The Mathematics of Star Trek

The Mathematics of Star Trek Lecture 5: Special Relativity

Topics • Galilean Relativity • Electricity and Magnetism • Maxwell’s Equations • The Michelson-Morley Experiment • Maxwell’s Equations Revisited • The Lorentz Transformations • Special Relativity

Galilean Relativity • The aim of science (of which mathematics is a part) is to describe and interpret reality. • When people compare notes on an observation, they find points of common agreement, which ultimately make up what is agreed upon as physically, or objectively, real. • Questions that need to be answered include: • “Do my calculations agree with yours?” • “Can I reproduce your results?”

Galilean Relativity (cont.) • Addressing the question of objectivity in connection with motion, GallileoGalilei (1564-1642) came up with the following principle of relativity: • “Two observers moving uniformly relative to one another must formulate the laws of nature in exactly the same way. • In particular, no observer can distinguish between absolute rest and absolute motion by appealing to any law of nature; hence there is no such thing as absolute motion, but only motion in relation to an observer.”

Galilean Relativity (cont.) • We can interpret Galileo’s principle of relativity in two ways: • Any physical law must be formulated the same way by all observers. • Anything formulated the same way by all observers is a physical law.

Galilean Relativity (cont.) • Suppose we have two “Galilean” observers, which we’ll call the Romulan Commander (R) and Captain Picard (P). • Each observer is on a spaceship with their own coordinate system. • For simplicity, we’ll assume motion in one space dimension. • Including time, we get a spacetime vector for each observer in their reference frame! • Such a vector is called an event. • Romulan coordinates for an event: (x, t) • Picard coordinates for the same event: (x’, t’)

Galilean Relativity (cont.) • Suppose that Picard’s ship, the Enterprise, is moving to the right at a constant velocity v and the Romulan ship, a warbird, is fixed in space. • Further, assume that at time t = t’ = 0, the ships are at the same point in space. • Using Galileo’s idea of relativity, if we know the coordinates of an event in one reference frame, we can figure out the coordinates of the event in the other reference frame!

Galilean Relativity (cont.) • If the Romulan’s coordinates for an event are (x,t), then Picard’s coordinates for this event will be (x’, t’) = (x - vt, t). (1) • If Picard’s coordinates for an event are (x’,t’), then the Romulan’s coordinates for this event will be (x, t) = (x’ + vt’, t’). (2) • We call (1) and (2) Galilean transformations. Romulan Picard velocity v P-axis x’ 0 R-axis x 0

Galilean Relativity (cont.) • If Picard measures an object’s velocity in his frame of reference as dx’/dt’ = w’, what does the Romulan Commander see? • Using Galilean transformation (2), the Romulan commander will see the object’s velocity (in his reference frame) as w = dx/dt = d/dt’[x’ + vt’] dt’/dt = (dx’/dt’ + v)(1) = w’ + v. Romulan Picard velocity v velocity w’ P-axis x’ 0 R-axis x velocity w’+v 0

Electricity and Magnetism • In 1820, Hans Christian Oersted (1777-1851) discovered that if an electric current is switched on or off in a wire near a compass needle, the needle will deflect. • This showed that electricity and magnetism were related phenomena. • The Oersted is a unit of magnetic field strength.

Electricity and Magnetism (cont.) • In 1826, after hearing about Oerstead’s experimental results Andre Marie Ampère (1775-1836) attempted to give a combined theory of electricity and magnetism. • Ampère formulated a circuit force law and treated magnetism by postulating small closed circuits inside the magnetised substance. • Ampère's theory, including Ampère's Law became fundamental for 19th century developments in electricity and magnetism. • The standard unit of current is called the Ampère.

Electricity and Magnetism (cont.) • Charles Augustin de Coulomb (1737 – 1806) developed a theory of attraction and repulsion between bodies of the same and opposite electrical charge. • He found that the force between a pair of charged particles obeys a law similar to Newton’s Law of Universal Gravitation. • Coulomb’s Law states that the force between a pair of charged particles is proportional to the square of the distance between the charges. • A fundamental unit of charge is the Coulomb.

Electricity and Magnetism (cont.) • Michael Faraday (1791-1867) was a self-taught experimentalist whose work led to deep mathematical theories of electricity and magnetism. • For example, Faraday’s Law says that any change in the magnetic environment of a coil of wire will cause a voltage to be "induced" in the coil. One way to do this is to move a magnet near a coil of wire. • Applications of Faraday’s Law include back-up generators and automobile engine ignition via sparkplugs and the Faraday Flashlight! • A Farad is a unit of capacitance.

