The Warped Point of View

Where “now” depends on how heavy you are

Go, wond'rous creature!
Mount where Science guides,
Go, measure earth, weigh air,
    and state the tides;
Instruct the planets in
    what orbs to run,
Correct old Time,
    and regulate the Sun.

Alexander Pope's
An Essay on Man

Einstein saw a beautiful idea in his notion of curved space and time. He saw that the falling astronaut wasn't being pulled or pushed by anything, but just moving along a straight line (a geodesic (Geodesic: Essentially the "straightest path" in a curved space or curved spacetime (Spacetime: A concept in physics which merges our usual notion of space with our usual notion of time.). This is the path followed by an object with no forces acting on it. In the curved spacetime 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 (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. ), 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 $G = 8 π T$. ), 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. )})) through curved spacetime. He realized that gravity could be reinterpreted, not as a force pulling on objects, but as a curvature of spacetime. Objects falling in a gravitational field — like around the Earth — aren't being pulled, but are simply moving along geodesics in the warped spacetime surrounding any heavy object. The Moon's orbit doesn't circle the Earth because of a pull, but because the straightest line through spacetime brings it back to the same point in space.

This bending of spacetime is particularly noticeable on Earth; throw a ball up in the air, and it follows its geodesic as it rises and falls. You can't miss the bending. To Einstein, the ball falls because spacetime is curving — not because there is a force pulling it back to Earth. While it is in the air, there is no force on the ball (except air resistance). We see it accelerate because there is a force acting on us: the force of the ground pushing up. Interestingly, Einstein's version of Newton's First Law of Motion (Newton's First Law of Motion: The first of Newton's Laws of Motion, which says that moving objects move in a straight line. Specifically, the Law says, "An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed (Speed: For a wave, the speed of a particular point (such as its crest).) and in the same direction unless acted upon by an unbalanced force.") (First Law of Motion: The first of Newton's Laws of Motion, which says that moving objects move in a straight line. Specifically, the Law says, "An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force.") says that we are not following a straight line (a geodesic) while standing still on solid ground.

Now, the space we are used to seems pretty flat — spheres act just as we learned they should in high-school geometry, for instance. So, we might think that spacetime must not be very warped. It turns out that it's very important to keep tabs on the warping of time, too. The motion of objects is very sensitive to this kind of warping, though most of us find it very difficult to understand this intuitively.

The essential idea here is that slices of spacetime are curved, and how they are curved changes the flow of time. Recall that a slice of spacetime is a set of points that an 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. ) picks out, and says that everything on that slice happens at the same time. This changing leads to another interesting effect. We saw earlier that astronauts moving relative to each other take different slices, and thus see things happen at different times.

A similar thing happens with slices warped by gravity. An astronomer on the Earth will move through the slices more slowly than an astronaut on the Moon, where the gravity is less intense. Moving through the slices more slowly means that the astronomer will get old more slowly than the astronaut. The effect is tiny and wouldn't ordinarily be noticed by humans, though it has been measured in extremely sensitive experiments.

All these descriptions of the effects of warped spacetime are useless, though, without any idea of how or why spacetime warps. To actually achieve his goal of predicting how objects move in their geodesics, or how time flows in different places, Einstein had to explain how massive objects curve spacetime. He devised an elegant — though extremely complicated — set of equations. We call these Einstein's equations (Einstein's Equations: A set of "tensor" equations Einstein devised to describe how mass warps spacetime. The set of equations may be written as $G = 8 π T$, where both $G$ and $T$ each represent a set of ten quantities. The $G$ quantities represent the warping of spacetime, while the $T$ quantities — the "Stress-Energy tensor" represent the mass.), appropriately. They can be written in a beautifully simple form:

$$G=8πT.$$

The $G$ on the left side represents all the curvature of spacetime at a point, while the $T$ on the right represents the mass at a point and its properties, and the same equation holds at every point in space and time. This is the elegant part.

The complicated part comes when we realize that this formula is almost completely useless for doing actual calculations. To use it, we have to expand it into at least ten different equations, each with dozens of terms. It is possible to solve the equations with pencil and paper in very special situations — when most of the dozens of terms happen to be zero — or in situations with low speeds, small masses, and large distances — when most of the dozens of terms happen to be very small and practically zero.

Most things in the Solar System, including on Earth, qualify for the low speed, small mass, and large distance situation. In this case, Einstein's theory gives exactly the same results as Newton's old theory of gravity. That's good news because Newton's theory worked very well in describing objects falling near the Earth, the planets orbiting the Sun, comets, and so on. Einstein's fancy warped spacetime theory would be all well and good, but needlessly complicated if it just reproduced what Newton had written down over three hundred years earlier. In cases with high speeds, small distances, or large masses, however, Einstein's theory is different from Newton's. This allows us to test the two theories, and decide which one is correct.

There was one exception to all the success of Newton's theory. For years before Einstein wrote down his equations, astronomers had known that the details of Mercury's orbit didn't perfectly match Newton's predictions. The difference was small, but clearly present. Mercury is the planet closest to the Sun. This combines a large mass (the Sun's) with small distances, which is one situation where Einstein's equations are not the same as Newton's. The very first thing Einstein did after discovering his equations was to solve them for the orbit of Mercury. He picked out the detail that Newton's theory couldn't account for, and saw that his own theory predicted it correctly. This was the first trial, and first success of Einstein's theory. He had shown that Newton was very nearly right much of the time, but wrong in the most extreme cases.

Einstein's theory doesn't just correct details in Newton's theory, but introduces completely new phenomena, like time warping and gravitational waves (Gravitational Wave: A gravitational disturbance that travels through space like a wave. This type of wave is analogous to an Electromagnetic Wave. Gravitational waves are given off by most movements of anything with mass. Usually, however, they are quite difficult to detect. Physicists are currently working hard to directly detect gravitational waves. Experiments like LIGO and LISA are designed for this purpose. ). Since his introduction of the General Theory of Relativity (General Theory of Relativity: Einstein's version of the laws of physics, when there is gravity. Building on the Special Theory of Relativity, 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 $G = 8 π T$. ), there have been many experiments testing his ideas and his equations. Every test so far has proven Einstein right and verified his new predictions, but the tests aren't over yet.

We know that Einstein's theory can't account for everything; we know that at some point there will be a deeper theory that will correct General Relativity, just as Einstein corrected Newton. Physicists are now embarking on a journey of discovery, and beginning to peer into the Universe in ways unlike ever before. They expect to see Einstein's theory fail so that they can gain a better understanding of the true laws of the Universe. They will make this journey with help from gravitational waves and numerical relativity (Numerical Relativity: The branch of Relativity research which deals with simulating the development of Spacetime, using computers. This is believed to be the only possible way to understand things like the merger of two Black Holes.).

Find out more about Gravitational Waves Read about Numerical Relativity