Egan, Greg - Foundations 1 - Special Relativity

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Foundations

by Greg Egan

1: Special Relativity

Anyone who reads science fiction will be familiar with some of the remarkable

predictions of twentieth-century physics. Time dilation, black holes, and the uncertainty

principle have all been part of the SF lexicon for decades. In this series of articles I'm

going to describe in detail how these phenomena arise, and along the way I hope to shed

some light on the theories that underpin them: special relativity, general relativity, and

quantum mechanics. The foundations of modern physics.

These articles are meant for the interested lay reader. If you can follow high

school algebra and geometry, and aren't afraid to take in a few new concepts — which is

the whole point, after all — nothing here should faze you.

Spacetime

The idea that we inhabit a four-dimensional spacetime is a very natural and intuitive one.

It's only because we take the duration of objects so much for granted that we tend to

gloss over it and refer to them as three-dimensional. Since most of the Earth's landscape

changes slowly, factoring out time from our mental models and paper maps is a very

pragmatic thing to do, but it's this unchanging space that we imagine for convenience

that's the abstract mental construct, not spacetime. Spacetime is simply what we live in,

all four dimensions of it.

Drawing a diagram of spacetime comes almost as naturally as making any other

kind of map; every historical timeline is halfway there, and placing a timeline for

Germany next to one for France, then sketching in the movement of armies between the

two, is as good a spacetime diagram as anything you'll find in particle physics. Of

course, a spacetime diagram in ink on paper has only two useful dimensions, so it

generally only shows time plus one dimension of space (though one more can be added,

using the standard techniques for drawing three-dimensional objects). Fortunately, many

problems in special relativity involve only one dimension of space; for example, a

spacecraft flying from here to Sirius would almost certainly travel along a straight line.

Figure 1 is a spacetime diagram for such a flight. Distances are in light years and

times are in years. For the sake of simplicity, the slight “proper motion” of Sirius relative

to the Sun, and any orbital manoeuvres and planetary take-offs and landings by the

spacecraft, are ignored. The spacecraft accelerates at the start of the journey, shuts off its

engines and cruises for the middle stage, then decelerates at the end, bringing it to a halt

just as it arrives. (There's no special reason for all three stages to cover equal distances;

this is just one possible flight plan of many.) Given that the distance to Sirius is almost

nine light years, it's reasonable to treat stars and spacecraft alike as mere specks, tracing

out one-dimensional world lines, rather than worrying about the fact that they're really

solid objects whose histories in spacetime are four-dimensional “world hypercylinders”.

When you draw a map, you have to choose a compass direction to point “up” on

the page. North is often convenient, but it's a completely arbitrary choice, and on a

house plan, say, it might be more useful to align the map so that the street frontage is

horizontal. Similarly, to draw a spacetime diagram you have to choose a reference

frame: you have to pick some object, such as the Sun, and treat it as fixed. The chosen

object's world line will then be vertical — it will be “moving” only in time, not space —

as will the world line of any other object at a constant distance from it. So the world lines

for the Sun and Sirius are vertical here because the diagram was drawn that way; it's a

matter of convenience, not a statement that the Sun is “truly motionless”, any more than

north is “truly up”.

However, some reference frames are different from others. Orienting a map so

that a given straight road runs vertically is one thing; arranging for a meandering river to

appear as a straight line is a much harder task. If we chose the spacecraft to be the fixed

point, everything we did would be complicated by the need to straighten out the curved

Egan: "Foundations 1"/p.2

sections of its world line when it accelerates and decelerates. To avoid this kind of

complication, special relativity deals only with inertial reference frames, which take

as their fixed point an object that is not accelerating. Unlike the idea of being motionless

(motionless compared to what?) this condition is easily defined in the middle of

interstellar space: if you're not firing your engines, and everything in the ship is

weightless, then you're not accelerating.

Of course, we can imagine a hypothetical second spacecraft which never

accelerates, but conveniently happens to match the first spacecraft's velocity for some

part of the journey — such as the entire middle stage, when the engines are shut off, or

even just for an instant during the acceleration or deceleration stages. That way, we can

analyse the first spacecraft's viewpoint at any given moment, without adopting a

reference frame in which it appears motionless from start to finish.

In a reference frame fixed to the Sun, the world line for the spacecraft starts out

being vertical, tips over as it accelerates, has a constant slope in the cruising stage, then

comes back towards vertical again as it decelerates. The world line for a pulse of light

that leaves the solar system at the same time as the spacecraft is shown for comparison; it

has a constant angle (45° in this diagram), because it travels at a constant velocity all the

way.

To be in motion relative to the Sun means tracing out a world line at an angle to

the Sun's world line. That might sound like nothing but a novel way to describe the

situation, but it's the key to all the relativistic effects of space travel. Two people facing

different directions in ordinary space see the same objects differently. Two people

driving between the same two towns will travel different distances, if one takes the most

direct route while the other takes a detour. In spacetime, the effects are analogous, but

not quite identical, because the geometry of spacetime is not quite the same as the

geometry of space.