Maxwell’s Equations • Using the work of Ampere, Coulomb, and Faraday, James Clerk Maxwell (1831-1879) found that the laws of electricity and magnetism were related to each other mathematically. • Maxwell’s Equationsare a set of four partial differential equations relating electric fields and magnetic fields, due for example to a current flowing through a wire.

Maxwell’s Equations (cont.) • In addition to showing that electricity and magnetism are related mathematically, Maxwell’s equations can be used to show that “electromagnetic waves” exist. • These waves travel through space at a constant speed, which turns out to be the speed of light! • From this, people deduced that light itself is an electromagnetic wave! • This “ended” the debate over whether light is a particle or a wave. • Equations (3) - (6) are Maxwell’s Equations. • Electric fieldE(t,x,y,z) and magnetic fieldB(t,x,y,z) are vector functions. • (t,x,y,z) is a scalar function that describes the electric charge density and J(t,x,y,z) is a vector function known as the current density. • k is a constant and c is the speed of light!

In the Star Trek universe, “sensors” are devices that are used to gather information about a planet, starship, life-form, etc. A primary example of this is the tricorder. One place where Maxwell’s equations are used in “real life” is to design sensors, such as magnetometers. Maxwell’s Equations (cont.)

Maxwell’s Equations (cont.) • A magnetometer is a device that is used to detect changes in a magnetic field. • Magnetometers have applications in areas such as industry, biomedicine, oceanography, space exploration, and law enforcement. • The Galileo Orbiter used a magnetometer to map the structure and dynamics of Jupiter's magnetosphere. • Magnetometers are used to detect metallic weapons such as handguns.

Maxwell’s Equations (cont.) Electric Field Magnetic Field HFSS Model of Magnetometer Component

The Michelson-Morley Experiment • If light is a wave, it should travel in some medium (think of water waves). • Since Aristotle’s time (384-322 B.C.), people had called this medium the aether. • Between 1881 and 1887, Albert Michelson (1852-1931) and Edward Morley (1828-1923) set up a series of increasingly more accurate experiments to detect the effects of the aetheron light. • Using Galilean relativity, if a beam of light travels in the same direction that the earth is traveling around the sun (at 30 km/sec) and another beam travels in another direction, the light beams should have different velocities.

The Michelson-Morley Experiment (cont.) • The Michelson-Morley Experiment uses an interferometer to measure the change in velocity of light beams traveling in different directions. • Unfortunately, Michelson and Morley were unable to detect any difference in the speeds of the light beams. • One possible reason is that there is no aether! • Another possibility is that the speed of light is the same for all Galilean observers, no matter what their relative motion! • If this is true, then light doesn’t transform properly under the Galilean transformations!

Maxwell’s Equations Revisited • Returning to Maxwell’s equations, let’s assume that we are in empty space, so that  = 0 and J = 0. • Further, suppose that our electric field depends only on t and x. • With these assumptions, it follows that the electric field is a scalar field, E, which satisfies the wave equation that we saw earlier in this course!

Maxwell’s Equations Revisited (cont.) • Recall that according to Galileo, any physical law must be formulated the same way by all observers. • Thus, if Maxwell’s equations are a physical law, then they should have the same form for both the Romulan commander and Captain Picard (in the setting as above). • Similarly, the wave equation should look the same in both reference frames.

Maxwell’s Equations Revisited (cont.) • In the Romulan’s coordinates, the wave equation is the same as what we saw above: • Using the Galilean transformation (2), with (t’,x’) = E(t, x-vt), in Picard’s coordinates, the wave equation becomes:

In 1904, mathematical physicist Hendrick Lorentz (1853-1928) proposed a modification to the Galilean transformations to mathematically “fix” the problem with Maxwell’s equations and Galilean Relativity. With the Romulan commander and Picard as above, the Lorentz transformations are as follows: The Lorentz Transformations

If the Romulan commander sees an event with coordinates (x, t), then Picard will see the event as If Picard sees an event with coordinates (x’, t’), then the Romulan commander will see the event as The Lorentz Transformations (cont.)

