Life in the fast lane: the kinematics of Star Trek

Hendrik Lorentz: a guy who was almost as smart as Einstein Maxwell’s highly successful equations… Gauss’ law In the 19th century, these equations were thought to hold only in the luminiferous ether! no magnetic monopoles Faraday’s law Lorentz wanted to make these equations covariant (the same in all reference frames). He came up with the Lorentz transformation: Ampere’s law The previous slides summarized key consequences of Special relativity. To go further we need math (unless we really want to stare into a vacuum and visualize experiments in your head – like Einstein). Let us start with the likely Einsteins mathematical motivation – Maxwells equations: Gauss law, no magnetic monopoles, Faraday’s law of induction, Ampere’s law. Lorentz was a rather mathy guy and he noticed that Maxwells equations were invariant under under so-called “Lorentz transformations” which have the factor:…. Contain a constant velocity! Recognize this?

Postulates of Relativity Postulate 1 – The laws of nature are the same in all inertial frames of reference Postulate 2 – The speed of light in a vacuum is the same in all inertial frames of reference. Let us analyze these postulates mathematically We have seen the postulates before – but let us now go mathy on them.

Galilean transformation from S to S’: x v x’ Galilean transformation from S to S’: x Guess that the relativistic version has a similar form but differs by some dimensionless factor G: (we know that this must reduce to the Galilean transformation as v/c ->0) The transformation from S’ to S must have the same form: It can be a rather mathematically intensive exercise to derive the Lorentz transformations in general. The main work there goes into proving that this is really the unique solution – but frankly I don’t think (you or even I) need to care about uniqueness as long as someone has taken care of this. Our goal is to fix the Galilean transformations – which worked for us at low velocities but got us into trouble with the speed of light. So let’s be lazy and try the minimal fix to Galileo – i.e. multiply by a factor G (to be determined) From first postulate of relativity-laws of physics must have the same form in S and S’ substitute into solve for t’, you get:

u=dx/dt=c u’=dx’/dt’=c Now we need an expression for the velocity dx’/dt’ in the moving frame: Take derivatives of: where u=dx/dt From second postulate of relativity- the speed of light must be the same for an observer in S and S’ u=dx/dt=c u’=dx’/dt’=c We use these pair of transformations to calculate the velocity – any guesses how we can use the postulates next to find G? Plug this into: and get: Solve for G:

The Lorentz transformations! The transformation: v x’ x To transform from S’ back to S: Behold: the simple generalization of Galilean transformation gives us a Lorentz transformation. Bonus is the velocity addition formula and we see that the speed of light is an invariant. Bonus velocity addition:

Coordinate systems Rotation transformations: Let’s look at this another way: thinking Newtonianly… x’ y y’ x Roll cat toy across the floor towards cat. The same cat, the same cat toy, different (arbitrary) choice of coordinate systems. This story should remind you some formulas from Galilean times – rotations! But usually Time isn’t shown. Newtonian relativity preserves space but it is assumed that time is absolute! You can use coordinates to quantify “geometry” i.e. turn geometry into algebra. In the Newtonian story distance was preserved – but now we know that this can’t be true. By Pythagorean’s theorem, the toy rolls: Easier visualization Rotation transformations: This is an example of rotating your coordinate axes in space. Distance is preserved.

Time, the fourth dimension? “Spacetime” In the previous slide, the two space dimensions were shown to be interchangeable. A similar relationship can be used to express the relationship between space and time in relativity. Light propagating in one dimension in a spacetime coordinate system as viewed from a frame S. The distance traveled is equal to the speed of light times the time elapsed. ct x=ct x’=ct’ Rotation equation interchanged coordinates of space i.e. x <-> y but left time untouched. Lorentz transformations transform time into space and vice-versa. So relativity forces us to write coordinates with both space and time. But what does this look like? What are invariants? We need some bearings! Well at least the line x=ct is an invariant – what? x

Space-time Diagrams Lets get a sense of what space-time looks like…..Let us think of the space-time diagram of a train….

Lorentz approach to Einstein Simultaneity: events simultaneous in one frame with distance D Time dilation: t0 = proper time = time in clock rest frame We can rederive all of what we learned from Einstein using Lorentz – and as a bonus we don’t need to think as hard!!! Length contraction: L= length of ruler in moving frame (measured at the same time)

Light cones The red and blue spaceships are at x=0, t=0 when one emits a pulse of light. At that instant one of the spaceships starts to move away from the other with velocity v. The lightcone- the distance light has traveled since x=0, t=0 as a function of time. Coming back to light: We saw that the light line was an invariant. Let us put this in a higher dimension – i.e. add both x and y direction – this gives you a cone. Other way to think is you emit a pulse at x=0,t=0 Who perceives him/herself to be at the center of the lightcone? A passenger on the red spaceship or a passenger on the blue spaceship?

They are both at the center of the light cone! The red spaceship’s reality. The blue spaceship’s reality. They are both at the center of the light cone! This can be achieved by rotating their coordinate axes as when playing with the cat, except one thing…notice that one set of axes is not orthogonal! You need to add a Lorentz boost. The Lorentz boost accounts for the fact that it is the speed of light that is constant.

ct ct’ x=ct x’ x The axes aren’t perpendicular but are scaled by some factor. They must be symmetric w/r to the light pulse. What is this factor?

Geometrically an event E is a “point” in space time Geometrically an event E is a “point” in space time. The trajectory of a particle is a curve in space-time called the “World line”.

Invariant “distances” in Spacetime! To find the space and time coordinates of an event in a specific frame, draw lines from the event parallel to the axis of that frame. x=ct ct’ E2 ct True, observers will differ on the length of objects or the time events occurred, however, there is a space-time “distance” that moving observers agree on. This distance has to be invariant under Lorentz transformations and we can guess based on analogy with the Galilean case E1 x’ How to determine coordinates in space-time from geometry? What about is there an analog of distance that is left invariant between observers? But what is a? x !

Causality x=-ct ct x=ct A B C x here,now O Could an event at O cause A? Yes, because a “messenger” at O would not have to travel at a speed greater than the speed of light to get there. where light that is here now may go in the future x=-ct ct x=ct A B Could an event at O cause B? C A light signal sent from O could reach B. x here,now O Could an event at O cause C? The space-time distance can be positive, negative or 0 – does this mean anything? Yes!! It has implications for Causality – let us look at the space-time diagram…In fact using Lorentz equations for space-like separation – we will see that we can change the sign of t’…..So Causality can’t mean anything. No, the spacetime distance between O and C is greater than could be covered by light. It would require time travel. where light that is here now may have been in the past

Another Einstein from Lorentz (Doppler) Light wave takes the form (in any frame): In a moving frame it looks like: Let us see how to get the Doppler effect from Lorentz by just doing algebra – Voila!! We get the same form as from thinking….

One more thought about the kinematics of Star Trek: (elaboration of homework) The enterprise is speeding along at some “realistic” velocity (less than the speed of light). Let’s pretend that these stars emit light of a single wavelength. Will a passenger on the enterprise perceive the star to be the same color as a passenger on a Klingon ship that is hovering near the star? (Hint, think about Doppler effect) How do we know that the universe is expanding? Redshifts?