Welcome! Here is the section where we discuss the curvature of space-time. Remember, one of the realizations that come with general relativity is that gravity is caused by curved space-time, not a particle like the other forces. For example: electromagnetic force has the photon as it’s force carrier. But gravity is much weaker than the other forces. The theorized force-carrying particle for gravity, the graviton, has not been detected and particle accelerators haven’t yet reached the energy levels where we believe these particles become visible. Until that day, our most accurate model of the force of gravity is curved space-time, and it does remarkably well.

With that preamble, I will jump right into this section with my favorite quote on the topic of space-time curvature:

“Matter tells space how to curve. Space tells matter how to move”- John Archibald Wheeler. To me, this sums up the phenomenon of motion in our four dimension space-time manifold. Before general relativity, it was thought that particles moved along straight lines until a force knocks them off. But now, gravity is represented as a curve of spacetime, not a force. To re-write the earlier statement, now “particles move along geodesics until forces knock them off”. More on geodesics later!

Imagine that space is a thin, rubbery sheet:

Place a Earth-like ball on this sheet, and it makes an indentation in the rubber:

Now, take a smaller (satellite) ball and roll it towards, but aimed just away from Earth. Anyone that has operated one of the coin donation funnels at a shopping mall knows what happens. The satellite-ball will roll past the Earth-ball, and then curve back towards the Earth-ball. A better visual is in the video below:

^{(PLEASE SEE “SPACE-TIME CURVATURE IMAGE RESOURCES” FOR VIDEO SOURCE)}

The coins in this video are analogous to planets/smaller bodies orbiting around a larger one (the cause of the funnel shape of space-time). The analogy holds well to get the general picture, but not all objects spiral towards their fate in the donation box. If an object has sufficient energy it can achieve a permanent orbit about the center, circling forever.

Remember the word “geodesic” that I mentioned earlier? It’s time we look at what this means!

**Geodesics:**

We (or I, at least) were always told that the shortest distance between two points is a line. Well, it appears this is true only when dealing with spaces that are Euclidean (flat planes). Have you ever heard of the Great Circle Route? Check it out here! When we plot the travel routes of airplanes, we see an arc, not a straight line. Straight lines on a curved surface are called geodesics.

Formally, according to Sean M. Carroll in his paper “A No-Nonsense Introduction to General Relativity” the geodesic “is a curve which extremizes the length functional”…and then he goes on with some higher level mathematics.

But here I just want to discuss the physical meaning of geodesics, for now. With general relativity, test bodies move along curved lines in space. A test body is any body with mass. Sean M. Carroll gives us some criteria for this:

- If bodies are massless, their geodesics are null.
- If they are massive their geodesics will be timelike.

“Bodies” refers to planets, stars, black holes, etc. A null geodesic will lead to no curvature of space, and objects will traverse a straight line. Timelike means that the space between two events is shorter than the distance light travels between them. So some of that space has been converted into time! We are dealing with space-time here, where one can transform into the other.

A wonderful video discussing geodesics exists on YouTube, from PBS Space Time:

^{(PLEASE SEE “SPACE-TIME CURVATURE IMAGE RESOURCES” FOR VIDEO SOURCE)}

**Einstein’s Field Equation**

Space and the energy/matter it contains are interlinked by Einstein’s equation. Just as a reminder, here is the equation again:

**MANIFOLDS:**

I tossed the word “manifold” around earlier without really offering much explanation. A great explanation comes from Sean Carroll’s paper. In the paper, he states “a manifold is a possibly curved space which, in small enough regions, looks like flat space”. The thing about manifolds is that in small regions space can seem Euclidean, but on larger scales that may be entirely different.

Carroll goes on to give a wonderful metaphor (which I will re-iterate here). The Earth (from sea level) looks flat. But when we go to high altitudes (upper atmosphere and beyond) we see that the Earth is in fact a sphere. So, locally (what your eyes can see on the ground) it is flat. But on large scale, it is really round.

The information about space-time curvature is contained in something called the Riemann curvature tensor (a four-component tensor). Let’s look at it below:

As we see, the tensor has a rank of 4 and exists in 4 dimensions. Given what we learned in the Math! section, we have N^{p},_{ }4^{p}=256 components. But thanks to the symmetry the tensor obeys, only 20 of these components are independent. Symmetry plays a major role in lots of complicated mathematics.

Now that we are at least familiar with the word Riemann, I can tell you that space-time is a curved pseudo Riemannian manifold. What can be said about this? A really great Wikipedia page exists for this topic, but a forewarning: it is quite heavy in theoretical jargon. Here is the link.

**Einstein Tensor: **

Remember in the Math! section, when I mentioned the left hand side of the Einstein equation can be considered a tensor? Let’s take another look.

The left hand side of this equation (with two tensors in itself), can be described as the Einstein Tensor. The common symbol used for this tensor is G^{µν}.

If we consider the left side as a tensor, as defined earlier, we have G^{µν}. The R_{µν} is the Riemann tensor. The G on the right side is Newton’s gravitational constant, and T^{µν} is a symmetric two-index tensor we call the energy-momentum tensor.

Space and the energy/matter it contains are interlinked by Einstein’s equation. The left hand side of the equation measures the curvature of space-time, and the right side describes the energy and momentum contained therein. The two are connected. How beautiful is that?

**WANT TO SEE SOME PHYSICAL EVIDENCE OF CURVED SPACE-TIME?**