Special Relativity • The Lorentz transformations were introduced to mathematically address the problems that arose from trying to apply Galilean relativity to the Michelson-Morley experiment and Maxwell’s equations. • In 1905, starting with the hypothesis that the speed of light is constant for all Galilean observers, Albert Einstein (1879-1955) was able to show that the Lorentz transformations must replace the Galilean transformations. • This work is known as the Theory of Special Relativity.

Here are some of the things that follow from Special Relativity, again with Picard moving at a velocity v, relative to the Romulan commander. All clocks on Picard’s ship will appear to be ticking more slowly to the Romulan commander. If Picard’s measures a time interval of T0, then the Romulan commander will see the time interval T as: All rulers on Picard’s ship will appear shorter in length to the to the Romulan commander. If Picard’s ruler has length L0 in his frame, then the Romulan commander will see the ruler’s length L as: Implications of Special Relativity

As the Enterprise’s velocity v increases, its mass will increase, approaching infinity as v approaches c. If the Enterprise has rest mass M0, then at velocity v, it will have relativistic mass M: Another result that follows from Einstein’s theory of Special Relativity is E = mc2 which relates energy to mass via the speed of light. This equation is the key to making nuclear reactors and nuclear weapons work. Implications of Special Relativity (cont.)

Hermann Minkowski’s Contribution to Special Relativity • In 1907, Hermann Minkowski (1864-1909) developed a new view of space and time and laid the mathematical foundation of the theory of relativity. • Minkowski realised that the work of Lorentz and Einstein could be best understood in a non-Euclidean space (in particular Hyperbolic space). • He considered space and time, which were formerly thought to be independent as vectors with four components - three for space and one for time. • This space-time continuum provided a framework for all later mathematical work in relativity.

Geometry is the study of figures in a space of a given number of dimensions and of a given type. Euclidean geometry, which is often taught in high school, is based on a thirteen volume book called the Elements written in 300 B.C. Starting with five postulates (axioms), Euclid (~325 - ~265 B.C.) shows how basic properties of triangles, parallels, parallelograms, rectangles, squares, circles, etc. follow. Hyperbolic Geometry

Hyperbolic Geometry (cont.) • Here are Euclid’s Postulates: • A straight line segment can be drawn joining any two points. • Any straight line segment can be extended indefinitely in a straight line. • Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. • All right angles are congruent. • If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. • Postulate 5 is equivalent to what is known as the parallel postulate: Given any straight line and a point not on it, there "exists one and only one straight line which passes" through that point and never intersects the first line, no matter how far they are extended.

Hyperbolic Geometry (cont.) • In 1823, Janos Bolyai (1802-1860) and Nikolai Lobachevsky (1792-1856) independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. • For example, we get hyperbolic geometry by keeping Euclid’s first four postulates the same and replacing the parallel postulate with: For any infinite straight line and any point not on it, there are many other infinitely extending straight lines that pass through and which do not intersect.

Hyperbolic Geometry (cont.) • In hyperbolic geometry, we find that: • The sum of angles of a triangle is less than 180 degrees. • Triangles with the same angles have the same areas. • Not all triangles have the same angle sum • There are no similar triangles in hyperbolic geometry.

One way to visualize hyperbolic geometry in the plane is via the Poincaré disk. In this model, A line is represented as an arc of a circle (diameters are permitted) whose ends are perpendicular to the disk's boundary. Two arcs which do not meet correspond to parallel rays. Arcs which meet with an angle of 90 degrees correspond to perpendicular lines. Arcs which meet on the boundary are a pair of limiting parallel rays. The artist M. C. Escher's used hyperbolic geometry to create the pattern Circle Limit III. Hyperbolic Geometry (cont.)

References • The Geometry of Spacetime, James J. Callahan, Springer Verlag (2000). • Hyper Physics: http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html • St. Andrews' University History of Mathematics: http://www-groups.dcs.st-and.ac.uk/~history/index.html • Star Trek Memory Alpha (tricorder): http://en.memory-alpha.org/wiki/Starfleet_tricorder • Wikipedia (Michelson-Morley Experiment): http://en.wikipedia.org/wiki/Michelson-Morley_experiment • Eugenii Katz’s Homepage (Famous Scientists): http://chem.ch.huji.ac.il/~eugeniik/history/oersted.htm • Math World: http://mathworld.wolfram.com/ • Euclidean and Non-Euclidean Geometries, Marvin Jay Greenberg, W.H. Freeman and Company (1979). • Doug Dunham’s Homepage: http://www.d.umn.edu/~ddunham/isis4/section6.html

The Mathematics of Star Trek