# Einstein’s Universe Final Exam

-The most general principle of relativity would be that the laws of physics are the same in all frames of reference.

-Another reason it is not consistent with special relativity is that no information can be transferred at a speed greater than the speed of light (in a vacuum). But that mathematics of Newton’s Theory of Gravity predicts that the gravitational force is propagated instantaneously: the Moon instantly knows if there is a change in the distance between it and the Earth. How can anything or any information be transmitted at infinite speed?

Gravitational mass is a measure of how strongly the Earth (or, by extension any planet or star) pulls on an object through the gravitational force. This is the mass that appears in the formula for Newton’s Law of Universal Gravitation: F = G m_1 m_2/r^2, where r is the distance between the two objects of mass m_1 and m_2.

-Newton achieved this in his theory (explaining Galileo’s result that all objects fall near the surface of the Earth with the same acceleration) by equating gravitational mass and inertial mass for every object.

-So the result of this would be that the same experiment done in the two frames shown in the figure would produce different results: in other words, the two frames of reference would be distinguishable by virtue of the different results of the same experiment done in the two frames of reference.

-So this comes back to the principle of equivalence, which asserts that inertial and gravitational mass are numerically equal for all bodies, results in a reference frame in free fall (a frame in which the objects within it are subject only to gravitation) is indistinguishable from a small reference frame subject to a uniform gravitational field.

-In this frame of reference the laws of special relativity hold true; the law of inertia holds true. While it may be impossible or difficult to determine if one is really in a uniform state of motion, it is one can determine if one is in a free-fall/float frame. Just determine if there are any non-gravitational forces present. Such forces do not affect all bodies equally: gravity does and that is the key to the equivalence principle.

“The word curvature is an analogy, a visual way of extending ideas about three-dimensional space to the four dimensions of spacetime. Travelers detect curvature – in both three and four dimensions – by the gradual increase or decrease of the “distance” between “straight lines” that are initially parallel. In three space dimensions, the actual paths in space converge or diverge. Think of two travelers who start near one another at the equator of Earth and march “straight north”. Neither traveler deviates to the right or to the left, yet as they continue northward they disoer that the distance between them decreases, finally reaching zero as they arrive at the north pole. They can use this deviation to describe the curved spherical surface on which they travel. Similarly, in four-dimensional spacetime, travelers detect the deviation from parallelism of nearby worldliness of free particles, each of which follows an ideally stragith spacetime path, often called a geodesic. This curvature can be measured by travelers and varies from place to place in spacetime.” – from “Exploring Black Holes: An Introduction to General Relativity” by Edwin F. Taylor and John Archibald Wheeler.

-When the light travels upwards, away from a star, the wavelength increases (the gravitational red shift). The person far from the star can compare the frequency of this gravitationally red shifted light to light emitted at her location by the same precise process. Since the light from the surface is gravitationally red shifted, the person far above the surface will say that the clock at the surface is running slow: gravity slows down the rate at which clocks keep time. The stronger the gravity the slower the clock will run.

Also note the difference between gravitational time dilation and the time dilation in special relativity. The latter is that the shortest time between two events is in the frame in which the two events happen at the same place. There is only one observer (one reference frame) who sees the two events as happening at the same place. Other observers in relative motion to this observer do not record the two events as happening in the same place.

What Wolfson calls “The principle of cosmic laziness” on page 197 refers to a pair of events with many observers who are in relative motion, but who are also present at both events, meaning that they all see the two events as happening in the same place, even though they are in relative motion (in different frames of reference.). The only way that one can have observers in different reference frames observing pairs of events that all agree happened at the same place is if one allows accelerating reference frames (acceleration due to a force, and remember gravity is not a force according to Einstein).

-From Wolfson: “Any massive body whose gravitation, or spacetime curvature in general relativity, bends light from more distant objects. The result may be brightened, distorted, or multiple images.”

-The orbital period is changing because the neutron stars’ orbit is shrinking because the stars are losing energy as predicted by general relativity. They are losing energy through the creation of gravitational waves, ripples in spacetime, that they create by moving so quickly through spacetime.

-General Relativity also predicts the neutron stars should be losing energy as they orbit each other and this has been measured in careful observations of the binary pulsar. Where does this energy go? Supposedly into gravitational waves, but those waves have never been detected. There are currently experiments under development to detect gravity waves.

-But the question remains are the galaxies moving away from us through a fixed space, or is the space itself between galaxies expanding, or stretching, which also produces this motion of all galaxies away from us?

-Note that some nearby galaxies were found to be blueshifted. Since the closer galaxies feel a stronger gravitational pull to our galaxy, these galaxies are moving towards us because of the local gravitational attraction with our galaxy. The question of whether the expansion will continue forever or reverse itself depends on the rate of the expansion of the Universe and the density of matter in the universe.

-Since the majority of the galaxies were moving away from us, as if we were at the center of a great explosion of matter in our Universe. This means that the farther back in time you go, the closer the galaxies, and all matter, are together. Keep going back farther in time and eventually all the matter seems to have been at the same place. At this place there was some sort of expansion or explosion that caused space to expand and carry matter away from other clumps of matter. We call this instant the Big Bang.

(2). When the average density of matter in the Universe is less than the critical density then the Universe expands forever with a non-zero limiting speed.

(3). When the average density of matter in the Universe is equal to the critical density then the Universe expands forever with a zero limiting speed.

(2). When the average density of matter in the Universe is less than the critical density then all of the matter in the Universe will cool down and get farther apart. The average density of matter will continue to decrease.

(3). When the average density of matter in the Universe is equal to the critical density then then all of the matter in the Universe will cool down and get farther apart. The average density of matter will continue to decrease.

A “photon sphere” is defined by the set of points surrounding a black whole where light which is emitted tangentially (at right angles) to the surface of such a sphere, would travel in a circle around the black hole, forever. If the light is emitted just slightly toward the hole, it will spiral in. If just slightly outward, it will escape. The radius of the photon sphere is three-halves the size of the event horizon (“Schwarzchild radius”), where no light can escape.