor for the Schwarzschild solution is:
ds^2 = -(1-2GM/rc^2)dt^2 + (1-2GM/rc^2)^-1dr^2 + r^2(dθ^2 + sin^2θ dφ^2)
where G is the gravitational constant, M is the mass of the object, r is the radial distance from the object, t is time, θ is the polar angle, and φ is the azimuthal angle.
Another significant solution is the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, which describes an expanding universe with different energy content. This solution provides insight into the cosmological implications of General Relativity. The FLRW metric is given by:
ds^2 = -dt^2 + a(t)^2 [ dr^2 / (1-kr^2) + r^2(dθ^2 + sin^2θ dφ^2) ]
Here, a(t) represents the scale factor that characterizes the expansion or contraction of the universe, and k indicates the curvature of space (k can take the values +1, -1, or 0, corresponding to closed, open, or flat universes, respectively).
In addition to exact solutions, numerical methods and approximation techniques are often employed to solve the Einstein field equations for complex scenarios, such as black holes or binary systems. Numerical relativity simulations, using powerful computers, are key for understanding various astrophysical phenomena involving strong gravitational fields.
Overall, the solutions to Einstein's equations of General Relativity are obtained through a combination of mathematical techniques, including advanced differential geometry, tensor calculus, and numerical methods. While exact solutions are limited, they have provided profound insights into the nature of gravity, space, and time.