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Tag Archives: Relativity

Changing coordinates for the wave equation

The following is based on section 3.3.2 of Electricity and Magnetism for Mathematicians by Thomas A. Garrity (Cambridge UP, 2015). See also blog post Relative Motion and Waves by Conrad Schiff.

The classical wave equation is consistent with the Galilean transformation. Reflected electromagnetic waves are also consistent with classical physics using the dual Galilean transformation, in which linear location is the independent variable. The dual Galilean transformation for motion in one dimension is: x′ = x; t′ = twx = tx/v, where w = 1/v is the relative pace of the moving observer, which is equivalent to their inverse speed.

Suppose we again have two observers, A and B. Let observer B be moving at a constant pace w with respect to A, with A’s and B’s coordinate systems exactly matching up at location x = 0. Think of observer A as at rest, with coordinates x′ for location and t′ for time, and of observer B as moving to the right at pace w, with location coordinate x and time coordinate t. If the two coordinate systems line up at location x = x′ = 0, then the dual Galilean transformations are

x′ = x and t′ = t + wx,

or equivalently,

x = x′ and t = t′wx.

Suppose in the reference frame for B we have a wave y(x, t) satisfying the wave equation

\frac{\partial^2 y}{\partial x^2}-k^{2}\frac{\partial^2 y}{\partial t^2}=0.

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Michelson-Morley experiment

This post relates to a previous post here.

The Michelson-Morley experiment is a famous “null” result that has been understood as leading to the Lorentz transformation. However, an elementary error has persisted so that the null result is fully consistent with classical physics. Let us look at it in detail:

The Michelson-Morley paper of 1887 [Amer. Jour. Sci.-Third Series, Vol. XXXIV, No. 203.–Nov., 1887] explains it using the above figures:

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Temporo-spatial light clock

This post builds on the post about the Michelson-Morley experiment here.

One “Derivation of time dilation” (e.g., here) uses a light clock, pictured below:

The illustration on the left shows a light clock at rest, with a light beam reflecting back and forth between two mirrors. The distance of travel is set at the beginning with the separation of the mirrors. It is like a race: one lap is a round-trip between the two mirrors. So the independent variable is distance, and the dependent variable is time. This is key to properly understanding the experiment.

In the stationary frame the round trip longitudinal distance between the two mirrors is 2L, the speed is c, and the time of one cycle is T = 2L/c.

The illustration on the right shows an observer moving with lenticity w transversely to the light clock. In this case there are two components of motion: the longitudinal axis and the transverse axis. These components are independent of one another since they are in different dimensions. The transverse axis is the same as the stationary case above: the total distance is 2L and the total time is 2L/c. The mean speed is 2cT/(2T) = c.

The longitudinal axis is simply the Galilean transform: t′ = t + wx since motion in different dimensions is independent. If the light clocks coincide at t = 0, this is t′ = wx, which is what was given.

Thus the distance, time, and mean speed are the same for both observers and independent of their relative velocity, which is what the Michelson-Morley experiment found.

Michelson-Morley re-examined

There are many expositions of the famous Michelson-Morley experiment (for example here) but they all assume the independent variable is time, which is not the case. As we shall see, distance is the independent variable, and so the experiment is temporo-spatial (1+3). Let us examine the original experiment as it should have been done:

The configuration diagrammed above is as follows: the apparatus is presumed to travel with pace w relative to the aether. In it a light source travels with pace k = 1/c to a beam splitter, whereupon part of it travels a distance L longitudinally and is reflected back, whereas another part travels a distance transversely and is reflected back. Part of the light is sent to an observer, who looks for an interference pattern.

Since the distance L is fixed, distance is the independent variable. In the stationary frame the round trip longitudinal distance between the beam splitter and the mirror is 2L. Let the time that light travels longitudinally from the beam splitter to the mirror be T1, and let the time for the return be T2. Then

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Principle of relativity

The relativity of uniform motion was stated by Galileo in the 17th century, though it was known to Buridan in the 14th century. Galileo’s statement of the principle of relativity is in terms of ships in uniform motion:

… so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still [Galileo Galilei, Dialogues Concerning the Two Chief World Systems (February 1632), Stillman Drake tr. (University of California Press, Berkeley, 1962, pp 186-8.]

This has been applied to constant speeds or zero accelerations, but it could just as well be applied to constant paces or zero retardations. In any case, if the speed is constant, so is its inverse, the pace. Let’s see how this operates.

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Reflected motion

This post was inspired by Chandru Iyer’s post here.

Consider a light ray sent a certain distance s that is immediately reflected back. According to Newtonian mechanics if a light ray travels at speed c, then for a body moving at speed v relative to the stationary frame, the light ray should travel at the speed c − v one way and at speed c + v the other way.

The total distance of the light ray is 2s. The total time of the light ray is

\frac{s}{c-v}+\frac{s}{c+v} = \frac{s(c+v)+s(c-v)}{c^2-v^2}=\frac{2sc}{c^2-v^2}=2s \gamma^2 /c

Then the mean speed is

\frac{2s}{\frac{s}{c-v}+\frac{s}{c+v}}=\frac{2s}{\frac{2sc}{c^2-v^2}}=\frac{c^2-v^2}{c}=c(1-(v^2/c^2))= c/\gamma^2

However, according to Einstein’s relativity theory, the mean speed of light is a constant, c. So the above speed needs to be multiplied by the gamma factor squared, γ². As Iyer notes, this is accomplished by contracting the moving rulers by the factor (1/γ) and dilating the moving clocks by the factor γ.

