Can anyone explain the Ricci curvature?

Question

I am 13 years old and love physics. Can anyone explain simply what it is?

Answer 1

Let's start with some mathematical basics. Some physical quantities are a single number; others are like a string of numbers, called a vector; still others expand this string to a rectangular array, called a matrix. A matrix is a $2$-dimensional arrangement of numbers; to say which of its entries you're talking about, you need two numbers to pick a row and column. The "shape" of spacetime is described by a matrix called the metric tensor, denoted $g_{ab}$. The value of $a$ specifies the row, and the value of $b$ specifies the column; and these indices can each have one of $4$ values, because spacetime has $4$ dimensions ($3$ of space, $1$ of time).

But there are other quantities with even more dimensions. They're harder to write down, because a sheet of paper only has a $2$-dimensional surface. However, if you wanted to specify a $4$-dimensional cousin of a matrix, say $X_{abcd}$, you just need to go through all choices of $abcd$ specifying the result.

Now to the physics. To understand the Ricci curvature, we must start with the Riemann tensor, denoted $R_{abcd}$. This collection of numbers measures the curvature of spacetime. On a flat surface, moving short distances in one direction and then another gets you to basically the same place if you change the order of the movements, but in curved spacetimes this doesn't work, and the Riemann tensor measures how much of a difference the order makes. It doesn't take $4^4=256$ independent numbers to specify this because of some constraints, such as $R_{abcd}=-R_{bacd}$; there are actually just $20$ degrees of freedom.

So what's the Ricci curvature? It could actually refer to two things, both expressed in terms of the Riemann tensor. The metric tensor I mentioned before is an invertible matrix (you can "multiply" matrices, so the inverse is a bit like a reciprocal), and its inverse is written $g^{ab}$. The Ricci tensor is defined as $R_{ac}=g^{bd}R_{abcd}$, where the rule is you sum over all possible values of any repeated indices (this is a bit like taking a scalar product of two vectors, of you've learned about that). Similarly, the Ricci scalar is $R=R_{ac}g^{ac}$.

So what are these two things named after Ricci, in short? They're quantified descriptions of how curved spacetime is. However, they don't provide a complete picture of curvature; you need the entire Riemann tensor for that. Some shapes spacetime can conceivably have result in $R=0$ even if the Riemann tensor contains some nonzero entries, which means the spacetime has to be curved.

Answer 2

Ricci curvature is the explanation of curvature Einstein opted for in his field equations. There are two types of Ricci curvature: the Ricci tensor and the Ricci scalar. I will explain the tensor as that features more heavily in the Field equations.

The Ricci tensor is a rank two tensor, found by contracting the Riemann tensor. It explains the curvature in terms of how the volume of a geodesic ball changes as it moves through the space. Imagine a sphere in some space, if you assign a vector to each point on this sphere then we can parallel transport this ball and all the vectors with it. We know that if space is flat then the vectors will remain unchanged- even after this translation, leaving the shape and volume of the sphere unchanged.

Alternatively, if the space is curved then the vectors will either diverge or converge after the parallel transport. These vectors, diverging or converging will change from the original spherical formation they were originally in. This change in shape will most likely come with a change in volume. Such a change in volume of the sphere can be measured and is representative of how curved the space is. In a very curved space; the vectors will diverge more and the volume will change a lot, and vice versa