The Mathematics of Star Trek

The Mathematics of Star Trek Lecture 6: General Relativity

Topics • Captain Picard and Einstein’s Elevator • Principal of Equivalence • Bending of Light • General Relativity and Curved Spacetime • Einstein’s Equations of General Relativity • Mercury’s Orbit • Faster than Light (FTL) Travel • Impulse Drive, Tractor Beams, Deflector Shields, and Cloaking Devices

Suppose that just as a phaser (light) beam is shot by a Romulan Warbird at Captain Picard’s yacht Calypso, the Calypso accelerates forward, perpendicular to the phaser shot. In the Romulans’ frame of reference, the phaser beam travels in a straight line. Thinking of light as a particle, it can be shown with vectors and the equations of motion that Picard will see the light beam travel along a curved path! Mathematica example! Captain Picard and Einstein’s Elevator Romulan Frame Picard Frame

If Picard’s acceleration is the same as that due to gravity at the Earth’s surface, then Picard will feel the same force pushing him back in his seat as he would due to the downward force of gravity at the Earth’s surface! Einstein argued that Picard (or someone in an elevator being accelerated upwards with the same acceleration) will never be able to perform any experiment to tell the difference between the reaction force due to accelerated motion and that due to the pull of gravity from some nearby heavy object outside the ship. Einstein concluded that whatever phenomena an accelerating object experiences would be the same as the phenomena an observer in a gravitational field experiences! Captain Picard and Einstein’s Elevator (cont.)

The Principle of Equivalence • Einstein’s idea, known as the Principle of Equivalence, says that inertial mass and gravitational mass are the same and is the starting point for the theory of General Relativity! • Here is one implication of the Principle of Equivalence: • Since Picard observes the light ray bending when he is accelerating away from it, it follows from Principle of Equivalence that the light ray would also bend in a gravitational field! • Matter produces a gravitational field, so matter must bend the path of a light beam.

In 1911, using the idea that light will be bent in a gravitational field, along with Newton’s Laws, Einstein predicted that light passing by the outer edge of the Sun should be bent by an angle of approximately 0.85 seconds. In 1919, Sir Arthur Eddington led one of two expeditions (his went to Sobral Brazil) to observe the apparent position of stars on the sky near the Sun during a solar eclipse. During the eclipse, rays of light from stars passing close to the sun would be bent. The amount by which the light is bent could be deduced by comparing the stars’ relative positions to those at some other time of the year. Eddington found that the light bends exactly twice as much as that predicted by Einstein! Bending of Light

The phenomenon observed by Eddington is the same as that of gravitational lensing, where a cluster of galaxies can produce multiple magnified images of a galaxy much farther away, which has been seen using the Hubble Space Telescope. A gravitational lens is a massive object that magnifies or distorts the light of objects lying behind it. For example, the powerful gravitational field of a massive cluster of galaxies can bend the light rays from more distant galaxies, just as a camera lens bends light to form a picture. Bending of Light (cont.)

In 1916, Einstein published a paper that introduced the world to his theory of General Relativity. Unable to incorporate gravity directly into the theory of Special Relativity, he was led to the idea that gravity is not a force, but a manifestation of the curvature of spacetime. Masses in space such as the Sun cause spacetime to be curved and the curved paths that we see objects (or light) following near these masses are simply the straightest possible paths in the curved spacetime! Adding in the idea that spacetime is curved, he was able to show that the predicted bending of light by the Sun during the 1919 eclipse was right! General Relativity and Curved Spacetime

Mathematically, General Relativity is built upon a set of ten coupled hyperbolic-elliptic nonlinear partial differential equations, which can be represented symbolically as shown at the right. (Click on the image for a link to the equations!) The equations boil down to this: CURVATURE (LHS) = MATTER AND ENERGY (RHS). This theory is hard to work with, because: The curvature of space is determined by the distribution of matter and energy in the universe. The distribution of matter and energy in the universe is determined by the curvature of space. Einstein’s Equations of General Relativity

Another way that the theory of General Relativity was shown to be correct was by answering a question about Mercury’s orbit! According to Newton’s Laws, the planet Mercury moves around the Sun in an elliptical orbit with the Sun at one focus of the ellipse. After one revolution around the Sun, Mercury should come back to its starting point, which isn’t what happens. It turns out that the perihelion (closest point to the Sun) of the orbit of the planet Mercury advances approximately two degrees per century (image is from Hyper Physics website). All but approximately 40 arc seconds of this advance can be accounted for via classical Newtonian physics, such as the force of gravity from other planets in the solar system! Mercury’s Orbit

Astronomers guessed that the last 40 seconds of arc might be due to another (unknown) planet in our solar system, which they named Vulcan. This is how the planet Neptune was discovered! Neptune was the first planet located through mathematical predictions rather than through systematic observations of the sky! After the discovery of Uranus in 1781, astronomers noted that Uranus was not faithfully following its predicted path. Uranus seemed to accelerate in its orbit before 1822 and to slow after that. Mercury’s Orbit (cont.)

