Chapter 12: General Relativity


by Austin Cole, Josefine Fabricius and Andrea Lavilles

The Equivalence Principle

The equivalence principle states that a gravitational field is equivalent to a uniformly accelerating reference frame. This means that an object inside of a spaceship accelerating at 9.8 m/s^2 would behave the same as an object on the Earth’s surface. From this principle, Einstein was able to deduce that an object in free-fall is really in a "local" inertial reference frame. This means that it does not accelerate. Instead, the time as measured by that object changes at an accelerating rate. This gives the appearance of acceleration, but the object really does not accelerate. This is why an accelerometer does not measure any acceleration when in free-fall. This is a sharp contrast to Newtonian physics, in which objects in free-fall were in an accelerating reference frame, while objects on the surface of a massive object such as a planet were in an inertial reference frame. In 1911, Einstein added a corollary to his equivalence principle that states, “Whenever an observer detects the local presence of a force that acts on all objects in direct proportion to the inertial mass of each object, that observer is in an accelerated frame of reference.” These ideas were the beginning of Einstein’s development of the theory of general relativity.

The metric

Geodesics

A geodesic is the shortest distance between two points in curved space. It is similar to a straight line, but it is bent along the local curve of space. For example, on a sphere such as the Earth the geodesic between two points would be an arc. In space, light travels along geodesics. This explains why light is bent in gravitational fields. Because gravitational fields bend space and time, the geodesics through those regions of space are also bent.

General Relativity

The general theory of relativity is a metric theory of gravitation. The "metric" is the object being studied, which in this case is the general gravitational field. The theory of general relativity was discovered in 1916 by Albert Einstein, a German physicist. It is the most accepted theory of modern physics, though it still bears many unsolved mysteries.

The theory of general relativity simplifies special relativity (its predecessor, discovered in 1905) and Newton's law of universal gravitation. Special relativity is the physical theory of measurement in an inertial frame of reference. Newton's law of universal gravitation states that every mass in the universe attracts every other mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them (F=G[(m1m2)/r^2]).

The theory describes gravity as a geometric property of space and time. The curvature of space-time is directly related to the mass energy and linear momentum of the matter present. This relationship can be explain with Einstein's field equations. The theory of general relativity was different than the ideas proposed by classical physics at the time in the ways it explained the passage of time, the geometry of space, the motion of falling bodies, and the transmission of light. It proposes the reality of black holes, the end state of massive stars in which space and time are distorted and from which nothing can escape.

Tests of General Relativity

  1. Precession of Mercury's Orbit
    According to the Newtonian theory of gravity, the orbits of planets should be ellipses that do not change with time. However, the orbits of planets have been observed to precess. That is, the point in space where the planet is closest to the sun changes with time, and the elliptical orbit of the planet rotates along with it. This effect was first discovered in the orbit of Mercury by Urbain Le Verrier in 1859. Einstein’s gravitational field equations were able to predict this precession, and it has since been accurately observed in the orbits on Mercury, Venus, and Earth.
  2. Bending of light due to the sun
    Part of the theory of relativity predicts that light is bent as it passes through a gravitational field. This occurs because light travels along what is known as a null geodesic. Far from any gravitational fields, the geodesic is a straight line, and light travels in a straight line. However, as spacetime is bent is gravitational fields, the geodesic becomes a curve. This effect is observable with light coming from distant stars as it passes near our Sun. This effect also explains how the massive gravitational fields of black holes are able to trap light and prevent it from escaping.
  3. Gravitational "time dilation"
    According to the theory of general relativity, time passes at different rates in different levels of gravitational potential. Time passes slower in regions of low gravitational potential than it does in regions of high gravitational potential. This theory predicts that clock at different altitudes would run at different rates. This prediction was first confirmed by the Pound-Rebka experiment in 1959. Even though the differences in these times are on the order of nanoseconds, the effect is significant enough that it has to be accounted for in many practical applications, such as GPS satellites. Time dilation occurs not only in gravitational fields, but also in accelerated reference frames. This is due to the equivalence principle of general relativity. Therefore, any kind of acceleration creates a time dilation for the observer in that reference frame.
  4. Gravitational Red Shift
    Robert Pound and Glen Rebka Experiment

    The last prediction of general relativity to be tested was the gravitational red shift. This was tested in the Pound-Rebka experiment that also confirmed the gravitational time dilation. The theory of gravitational red shifting predicts that as light moves from an area of high gravitational potential to an

    area of low gravitational potential, the frequency is shifted toward the red side of the spectrum. This was tested using light emitted by an excited electron in an atom. The photon emitted can be absorbed by another identical atom, but only if the frequency and energy of the photon is the same as when it is emitted. In the experiment, two samples of iron were placed at the top and bottom of a tower, and the sample at the top was excited so that it would emit photons. The sample at the bottom of the tower did not absorb these photons, proving that the frequency of the light was in fact shifted. The magnitude of this shift was able to be measured by creating a Doppler effect that cancelled out the gravitational shift. The original data agreed with the predictions of general relativity within 10%, and successive experiments have agreed within 1% of the predicted values.

