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Einstein Field equations explained intuitively and visually: Isaac Newton changed our paradigm by connecting earthly gravity, with the movement of heavenly bodies. He formulated an equation that is still used today – the law of universal gravitation. But it had problems – action at a distance and the the rate of precession of Mercury
250 years later, Einstein, with the General theory of relativity, solved the riddle of Mercury’s precession, and showed that gravity was due to a bending of space-time itself. Today, we find holes in Einstein’s theory, such as its inability to explain the singularity inside black holes, and the big bang.
Newton’s and Einstein’s equation are similar . They both have Newton’s gravitational constant. In Newton’s equation, a force is on the left side, created by mass on the right side. In Einstein’s equation, the analog of force, the curvature of space-time is on the left side. And mass-energy-momentum tensor is on the right side.
A tensor can be an array of vectors, scalars, or other tensors represented by an N x N matrix.
Although Einstein’s equation looks simple, it is actually 10 equations, and is very complicated. The equations describe curvature of space-time by treating it as being flat at infinitesimally small distances. So general relativity behaves like special relativity at these small distances. But overall curvature is taken into account. The Ricci curvature tensor tells us how space-time is deviating from flat.
The second term on the left side is composed of R, the scalar curvature, which is how much the space is changed at a point, such that you know how to correctly measure distances. Little g is the metric tensor. It tells you the geometry and structure of spacetime. Together this term defines how distances are calculated, given a curvature at a point.
Sometimes a third term is added, lambda. the cosmological constant. This describes the intrinsic energy density of empty space. It is the mathematical expression for dark energy – the accelerating expansion of the universe.
The right side has a constant which is the Einstein gravitational constant. It is a conversion factor to make sure we get the proper units.
On the right side – T is the stress energy momentum tensor, which tells us the density of energy and momentum at each point in space time. It is a source of the curvature.
The way these equations are formulated is by treating space-time in 4 dimensions. Three spatial dimensions, and one dimension of time. This is incorporated in the mu and nu subscripts.
When mu and nu are zero and zero, the left side describes the speeding up or slowing down of time at a point in space. The right side describes the energy at that point.
If the mu and nu are zero and one, the left represents the stretching of time within one spatial dimension. The right side is the momentum.
If mu and nu are one and one, the left side describs the stretching of space in one of the dimensions. The right side is the pressure at a point in space.
General relativity is generally true – it predicts bending of light around massive objects, which has been observed, and it predicts that time will run more slowly on the surface of earth than a mountaintop, which has been confirmed.
But it is incomplete. The problem is that it does not fit with an even more accurate theory – quantum mechanics.
For example, quantum theory says that the electron in an atom is in a superposed state, meaning it is in multiple positions from the nucleus at the SAME time. We only know the probability of finding it at particular radius, if we measure it. Since the electron has mass, general relativity says it must curve space-time. But if it is in multiple locations at the same time, then where is the curvature? Is it also at multiple locations at the same time? We don’t know. There is nothing in general relativity akin to superposition.
GR also predicts matter and energy being compressed to an infinitely small point with infinite curvature in a black hole. But since the equations treat space-time mathematically as being flat at infinitesimally small distances, a problem occurs when space is not flat at infinitesimally small distances, at a singularity.
Mathematical infinities are usually wrong, so there is probably something else going on. There are two theories that are promising, Loop Quantum Gravity and String theory – the subject of my next video.