One possible explanation was that the gravity of an undiscovered planet was affecting the orbit of Uranus. Starting in 1841, British astronomer John Couch Adams and the following summer, French astronomer Urbain Jean Joseph Le Verrier without knowledge of each other independently calculated where the new planet should be. At first, neither was taken seriously, but by 1846, based on their work, Neptune was discovered! Mercury’s Orbit (cont.)

Einstein suggested that the extra advance of Mercury’s perihelion could be due to the curvature of spacetime near the Sun. He predicted that the amount by which the Mercury’s perihelion should advance is given by: where a is the length of the semi-major axis of Mercury’s elliptical orbit, e is the eccentricity of the ellipse, c is the speed of light, and T is the period of revolution. HW: See if this formula works! Experimental Verification: Mercury’s Orbit (cont.)

One of the implications of curved spacetime is the idea that what we perceive as a straight line need not be the shortest path between two points! Krauss gives an explanation of how this might occur in two-dimensions! Consider a piece of elastic material, as shown on p. 44 of our textbook. If the material is laid flat and a circle drawn on the sheet, the shortest path between two opposite points on the circle, A and B, would be a straight line through the center. Faster Than Light (FTL) Travel

If the center is pushed down and the material stretched, then the shortest path may be along the circle. The sheet has been curved (in 3-space), but to a bug walking on the “line” from A to B, it thinks it is moving in a straight line. This is the idea we want to extend to spacetime - we are the bug and cannot perceive the curve of spacetime in 3-space! It may be possible to traverse what appears to be a huge distance (line-of-sight wise) by finding a shorter route through spacetime! If spacetime itself can be manipulated, then objects can travel locally at low velocities, yet an accompanying expansion or contraction of space could allow huge distances to be traversed in short time intervals! FTL Travel (cont.)

An example of this idea has been developed by physicist Miguel Alcubierre, who has shown that mathematically “warp drive” could be possible in the theory of general relativity. According to Alcubierre, a spacetime configuration can be created in which a spacecraft could traverse a distance between two points in an arbitrarily short period of time. Throughout this journey, the spacecraft would move with respect to its local surroundings at speeds much less than the speed of light! Therefore clocks on the ship would stay synchronized with the outside world. FTL Travel (cont.)

The idea works like this: warp spacetime so it expands behind the ship and contracts in front of it, thus propelling the ship along with the space surrounding the ship (like a surfboard and surfer on a wave). The spaceship will never travel faster than the speed of light, as the light near the ship will be carried along with the ship! In this theory, it would be possible to arrange for the huge gravitational fields needed to be somewhere far from the spaceship or star bases or planets the ship may travel to as a destination, thus avoiding problems with slow clocks. FTL Travel (cont.)

The same idea that works for warp drive, namely warping spacetime, would allow for travel at impulse speeds! The crew wouldn’t be subjected to large accelerations, so inertial dampers would no longer be needed! Warping spacetime could also be used to move a planet – contract space behind the asteroid and expand space in front of it! This could be how the tractor beam really works – if so, then Newton’s Third Law doesn’t apply any more! Two other applications of warping spacetime might be deflector shields and cloaking devices! Deflector shields are force fields that prevent phaser beams (light rays) from hitting a starship. Cloaking devices make a ship invisible. In each case, space would be warped to either deflect (bend) the light rays away from the ship or cause the light rays to bend around the ship (cloak it). Impulse Drive, Tractor Beams, Deflector Shields, and Cloaking Devices

References • Relativity: The Special and General Theory by Albert Einstein • The Geometry of Spacetime by James Callahan • The Physics of Star Trek by Laurence Krauss • http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html • http://www-groups.dcs.st-and.ac.uk/~history/index.html • http://hubblesite.org/newscenter/newsdesk/archive/releases/2004/08/text/ • http://archive.ncsa.uiuc.edu/Cyberia/NumRel/EinsteinEquations.html#intro • http://pds.jpl.nasa.gov/planets/captions/neptune/fullnep.htm • http://members.aol.com/nogravityguy/book02.htm • http://marge.uvm.edu/Sdempse/images/TV_Movies/Star_Trek/borgtrac.gif • http://www.thasos.ukgateway.net/images/Ent_Warp_Small.jpg • http://www.exn.ca/mini/startrek/warpdrive.cfm

The Mathematics of Star Trek