# Section 6: The Theory of General Relativity

The law of universal gravitation describes a force that acts instantaneously between objects separated by arbitrarily large distances. This behavior is in conflict with the theory of special relativity, which forbids the transfer of information faster than the speed of light. So how does one construct a theory of gravity that is consistent with special relativity?

Einstein found the key: the apparent equivalence between gravity and acceleration. Imagine that you are in a windowless rocket ship far from any stars or planets. With the rocket engines turned off, you and everything else not secured to the rocket float freely in weightlessness within the rocket's cabin. When turned on, the rocket engines provide a constant acceleration—9.8 m/s^{2}, say—and you stand firmly on the floor directly above the engines. In fact, the floor pushes against your feet with the same force that the ground pushes against your feet when you stand on Earth. Einstein posed the question: Is there any experiment you could perform within your sealed rocket ship that could distinguish between being in a rocket with a constant acceleration of 9.8 m/s^{2}, or a rocket at rest on the launch pad on Earth?

**Figure 15:** A person in an accelerating rocket feels the same downward pull as a person on Earth feels from gravity.

**Source: **© M. Poessel/Einstein Online/Max Planck Institute for Gravitational Physics. More info

Einstein concluded that the answer was no: There is no way to tell the difference between the presence of a uniform gravitational field and a frame of reference that has a constant acceleration. This observation embodies Einstein's principle of equivalence, the equivalence of gravity and acceleration, on which he built the theory of general relativity.

## Gravitational lensing and the bending of light

We can use the equivalence between an accelerated reference frame and a frame with a uniform gravitational field to infer the behavior of light in a gravitational field. Imagine a beam of light traveling horizontally from one side of the rocket cabin to the other. With the rocket engines off, the light follows a straight path across the cabin in accordance with the laws of special relativity. With the engines on, causing constant acceleration, the cabin moves slightly upward in the time it takes the light to travel across the cabin. Hence, the light beam strikes a point lower on the cabin wall than when the engines were off. In the frame of the accelerating rocket, the light beam follows a curved (parabolic) path. Because an accelerating rocket is equivalent to a rocket at rest in a uniform gravitational field, a light beam will follow a curved path in a gravitational field; in other words, light is bent by gravity. A famous observation during a solar eclipse in 1919 confirmed that prediction: Measurements showed that starlight passing near the edge of the eclipsed Sun was deflected by an amount consistent with the principle of equivalence.

In the absence of gravity, a distant galaxy will appear to an observer on Earth to be a tiny source of light. However, if there are mass distributions such as other galaxies or clouds of dark matter near to the line sight between the Earth and the distant light source, the gravity from these mass distributions will bend the light from the distant galaxy. The image of the distant galaxy on Earth can then become a ring, one or multiple arcs, or even appear as several galaxies depending upon the location and distribution of the intervening mass. This distortion of light from distant sources is called gravitational lensing and is well established in observations from modern telescopes. The observed gravitational lensing is used to infer what sources of gravity lie between the Earth and distant light sources. A related phenomenon is an increase in intensity of the light observed from a distant source due to the passage of a massive object near to the line of sight. The gravitational field of the moving object acts as a lens, focusing more light into the telescope during the time that the massive object is near to the line of sight.

## Gravitational time dilation

Returning to our rocket ship thought-experiment, imagine that a light beam travels from the ceiling to the floor of the accelerating rocket. In the time the beam takes to traverse the cabin, the cabin floor has acquired a larger velocity than it had when the light left the ceiling. A device on the floor measuring the frequency of the light would find a higher frequency than that of the emitted beam because of the Doppler shift, a phenomenon noticed most commonly in an ambulance siren that has a higher pitch as the ambulance approaches and a lower pitch as it recedes. The principle of equivalence then asserts that, in a gravitational field, a light beam traveling opposite to the field acquires a higher frequency, shifted toward the blue end of the spectrum; while a light beam shining upward from the Earth's surface decreases in frequency as it rises, the effect that we know as the gravitational redshift. Again, experiments have confirmed this phenomenon.

