Centre of the Lightcone

This page describes Einstein's solution to The Paradox of Special Relativity set forth on the previous page.

Einstein's solution to the paradox is that Cerulean's spacetime is skewed compared to Vermilion's.

The thing to notice in the diagram is that Cerulean is in the centre of the lightcone, according to the way Cerulean perceives space and time.

Vermilion remains at the centre of the lightcone according to the way Vermilion perceives space and time.

In the diagram Vermilion and her space are drawn at one “tick” of her clock past the point of emission, and likewise Cerulean and his space are drawn at one “tick” of his identical clock past the point of emission.

How do Vermilion and Cerulean know their clocks are identical? Because both clocks were calibrated at NIST.

Of course, from Cerulean's point of view his spacetime is quite normal, and it's Vermilion's spacetime that is skewed.

Here's a movie which shows Cerulean's spacetime transforming into Vermilion's spacetime, and back again (50K GIF); or same movie, double-size on screen (same 50K GIF).

In special relativity, the transformation between the spacetime frames of two inertial observers is called a Lorentz transformation.

In general, a Lorentz transformation consists of a spatial rotation about some spatial axis, combined with a Lorentz boost by some velocity in some direction.

Only space along the direction of motion gets skewed with time. Distances perpendicular to the direction of motion remain unchanged.

Why must this be so? Consider two hoops which have the same size when at rest relative to each other. Now set the hoops moving towards each other. Which hoop passes inside the other? Neither! For suppose Vermilion thinks Cerulean's hoop passed inside hers; by symmetry, Cerulean must think Vermilion's hoop passed inside his; but both cannot be true; the only possibility is that the hoops remain the same size in directions perpendicular to the direction of motion.

Cottoned on? Then you've understood the crux of special relativity, and you can now go away and figure out all the mathematics of Lorentz transformations. Just like Einstein. The mathematical problem is: what is the relation between the spacetime coordinates \(( t , x , y , z )\) and \(( t' , x' , y' , z' )\) of a spacetime interval, a 4-vector, in Vermilion's versus Cerulean's frames, if Cerulean is moving relative to Vermilion at velocity \(v\) in, say, the \(x\) direction? The solution follows from requiring

1. that both observers consider themselves to be at the centre of the lightcone, and
2. that distances perpendicular to the direction of motion remain unchanged,

[An alternative version of the second condition is that a Lorentz transformation at velocity \(v\) followed by a Lorentz transformation at velocity −\(v\) should yield the unit transformation. This is illustrated in the Lorentz transformation movie above (go back up to Cerulean's point of view), where the transformation by negative velocity −\(v\) is accomplished by rotating space by 180° and then transforming by velocity \(v\).]

Note that the postulate of the existence of globally inertial frames implies that Lorentz transformations are linear, that straight lines (4-vectors) in one inertial spacetime frame transform into straight lines in other inertial frames.

A solution to this problem is given in the next section but two, Construction of the Lorentz Transformation. As a prelude, the next two sections discuss Simultaneity in Special Relativity and Time Dilation.