How Mass WARPS SpaceTime: Einstein's Field Equations in Gen. Relativity | Physics for Beginners

How does the fabric of spacetime bend around objects with mass and energy?

Hey everyone, I'm back with another video! This time, we're looking at the Einstein Field Equations. These are some of the most important equations found in the theory of General Relativity.

The first thing worth mentioning is that the common way to write the Field Equations is as one single tensor equation, where the subscripts added after different terms can take the values 0,1,2,3. The meaning of these subscripts is discussed in the video, but it's important to note that what looks like a single equation is actually a convenient way of writing multiple equations.

We have already mentioned tensors. The terms in the Einstein Field Equations refer to these tensors, which are rather interesting mathematical objects. They can be represented by matrices, which are just a particular way of displaying information.

The first tensor worth mentioning, labelled T_mu,nu is known as the Stress-Energy tensor. This contains information about the distribution of mass, energy, momentum, pressure, and so on, within the region of spacetime we happen to be considering. Often, this region happens to be the entire universe. Some of us may be familiar with the idea that the existence of mass and energy can cause the "spacetime fabric" around it to warp and bend. And this is exactly what the Einstein Field Equations are about.

The G_mu,nu tensor seen on the left hand side of the equation, is known as the Einstein Tensor. It contains information about the curvature of spacetime. This allows us to understand how the region of spacetime we happen to be considering is curved and warped.

In other words, the Einstein Field Equations can be simply thought of as relating a mass / energy distribution to the warping it causes to spacetime.

The final term on the left hand side of the equation is a product of lambda, the "Cosmological Constant" and g_mu,nu, the "Metric Tensor". The metric tensor simply contains information about the exact shape of the spacetime fabric. Whether it is flat, or curved, and exactly how it curves. It is a very important tensor in relativity.

The cosmological constant is simply Einstein's way to encode the fact that the universe is expanding, into these equations. In other words, the behaviour of the spacetime fabric is not only dependent on the mass / energy inside it, but also on the fact that it just seems to be inherently stretching. We have observed the accelerated expansion of the universe - galaxies very far away from us are moving away from us faster and faster, as compared with galaxies much nearer to us. The positive cosmological constant accounts for this expansion of the universe.

The cosmological constant has an interesting history. Initially, Einstein published his field equations without the constant. Then he realised it was necessary in order to combat the probable collapse of the universe due to gravity causing everything to eventually attract everything else. Then, he called it his biggest blunder. But now, we see that the universe is not just expanding, but it is expanding at an ever-accelerating rate. Therefore, the constant is indeed necessary in our mathematics.

Finding solutions to the Field Equations is hard - but we have found a few. For example, we know that flat spacetime is a solution of the field equations - we can have a region of spacetime in our universe where there is no mass or energy, and this is a flat spacetime. Also, we have the Schwarzschild and Kerr solutions that look at stationary and rotating black holes respectively - check out my Black Holes playlist here:    • Black Holes EXPLAINED  

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