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Special Relativity

The ‘principle of relativity’ states that the laws of physics should be the same for all observers. This makes a lot of sense since if they weren’t the same then we should argue that our laws are incorrect and not general enough.

 

Albert Einstein produced two theories of relativity:

 

·         The Theory of Special Relativity (1905). This looks at observers travelling at a constant speed in a straight line[1] (including being at rest) in the absence of a gravitational field. The technical term for these observers is ‘inertial frames of reference’.

 

·         The Theory of General Relativity (1916). This looks at observers who are in non-inertial frames of reference. This includes ones accelerating or decelerating, rotating ones, and ones in a gravitational field (see Inertia). His ‘equivalence principle’ asserted that these were basically the same and that the gravitational force is actually just a pseudo-force experienced by an observer in a non-inertial accelerated frame of reference.

 

It is often said in non-scientific literature, and in fiction, that according to Special Relativity, as you get closer to the speed of light then you start to shrink, that time goes more s..l..o..w..l..y, and that you get heavier. In fact none of these are true statements.

 

Before we explain this in more detail, let’s just go over the history of where the Theory of Special Relativity came from.

 

In 1864, James Clerk Maxwell devised a set of equations demonstrating that electricity, magnetism and light were all manifestations of the same phenomenon, i.e. the electromagnetic field. Maxwell's equations, as they’re now known, had a problem though: They referenced the speed of light but there was no indication of what it was relative to. Do you see the problem? If all motion is relative then how can these equations be valid for all inertial observers if they involve the speed of light?

 

In 1887, Albert Michelson and Edward Morley performed an experiment designed to measure the speed of the Earth through the “luminiferous aether". This was a hypothetical medium through which light waves supposedly travelled, analogous to sound waves through air. The result was technically a failure because of the unexpected observation that the speed of light was always found to be the same, irrespective of the speed of the observer.

 

It is debated how much Einstein was affected by the Michelson-Morley result but he was certainly influenced by the problems of Maxwell’s Equations. He considered that the speed of light must be a universal constant and so must be the same for all inertial observers, irrespective of whether it’s travelling towards them or away from them.

 

In summary, his basic two precepts were that the laws of physics must be the same for all inertial observers (‘principle of special relativity’) and the constancy of the speed of light.

 

This section has to use a little mathematics now to derive all the predicted effects. You can skip to the last section if you simply want the “executive summary”.

 

In pre-relativity physics, translating the coordinates from one inertial frame of reference (S) to another one having parallel axes (S') involved an equation of the form:

 

x'i = xi – (ai + vi t)

t' = t – b

 

This is called the Galilean Transformation. The quantities x1 to x3 are the normal three spatial coordinates. The terms ai and b represent the difference in origins between the two frames and can be ignored if they have the same origin. The terms vi are the relative velocities[2] of the two frames in the three spatial directions.

 

A more familiar form, using traditional x, y, z, and t, that represents a relative velocity along the x axis only is:

 

x' = x – v t                   . . . . . (1)

y' = y

z' = z

t' = t                             . . . . . (2)

 

The Galilean Transformation makes two important assumptions: (i) that time is absolute and so the time of an event is the same in both frames, and (ii) that length is absolute and so the distance between two points is the same in both frames. In other words, using a bit of Pythagoras, that ∑(x'i - xi)2 is invariant.

 

The transformation works fine for Newtonian physics but it had a big problem with Maxwell’s equations. It implied that the speed of light (c) depended on the speed of the observer relative to the source of the light. For instance, if S' sent a light signal to S then S would measure the speed as (c – v) because S' is travelling away from S. Maxwell’s Equations required that the speed be a fundamental constant of nature and so should be the same for all inertial observers.

 

If all inertial frames are to agree that light travels the same distance in the same time interval (t0 to t) then the following must always be true:

 

(x – x0)2 + (y – y0)2 + (z – z0)2 = c2(t – t0)2

 

We can replace this equation with the following much simpler one if we substitute a fourth coordinate, x4, for ict where i represents √(-1):

 

∑ (xi – xi0)2 = 0

 

The value of the subscript i now ranges from 1 to 4 so that x1=x, x2=y, x3=z, and x4=ict. What this has done is to treat time and space coordinates in the same consistent way to yield a single space-time interval that is the same for all inertial observers. The fact that it is zero is convenient since it avoids introducing an arbitrary constant. The square root of -1 is unfortunately referred to as an imaginary number since (-1)2 = (+1)2 = +1 implying that -1 has no square root. There’s nothing supernatural about it, though, and it occurs in many mathematical situations.

 

The four-coordinate form is known as Minkowski space-time after Hermann Minkowski. When applied back to Maxwell’s equations, they take on a very much simplified and beautiful form - which is always a pleasing result to physicists.

 

Now that we have a four-dimensional space-time continuum to play with, let’s consider a rotation of one inertial frame relative to another. This is known as the Special Lorentz Transformation.