An inertial (nonaccelerating) observer sees no change in the light's frequency—the frequency associated with the atomic transition generating the light—as the light moves across the cabin, because it is traveling freely through empty space. Yet, an observer on the rocket floor, accelerating with the rocket, can use the same, now accelerating, atoms and atomic transitions as a clock (see Unit 5 for details); the observer defines a second as the time required for the fixed number of oscillations of a specific atomic transition. We concluded in the last paragraph that this accelerating observer will see the frequency of the light beam to be higher than the frequency of the same atomic transition in the measuring device. The inescapable conclusion is that the atomic clock (like all clocks) ticks more slowly in the accelerating frame of reference.

By the principle of equivalence, clocks in a gravitational field tick more slowly than in the absence of the field; the stronger the field, the more slowly the clock ticks. An atomic clock at sea level loses five microseconds per year relative to an identical clock at an altitude of 5,000 feet. We age more slowly at sea level than on a mountaintop. The global positioning system (GPS) relies heavily on the accuracy of clocks and corrects for the gravitational time dilation to achieve its fantastic precision.

## Curved spacetime

The second key ingredient of general relativity is the notion of curved spacetime. Special relativity combines space and time into a four-dimensional spacetime, often referred to as "flat spacetime" or "Minkowski space." In flat spacetime, Euclidean geometry describes the spatial dimensions: Parallel lines never intersect, and the sum of the interior angles of a triangle is always 180 degrees. The two-dimensional analogue of flat space is the Cartesian plane, familiar from high school geometry class.

The surface of a sphere is also a two dimensional surface, but one that must be described by non-Euclidean geometry. Lines of constant longitude are parallel at the equator yet intersect at the poles. If you start at the equator and walk due north to the pole, turn right by 90 degrees, walk south to the equator, and then turn right again and walk along the equator, you will return to your original position having taken a triangular path on the Earth's surface. The sum of the interior angles of your triangular path is 270 degrees. A spherical surface is said to have positive curvature; a saddle-shaped surface has a negative curvature; and the sum of the interior angles of a triangle drawn on a saddle is less than 180 degrees.

Viewed from three dimensions, the shortest path between two points on a spherical or saddle-shaped surface is a curved line. This "geodesic" is the path that light would follow in that space. Three-dimensional space and four-dimensional spacetime, described by Riemannian geometry, can also be curved. The geodesics in spacetime are the paths that light beams follow—or, equivalently, the paths that observers in free fall follow. The Earth is in free fall about the Sun. We can construct a curved spacetime in which a circular orbit about the Sun is a geodesic. In such a spacetime, the Earth's orbit would stem from the curvature of spacetime rather than from a force acting between the Earth and the Sun.

The theory of general relativity takes the equivalence between motion in a gravitational field and motion in curved spacetime one step further. It asserts that what we call gravity is the bending of spacetime by matter. Rather than viewing gravity as a force acting between objects in flat spacetime, we should understand gravity as the interaction between matter and spacetime. The field equations of general relativity specify how matter and energy determine the curvature of spacetime. In turn, the spacetime curvature determines how the matter and energy will evolve.

## Black holes

If enough matter and energy are concentrated in a small enough volume, general relativity predicts that spacetime can become so highly curved that a black hole is formed. A black hole is characterized by an event horizon, a surface surrounding the enclosed matter, through which nothing can escape; the event horizon represents a surface of no return. To an outside observer, the black hole is completely described by just three numbers: its mass, its electric charge, and its angular momentum.

**Figure 19:** As gas falls into this supermassive black hole, it emits x-rays.

**Source: **© NASA, ESA, A.M. Koekemoer (STScI), M. Dickinson (NOAO), and the GOODS team. More info

Black holes can be created when a star of sufficient mass, after having burnt its nuclear fuel, collapses under its own weight. The black hole grows by capturing nearby matter and radiation that is pulled through the event horizon and by merging with astronomical objects such as stars, neutron stars, and other black holes. Massive black holes, millions to billions times more massive than our Sun, have been found near the center of many galaxies, including our own Milky Way. The black holes become visible when they accrete gas from the surrounding regions; the gas is accelerated and heated, producing observable radiation, before falling through the event horizon. The presence of a black hole can also be inferred from its gravitational influence on the orbits of nearby stars.