But that is not the correct approach.

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Space and time reciprocity

This post is related to others, such as here. Consider an analogue clock:

Analogue clock

The movement of the hand clockwise relative to the dial is equivalent to the movement of the dial couter-clockwise relative to the hand. That is, the motion of the hand relative to the dial corresponds to the opposite motion of the dial relative to the hand. This is the reciprocity principle of relative motion.

This means that motion in the space frame and the time frame are opposites. The displacement direction and the dischronment direction move in opposite directions. The velocity of A relative to B in space corresponds to the velocity of B relative to A in time, and vice versa.

The interchange of space and time needs to take this into account. Their vectors are in opposite directions.

Inverse units, inverse arithmetic

The use of space (stance) as an independent variable and time as a dependent variable leads to inverse ratios. There is pace instead of speed, that is, change in time per unit of distance instead of change in length per unit of time. But a faster pace is a smaller number, which is counterintuitive and contrary to speed, for which faster speeds are larger numbers. There are two approaches to dealing with this:

  1. Use two kinds of arithmetic: the usual one, in which zero signifies the smallest quantity, and an arithmetic in which zero signifies the largest quantity (or infinity). Then they are isomorphic, with their extremes corresponding inversely. For example, speed and pace both measure how fast a body is going, but they use different arithmetics. A large number for speed corresponds to a small number for pace, and vice versa.

  2. Use the same arithmetic for both but invert one when making a comparison. For example, speed and pace are effectively inverses (apart from which is the independent and which the dependent variable). Given a speed and a pace for some body, to compare them requires inverting either the speed or the pace. An arithmetic means of one corresponds to the harmonic mean of the other.

The first approach inverts the arithmetic, whereas the second approach inverts the units. The second approach is preferable because we are so accustomed to ordinary arithmetic that introducing an alternative would be unnecessarily difficult. It’s much easier to change units than to change arithmetic.

The second approach is found in the comparison of time mean speed and space mean speed: the time mean speed is the arithmetic mean of speeds with a common time unit, and the space mean speed is the harmonic mean of speeds with a common length unit. The space mean speed is essentially the mean pace inverted, which is a subcontrary speed.

Relativity uses a quotient, β, which is the speed of a body divided by the speed of light. Since the speed of light is the highest speed, β is always between zero and one, inclusive, in which zero signifies rest and one signifies the speed of light. What corresponds to β for pace? At first it seems to be the pace of a body divided by the pace of light. But such a quotient is the inverse of β, which would require an inverse arithmetic. The second approach is to take the inverse of this, which equals β. This is consistent with the inverse correspondence between space and time.

Speed of light

Speed is defined as “The time rate of change of position of a body without regard to direction; in other words, the magnitude of the velocity vector.” (Dictionary of Physics, 3rd edition, McGraw-Hill, 2002.

This is ambiguous, however. Consider a light beam reflected off a surface:

light clock at rest(1) Since the light returns to its starting point, the total travel distance is zero, so the overall velocity is zero and the speed is zero.

(2) However, the interest is in each leg of the journey. In that case, in the first leg light travels +L in time t, and in the second leg light travels –L in time t. The mean velocity in the first leg is v1 = +L/t, and the mean velocity in the second leg is v2 = –L/t. The mean velocity for both legs is the harmonic mean of these two velocities because what is fixed and independent is the length, not the duration.

1/((1/v1) + (1/v2)) = 1/((1/L) – (1/L)) = 1/0 = ∞.

Thus the mean velocity is infinite, and the mean speed of light is infinite.

(3) Another approach looks at length of each leg apart from direction. In that case, in the first leg light travels L in time t, and in the second leg light travels L in time t. The speed in each leg is L/t, so the mean speed of light is L/t. This is the best known approach to the speed of light.

It’s interesting that (2) leads to the Galilean transformation, and (3) leads to the Lorentz transformation.

Lorentz transformation derivations

What follows are four derivations of the Lorentz transformation from the complete Galilei (Galilean) transformations in space with time (3+1) and time with space (1+3). Their intersection is linear space and time (1+1), which is the focus of the derivations. The other dimensions may be reached by rotations in space or time.

I. Space with Time (3+1)

Consider two inertial frames of reference O and O′, assuming O to be at rest while O′ is moving with velocity v with respect to O in the positive x-direction. The origins of O and O′ initially coincide with each other. A light signal is emitted from the common origin and travels as a spherical wave front. Consider a point P on a spherical wavefront at a distance x and x′ from the origins of O and O′ respectively. According to the second postulate of the special theory of relativity the speed of light, c, is the same in both frames, so for the point P:

x = ct, and x′ = ct′.

A. Time velocity

Define velocity v as the time velocity vt = ds/dt. Consider the standard Galilean transformation of ct = x with a factor γ, which is to be determined and may depend on β, where β = v/c:

x′ = γ(x − vt) = γ(x − βct) = γx(1 − β).

The inverse transformation is the same except that the sign of β is reversed:

x = γ(x′ + vt′) = γ(x + βct) = γx′(1 + β).

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