 

We’re leaving out x2 and x3 here for simplicity. The standard equation for a rotation of Cartesian coordinates in the (x1, x4) plane yields:

 

x'1 = x1 cos α + x4 sin α

x'2 = x2

x'3 = x3

x'4 = -x1 sin α + x4 cos α

 

Reverting back to x and t representations, this becomes:

 

x' = x cos α + ict sin α                       . . . . . (3)

ict' = -x sin α + ict cos α                   . . . . . (4)

 

The rotation actually describes one inertial frame (S') moving at a constant velocity relative to the other (S). OK, so what’s the difference between this diagram and a traditional “travel graph” that we may have used at school? Well, it is true that the axis x'4 represents a body moving in the coordinate system of S. However, that’s not the same as being able to transform to-and-fro between the two coordinate systems as we’re doing here. The essential difference here is that we have a single set of space-time coordinates that can be interrelated – space and time are no longer independent of each other.

 

If v is the conventional velocity, x/t, then:

 

tan α = iv / c                                        . . . . . (5)

 

cos α = 1 / √(1 – v2/c2)

 

sin α = (iv/c) / √(1 – v2/c2)

 

Substituting these into equations (3) and (4) gives:

 

x' = (x – vt) / √(1 – v2/c2)                   . . . . . (6)

 

t' = (t – vx/c2) / √(1 – v2/c2)                . . . . . (7)

 

If v is much less than c – that is, we’re nowhere near the speed of light – then these equations are approximately the same as the Galilean ones: (1) and (2). However, the new transformation now means Maxwell’s Equations are then true for all inertial observers. Note that v must always be less than c or the equations result in infinities and imaginary numbers. Also, if we solved for equations transforming S' to S, rather than S to S', then they would have an identical form except that the sign of v would be reversed. In other words, the predicted effects of Special Relativity are completely reciprocated between S and S'. They are the result of a different perspective of the observer – not a physical change in the frame being observed.

 

Let’s look at a spatial interval (x' – x'0) in S'. How would S see this at a given instant in time of t? Well subtracting equation (6) from itself for the two ends of the interval gives:

 

(x – x0) = (x' – x'0) √(1 – v2/c2)

 

S would see the interval reduced by the factor √(1 – v2/c2) which is always ≤ 1. This is called the Fitzgerald Contraction although no physical contraction occurs. S' sees the length correctly, and also measures a similar contraction for a spatial interval in S. It’s simply a matter of perspective between the two inertial frames.

 

Now lets look at a time interval (t' – t'0) in S' measured at a given location x'. If we use the inverse form of equation (7) to map S' to S then we have:

 

(t – t0)  = (t' – t'0) / √(1 – v2/c2)

 

S sees the same time interval as being longer. This is called Time Dilation[3] although time has not slowed down. S' measures the time interval correctly, and also measures a similar dilation for a time interval in S.

 

Note that we had to use the inverse of the Lorentz Transformation equation (7) to calculate this. The reason is that although the two events were at the same location in S' (i.e. x'), S would not see them at the same location.

 

In general, if S had two simultaneous events at (t, x0) and (t, x) then S' would see them separated by the following time interval:

 

(t' – t'0) = (v / c2) (x – x0) / √(1 – v2/c2)

 

Note that t' ≠ t'0 unless x = x0 and so there’s no such thing as absolute simultaneity.

 

Let’s briefly go back to the suggestion earlier that the speed of light is always the same – no matter whether the source is moving towards you or away from you. In pre-relativity physics, the closing speed of two objects is simply the sum, w = (v + u), but that is different now. Equation (5) relates the velocity to the angle of rotation in Minkowski space-time. Let’s use it to calculate the closing velocity, w, of v and u by considering two successive rotations by angles of α and β.

 

iw / c   = tan (α + β) = (tan α + tan β) / (1 – tan α tan β)

            = (iv/c + iu/c) / (1 – (iv/c)(iu/c))

 

Hence:

 

w = (v + u) / (1 + vu/c2)

 

Note that even if v=c and u=c then w is still only c, not 2c.

 

OK, so let’s take stock of what we now know. Special Relativity was based on the ‘principle of special relativity’ and that the speed of light is a fundamental constant of nature. Out of the constancy of the speed light, we’ve found that space and time are actually part of a single space-time continuum. Also by looking at rotations in Minkowski space-time, we’ve identified all the predicted phenomena of Special Relativity as glorified ‘parallax error’. Parallax error, for anyone who’s forgotten, is a difference of perspective. If two observers are looking at a house from different angles then one will see more of the length and less of the width than the other. The house itself is physically unchanged. We take this for granted and our brains compensate for it. In Minkowski space-time, the different angles have been shown to represent the different velocities of two inertial observers. The predicted effects are simply due to observers looking at the same space-time interval from those different angles – they each measure differing amounts of space and time for the same interval.



[1] What a ‘straight line’ actually is was generalised in the Theory of General Relativity. The mathematical term is really a ‘geodesic’.

[2] In physics, the term speed is usually reserved for the rate of movement irrespective of the direction, and so is represented by a scalar. Velocity, on the other hand, has components in each of the three spatial directions and so is represented by a vector.

[3] There is another form of Time Dilation called Gravitational Time Dilation but that is not discussed here.