Describing How Mass Warps Spacetime

Review the meaning of warped space and time.

See Einstein's Equations in action.

One of the central challenges of physics is—and has always been—to predict how things move. The earliest astronomers were really nothing more than astrologers, trying to discern when stars would appear on the horizon, or where the Sun would be found at a certain date. Eventually, this turned into true, scientific astronomy, and physicists used their laws to predict how all sorts of bodies moved through the heavens. Newton showed us that these same laws could be used to predict more earth-bound things, such as how an apple falls from a tree. Modern physicists are concerned with the motions of the tiniest particles in the atoms around us, and the motions of the heaviest objects in the heavens above.

We've mentioned that differential geometry gives us tools to understand the motion of particles when spacetime is curved. It doesn't say anything about why spacetime should be curved, though. Einstein took the tools of differential geometry, and showed us how and why spacetime curves. In doing this, he gave us very powerful tools to predict the motion of particles.

Pythagoras's Theorem

The Pythagorean Theorem: A2+B2=C2

Understanding Einstein's laws begins with another old theorem you might remember from high-school geometry. The Pythagorean theorem tells us the length of one side of a triangle, given the lengths of the other two sides. This is a very basic, and straightforward theorem that applies to any triangle with one right angle—that is, one angle of 90 degrees.

The theorem is not very difficult to use. Suppose we have a triangle with a shortest side of 3 feet, and a second-shortest side of 4 feet. We take these numbers and square them (multiply them by themselves), giving 9 and 16 square feet. We add these together, and get 25 square feet. Now, whatever the longest side is, its square must be 25. The correct length, then, is 5 feet, since 5 squared is 25.

Of course, not every triangle has a right angle. We can extend Pythagoras's theorem to this more general case by subtracting a term. Suppose that we know the length of two sides of the triangle, and the angle between them. If that angle is α, then the "Law of Cosines" is as given below. This rules is just like the Pythagorean theorem, except that we also need to measure at least one angle of the triangle.

The Law of Cosines: A2+B2-2×A×B×cos(α)=C2

You might remember the cosine function here written as "cos"; it takes the angle, and gives back another number, as shown here.

A plot of the cosine function. Notice that it is zero for 90°.

When α is 90 degrees, the cosine is 0, so the extra term in the Law of Cosines drops away, and it reduces to the same thing as the Pythagorean Theorem. When α is less than 90 degrees, we see that the extra term makes the side with length C smaller, which we would expect. Similarly, if α is greater than 90 degrees, the extra term makes C larger.

The Pythagorean Theorem—and, more generally, the Law of Cosines—gives us an indispensable way to measure geometric objects. We also have another tool, which gives us an indispensable way to keep track of where those geometric objects are.

Coordinates

Suppose you want to keep track of an astronaut who is somewhere along a single line, like in our examples from the section on Relativity. You could choose a point, and measure how far the astronaut is from that point. If the astronaut is, for example, ten feet to the right of the special point, you might say that she is at the coordinate x=+10. Ten feet to the left, and you could call it x=-10. She could even be at a fractional coordinate like x=+6.78.

An Astronaut at x = +10

Of course, coordinates are just numbers that we place at each point to label that point, and to keep track of what happens at each point. We can lay those numbers down in any way we want. We might, for example, put in three coordinates for every foot. Then, if the astronaut is at a coordinate of x=+30, she would still be 10 feet to the right of the origin. We say that there is a factor of proportionality—three, in this case. Then, if we want the actual distance, we just divide the coordinate x by the factor of proportionality:

 

An Astronaut at x=+4, and y=+ 3

In a similar way, you may want to keep track of an astronaut somewhere in two dimensions. Again, you could choose a special point—called the origin—then set up a grid. Move to the right or left along the grid until you are directly under or above the astronaut to find the first coordinate, then move up or down to find the second. For example, our second astronaut might be four feet to the right, and three feet up. His coordinates would be x=+4 and y=+3. If he were three feet down, his coordinates would be x=+4, and y=-3. We can see this in the figure at the right.

Using the Pythagorean Theorem with these coordinates is a breeze. To find how far the second astronaut is from the origin, we take the x coordinate and square it, then take the y coordinate and square it, then add the two together.

