## Curving Space and Time

### Warping Slices of Reality

Dark-heaving—boundless,
endless, and sublime,
The image of eternity—
the Throne of the Invisible...

From Lord Byron's Childe Harold's Pilgrimage
Canto IV, Stanza 183

For his first revolution, Einstein unified space and time, and showed that a given observer (Observer: A person or piece of equipment that measures something in physics. Frequently, we speak of an observer measuring time or a distance in a particular place. )}) just took a slice out of this spacetime (Spacetime: A concept in physics which merges our usual notion of space with our usual notion of time.). Each of the slices was flat: a one-dimensional slice was just a straight line; a two-dimensional slice was a flat sheet. In a similar way, the three-dimensional slices Einstein took were also flat, in a way we will explore below. For his second — and greater — revolution, Einstein allowed the slices to be curved and warped. One-dimensional straight lines became one-dimensional curvy lines; two-dimensional flat sheets became two-dimensional curvy sheets; three-dimensional flat spaces became three-dimensional curvy spaces. To understand this warping, it helps to think about warping in two-dimensional space.

We are all familiar with two-dimensional surfaces. Movie screens, pages in a book, and the surface of an apple are all examples. We also recognize a curved surface when we see it. Turn up half the page in a book, and you will see that it is curved. Leave the page down, and you see a flat surface. In high-school geometry class, we typically explore two-dimensional space, which is almost always assumed to be flat.

One familiar fact from this exploration is the formula for the circumference of a circle in terms of its radius. We remember the number π — pi (pi: or π (Pronounced as "pie".) An important number in geometry. This is defined to be the ratio of the circumference of a circle to the diameter of that circle, in flat space. π is an irrational number, which means that its exact value cannot be written down, though it can be calculated as precisely as necessary. Its value is approximately 3.14159265358979323...), roughly 3.1416 — showing up in the formula

$$Circumference = 2 \times \pi \times Radius$$

To use this formula, we make a circle in a particular way. Choose any point, and tie one end of a length of rope to that point. Stretch the rope taut, and rotate around the center point, tracing out the position of the free end of the rope. This will leave us with a circle. If we measure around the circle — its circumference — we should find that it is 2 × π times the length of our rope.

Now imagine that you try to use this formula in a surface that is not flat. Choose the center point of the circle to be the top of a hill and pretend that you are trapped on the hill; you and your rope can't rise up off of it, or dig down inside of it. Draw the circle on the lower parts of the hill. If you now measure around the circumference of the circle you drew, you get a number that is less than 2 × π times the length of the rope! If we tried to make a circle centered on a mountain pass (or in the middle of a horse's saddle), we would get a circumference that is greater than 2 × π times the length of the rope. The classic formula doesn't apply in these curved space examples.

Now, to think about curvature in three dimensions, we recall another high-school formula that gives the surface area of a sphere in terms of its radius:

$$Area = 4 \times \pi \times Radius \times Radius = 4 \times \pi \times (Radius)^2.$$

To use this formula, we make a sphere just as we made a circle, but we move everywhere the rope will let us. The places where the free end of the rope reach define a sphere. If we want to cover this sphere in fabric, for instance, we will need a piece 4 × π (about 12.566) times as big as a square with edges as long as our piece of rope, cut up into little patches, and sewn together just right. That piece of fabric should perfectly cover the sphere — assuming we have flat three-dimensional space.

In the bizarre world of curved three-dimensional space, however, we might need much less or much more fabric than we would think, just using the formula above. Again, the standard formula we learned in high-school (flat-space) geometry doesn't apply. It is not that this geometry is wrong, it's just that it only applies to flat space. We need a more sophisticated type of geometry to deal with curved space. That sophisticated geometry was developed by mathematicians, and called differential geometry (Differential Geometry: The branch of geometry which deals with curved surfaces and spaces. Calculus is used to analyze the shape of these surfaces and spaces. Differential geometry is a key tool used in the study of General Relativity (General Theory of Relativity: Einstein's version of the laws of physics, when there is gravity. Building on the Special Theory of Relativity (Special Theory of Relativity: Einstein's version of the laws of physics, when there is no gravity. The two fundamental concepts in the foundation of this theory are equality of observers, and the constancy of the speed of light. The first of these means that the laws of physics must be the same, no matter how quickly an observer is moving. The second means that everyone measures the exact same speed of light. This theory is useful whenever the effects of gravity can be ignored, but objects are moving at nearly the speed of light. It has been successfully tested many times in particle accelerators, and orbiting spacecraft. For objects moving much more slowly than light, Special Relativity (Special Theory of Relativity: Einstein's version of the laws of physics, when there is no gravity. The two fundamental concepts in the foundation of this theory are equality of observers, and the constancy of the speed of light. The first of these means that the laws of physics must be the same, no matter how quickly an observer is moving. The second means that everyone measures the exact same speed of light. This theory is useful whenever the effects of gravity can be ignored, but objects are moving at nearly the speed of light. It has been successfully tested many times in particle accelerators, and orbiting spacecraft. For objects moving much more slowly than light, Special Relativity becomes very nearly the same as Newton's theory, which is much easier to use. ) becomes very nearly the same as Newton's theory, which is much easier to use. ), this theory generalizes Einstein's work so that the laws of physics must be the same for all observers, even in gravity. Einstein showed that gravity is best understood as a warping of the geometry of spacetime, rather than as a pulling of objects on each other. The crucial idea is that objects move along geodesics — which are determined by the warping of spacetime — while spacetime is warped by massive objects according to the formula $$G = 8 π T$$. ).).

With the second revolution, Einstein let his slices be warped in the ways we've just seen. Since he had brought space and time together in spacetime, this meant that time could be warped, too. Understanding warped space and time didn't come easy to Einstein, and he resisted as much as he could. Eventually, though, he came to realize that this warping was simply how nature worked, and the differential geometry describing it was the only way to deal with a fundamental concept in his theory: the geodesic (Geodesic: Essentially the "straightest path" in a curved space or curved spacetime. This is the path followed by an object with no forces acting on it. In the curved spacetime of General Relativity, these paths may seem to be very curved — even appearing as circles or ellipses, for example. A geodesic is easily understood by looking at a very small region around the object. Even in highly curved spacetime, a small enough region will seem flat, so there is a natural idea of a "straight path". By following short segments, the whole geodesic is built up into one long path. )}).