Karl Schwarzchild Karl Schwarzschild was a German astronomer and physicist born on October 9, 1873 in Frankfurt, Germany. A child prodigy, Schwarzschild wrote a paper on celestial mechanics at the age of sixteen, which was then published. An early interest in astronomy sparked his love of science and math, leading Schwarzschild to make many important discoveries. Much of his early education can also be attributed to Paul Epstein, an older friend and son of Schwarzschild's professor. He studied at the University of Strasburg and the University of Munich, where he got his doctorate. After this, he was hired to work at an observatory in Vienna, Austria. In 1901 Schwarzschild was a professor at the University of Gottingen, which he then left
in 1909 after his marriage. He was soon after appointed as director of the observatory in Potsdam, Germany. Then in 1914 with the outbreak of the first world war, Schwarzschild volunteered with the military and was sent to Belgium, France, and Russia.

It was while he was serving in the war that Karl Schwarzschild made his most significant discovery: the first two solutions of Einstein's field equations of general relativity (general gravitational equations). One was in static, isotropic empty space surrounding a massive body (the beginning of the concept of the black hole), and the other was inside a spherically symmetric body of constant density. He did this in 1915, only shortly after Einstein came up with the theory itself. These solutions gave an understanding of the geometry of space near a point mass. Einstein himself was surprised at the simplicity of Schwarzschild's results.

Though it is not certain, it is believed that Schwarzschild died of a rare skin disease called pemphigus on May 11, 1916 in Potsdam.

Arthur S. Eddington
Arthur Eddington was born into a Quaker family on December 21, 1882 in Westmorland, England. His father passed away during the typhoid epidemic of 1884 and his mother was left to raise Arthur and his sister alone on a low income. Eddington was home educated until he was sent to preparatory school. In 1893, Eddington began to attend Brymelyn School where he did well in math and literature; although, his education in mathematics did not progress past differential and integral calculus. Eddington was awarded a scholarship in 1898 of 60 pounds a year for 3 years at Somerset County. He was under the age of 16 at the time and was unable to attend the university due to age restrictions. As a result, Eddington chose to attend Owens College in Manchester, where he studied from 1898 to 1902. Eddington mainly studied physics during his time there and was influenced by Horace Lamb, a math professor. The financial hardships from his family status did not hinder him from attaining an education because of the many competitive scholarships he won during his time there.

After being awarded a Natural Science scholarship of 75 pounds a year to Trinity College he won the Mathematics Scholarship of 100 pounds a year. He graduated with a M.A. in 1904 and went on to start a research project in the Cavendish Laboratory which eventually failed. His research in mathematics did not go well but he was able to salvage some of his work later on when applying them to an astronomy problem. In 1905, Eddington began to study astronomy and received a post at the Royal Observatory in Greenwich. When he arrived, he began to partake in a research project in which he used photographic observation to determine accurate values for the solar parallax. His analysis was based on two star-drifts and concentrated on the motions and distribution of stars. He was awarded a Trinity college Fellowship for his essay on the proper motions of stars.

Eddington was promoted to the Professor of Astronomy in 1913. When his fellow co-chair of Astronomy passed away, he became the director of the Cambridge Observatory and the head of theoretical and experimental astronomy at Cambridge. Shortly thereafter he was elected a Fellow of the Royal Society. Soon after these events World War I began. His Quaker heritage led him to avoid service during the war and he was able to do research during 1914-1918.

Arthur Eddington is best known for his work with general relativity. In 1915, Eddington received papers from Einstein and de Sitter which sparked his interest in the topic. General relativity offered a possible explanation for his observation of an advance of the perihelion of Mercury. He lectured on relativity in 1916 and wrote a report on the topic for the Physical Society in 1918. In 1919, Eddington took an eclipse expedition in West Africa to verify the bending of light that passed close to the sun; a prediction of general relativity. The stars close to the sun can only be observed during a total eclipse, unfortunately, it was cloudy and rainy on the day they observed the eclipse. When the rain stopped, the researchers took 16 photographs of the sun and the surrounding sky. The clouds made the star locations difficult to measure. His results however were enough to provide confirmation of the theory that gravity bends the path of light when it passes near a massive star. Eddington continued to lecture on relativity at Cambridge and provided more of a mathematical approach to the subject. He then published Mathematical Theory of Relativity in 1923.

Eddington also was interested on the internal structure of stars. His work included the mass-luminosity relationship for stars, the calculation of the abundance of hydrogen, and a theory of the pulsation of Cepheid variable stars. His early research on stars was published in The Internal Constitution of Stars in 1926. Eddington also included philosophy in his analysis of astronomy with books like The Philiosophy of Physical Science. He expressed in these texts that stressing the history of a scientific subject will hinder creativity in future research. In Fundamental Theory, Eddington tried to identify fundamental constants of nature of the relation of sixes within different physical systems. This was Eddington’s effort to unite quantum mechanics and general relativity. Overall Eddington is known for using his mathematical prowess in physics and astronomy and was able to clearly communicate difficult concepts in a simpler manner.