That is, the distance squared is 25, so the astronaut's distance from the origin is just 5 feet. Just as in the one-dimensional case, however, our coordinates could be stretched or squished compared to the actual physical distances. In this case, though we might have two different stretches in the different directions. To take an example, for every three x coordinates, there might be one foot of distance, while one foot of distance might correspond to seven y coordinates. In this case, we have to do something like we did for the one-dimensional problem:

 

An Astronaut at x = +5.73, and y = + 3.46

Another way our original grid might be distorted is if it were skewed. That is, we might keep all the line intervals fixed, but simply push the vertical lines over at an angle of 60 degrees, for example. In this case, we have to go farther over to be "under" the astronaut, so his new x coordinate is x=+5.73. This also gives us farther to go "up" to reach him, so his new y coordinate is y=+3.46, as we see to the right. Now, we can't use the Pythagorean Theorem with these coordinates; we have to use the Law of Cosines instead. The cosine of 60 degrees is 0.5, so we have

Again, this says that the distance squared is 25, so the astronaut's distance from the origin is still 5 feet. We expect that to be the case, since the real physical distance shouldn't change if we just change the way we write our coordinates.

The Metric

When laying down a grid for coordinates, we could even combine the stretch-squish with the skew. In general, then, we would need a formula relating distance to coordinates like

Obviously, this quickly gets complicated—and tiring to write out. We can save time and effort by grouping some of the terms together and writing this same formula as

That is, we group those big terms together, giving them new name. Specifically, we define

These numbers we've defined—g_xx, g_yy, and g_xy—are very important in physics. Together, they form the metric, which relates physical distances to whatever coordinates we decide to use.

In general, the metric is just a little more complicated than the one we have shown here. First, we could be dealing with more than two dimensions. In three dimensions, we would add a z coordinate, and we would need g_zz, g_xz, and g_yz for the metric. We could even be working with time by adding a t coordinate. With all four dimensions, the metric involves the numbers

More importantly, the metric could change from place to place. If our coordinates were warped, we might have a grid that looks like our first grid in one place, but is bent over like the second grid in another place.

It is possible to draw warped grids on a flat piece of paper. In this sense, we would get a warped metric in a space that is really flat. On the other hand, it is impossible to draw a nice, straight grid in a space that is really curved. By very carefully examining exactly how the metric changes from point to point, we can tell if we've just drawn curvy coordinates in a flat space, or if we've drawn our coordinates in a truly curvy space.

Now we are getting very close to Einstein's Equations. Einstein had these warped pieces of spacetime that he needed to describe in some quantifiable way. He saw that a careful examination of the metric could describe the true geometry of any spacetime, whether curved or flat, so he used it for his theory. Einstein combined certain numbers describing the metric's changes from place to place into what is now called the Einstein tensor. Just like the metric, the Einstein tensor is a set of numbers. For four-dimensional spacetime, we have

These numbers describe what is physically interesting about the geometry of spacetime. Understanding the geometry of spacetime allows us to see how particles will move, bringing us one step closer to the ultimate goal of physics. There is just one more ingredient left.

Energy, Matter, and the Curvature of Spacetime

We've gotten a sneak preview of Einstein's equations before: G=8πT. The G on the left stands for the different numbers in the Einstein tensor. But, the Einstein tensor represents the geometry of spacetime, so this is what the left side really represents. We also know that the curvature of spacetime is caused by matter, so the T on the right must represent matter.

Just like G, the symbol T stands for a set of numbers:

These numbers measure different things about matter. Together, they make up the Stress-Energy Tensor.

As we see from the table, things like stress, pressure, and momentum come into Einstein's equations. That is, stress, pressure, and momentum all have some effect on the warping of spacetime. This is related to Einstein's most famous equation, E=mc², which says that energy has mass.

Warped spacetime affects how matter moves by changing its geodesics. On the other hand, Einstein's equations show us how matter—and its movement and pressures—affect the shape of spacetime. Thus, Einstein solved the fundamental problem in Physics—in principle. Of course, solving something in principle is very different from solving in practice. Finding real solutions has proven to be very difficult. Often, it is a job best left to computers.