Rotations in Space

Despite the differences, analysing the effects of rotating your angle of view in ordinary

space makes a useful rehearsal for tackling the problem in spacetime. It's easier to deal

with ordinary space, where we can rely on everyday geometrical intuition, and then the

results can be carried over to spacetime with only a few small changes.

Egan: "Foundations 1"/p.3

First, a quick review of the geometry we'll need. Pythagoras's theorem says

that the square of the hypotenuse (OP in Figure 2) of any right-angle triangle equals the

sum of the squares of the other two sides (OQ and PQ).

OP2=OQ2 + PQ2

The sine of the angle marked A is equal to the ratio between the side opposite it

(PQ) and the hypotenuse (OP).

sin A =PQ / OP

PQ =OP sin A

The cosine of A is equal to the ratio between the side adjacent to it (OQ) and the

hypotenuse (OP).

cos A =OQ / OP

OQ =OP cos A

The tangent of A is equal to the ratio between the side opposite it (PQ), and the

side adjacent to it (OQ).

tan A =PQ / OQ

= (OP sin A) / (OP cos A)

= sin A / cos A

Egan: "Foundations 1"/p.4

There's a simple relationship between the sine and cosine of an angle, which

comes straight from the definitions and Pythagoras's theorem:

(cos A)2 + (sin A)2=(OQ / OP)2 + (PQ / OP)2

= (OQ2 + PQ2) / OP2

= OP2 / OP2

= 1

The notation “(x,y)” beside the point P is a reminder that points can be referred to

by their x- and y-coordinates, written as an ordered pair. The arrow drawn from O to P

is a reminder that every point can be thought of as defining a vector from the origin to

the point. The advantage of dealing with vectors, rather than just points in space, is that

the same geometry can then be applied to other vectors, like velocity and acceleration.

To make it easier to carry things over from Euclidean geometry to spacetime

geometry, it will help to restate some of these familiar ideas in slightly different language.

In both Euclidean and spacetime geometry, there's a formula for taking two vectors and

calculating a number from them which depends on the length of the vectors and the angle

between them. This formula is known as the metric for the geometry. (You might also

have come across it as the “dot product” of two vectors.) It's usually written as g:

g[(x,y),(u,w)] =xu + yw (1)

Eqn (1) defines the Euclidean metric. Eqns (2a)-(2c) demonstrate some of its

properties: it's symmetric (swapping the two vectors leaves the value unchanged), and

it's linear (its value is simply multiplied and added as shown, if you apply it to a vector

that's been multiplied by a factor, or had another vector added to it).

g[(u,w),(x,y)] =ux + wy

= xu + yw

= g[(x,y),(u,w)] (2a)

g[a(x,y),(u,w)] =g[(ax,ay),(u,w)]

= axu + ayw

= a (xu + yw)

= a g[(x,y),(u,w)] (2b)

g[(x,y)+(p,q),(u,w)] =g[(x+p,y+q),(u,w)]

= (x+p)u + (y+q)w

= (xu + yw) + (pu + qw)

= g[(x,y),(u,w)] + g[(p,q),(u,w)] (2c)

Egan: "Foundations 1"/p.5

Eqn (3) is just a restatement of Pythagoras's theorem; the notation |(x,y)| means

the length of the vector (x,y) — also referred to as its magnitude — or if you prefer to

think in terms of the coordinates of a point, |(x,y)| is the distance from the origin (0,0) to

the point (x,y).

|(x,y)|2=g[(x,y),(x,y)]

= x2 + y2(3)

Eqn (4) states that the cosine of the angle between two vectors (x,y) and (u,w)

is equal to the metric function applied to the two vectors, divided by both their lengths.

Eqn (4) will take a bit of work to prove, but in doing so we'll solve the whole problem of

rotations in space.

cos B = g[(x,y),(u,w)] / (|(x,y)||(u,w)|) (4)

where B is the angle between (x,y) and (u,w).

If you want to know the x-coordinate of a point like P in Figure 3, you draw a

line through P at right angles to the x-axis, and see where it hits the axis. In the process,

the vector OP is shown to be the sum of two vectors: OQ, which is parallel to the x-

axis, and QP, which is perpendicular to it.

The same thing can be done with any other vector in place of the x-axis. If a line

from P to OG meets OG at a right angle, at point S, then OS is called the projection of

Egan: "Foundations 1"/p.6

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FoundationsbyGregEgan1:SpecialRelativityCopyright©GregEgan,1998.Allrightsreserved.Anyonewhoreadssciencefictionwillbefamiliarwithsomeoftheremarkablepredictionsoftwentieth-centuryphysics.Timedilation,blackholes,andtheuncertaintyprinciplehaveallbeenpartoftheSFlexiconfordecades.InthisseriesofarticlesI'm...

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