Lecture 6:

We reviewed Lorentz transformations and the invariance of the proper distance (also called spacetime separation), as well as the proper time. We then considered how to write physical laws that are manifestly true in all Lorentz frames, even though they may relate physical quantities that look different in each frame.

We compared the mixing of space and time components under Lorentz transformations to the more familiar mixing of spatial components under rotation of coordinate axes, writing simple examples of both in matrix notation. We mentioned how both leave invariant the norm of a vector. We noted that mixing space and time components will cause mixing of other physical quantities which depend on implicit spatial measurements (on volumes, distances or times) -- such as densities, momenta, and currents. We claimed that such observables can be grouped into 4-vectors whose coordinates transform according to the same matrix $\Lambda$ as does the position 4-vector $x^\mu$, the column vector $(ct, x, y, z)$. We mentioned two important examples -- the momentum 4-vector $p^\mu$ and the wave vector 4-vector $k^\mu$.

We returned to the question: how can we write sensible physical laws, given that observers in different Lorentz frames view the world differently?

We considered first a simple possibility: we could restrict ourselves to physical laws that concern only Lorentz-invariant quantities, which appear the same in all Lorentz frames. For example, we could restrict ourselves to statements about an object's trajectory which depend only on the spacetime separation $(\Delta s)^2$ between two subsequent positions $x^\mu$. We discussed the physical and causal content of such statements, of which there can be three:

Null separation, $(\Delta s)^2 = 0$

We showed (for $\Delta t \rightarrow 0$) that null separation implies the object travels at speed $c$. Because $(\Delta s)^2 = 0$ for all observers, an object which appears to have speed $c$ to one observer has speed $c$ for all other Lorentz-boosted observers. Since light travels at speed $c$, this result is often stated ``light travels at a fixed speed $c$, as measured by all observers.'' We have shown that it may be stated more geometrically (in a form more generalizable to general relativity) as ``light travels on null geodesics.''

Timelike separation, $(\Delta s)^2 > 0$

We showed (for $\Delta t \rightarrow 0$) that timelike separation implies the object travels at speed less than $c$, in all Lorentz frames (although the specific speed $\dot{\bf {x}}$ is frame-dependent). Since we observe massive objects with speed less than $c$ (particularly in their own rest frame), this gives the common result ``massive objects travel with speed less than $c$, as measured by all observers.'' This is equivalent to the geometric statement ``massive objects follow time-like geodesics.''

Spacelike separation, $(\Delta s)^2 < 0$

We showed (for $\Delta t \rightarrow 0$) that spacelike separation implies the object travels at speed greater than $c$, in all Lorentz frames (again the specific speed $\dot{\bf {x}}$ is frame-dependent). Since neither massive nor massless objects fall into this category (having an observed speed greater than $c$), we have the geometric result that ``spacelike-separated events cannot lie on the worldline (trajectory) of any particle.'' This means that no signal can propagate between the two events, so they can have no causal influence on each other.

We summarized these results by discussing spacetime diagrams, and an observer's future and past light cones. We noted that this basic causal structure for spacetime has consequences which will restrict cosmology, as the universe has a complicated history of superluminal and subluminal expansion. This requires us to keep careful track of regions whose lack of causal contact prevents them from equilibrating.

We then continued our more ambitious program: while discussing Lorentz-invariant objects has given us some notion of the spacetime structure which physics must obey, it hasn't given us anything as prescriptive as, say, Newton's second law. To find such physical laws of motion, we have to learn how to write Lorentz-covariant equations: those relating Lorentz frame-dependent measurements, but in a way that remains true in each frame, since each side of the equation transforms in the same way.

This is just like writing vector equations in ordinary space, and we noted that it can be easy (generalizing deBroglie's relation). But it can be very complicated; we noted why Newton's law of gravitation relates non-invariant objects whose transformation properties are hard to see.

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