Special Relativity, part two:
The Minkowski Interval and the Lorentz Transformations.

Links to sections:

• The Minkowski Interval ( 1908 ): described.
• The Interval: diagram explained.
• The Interval: deriving the Lorentz transformations.
• The Interval: Phase-different frames of reference.

The Minkowski Interval ( 1908 ): described.

'The Interval' of Minkowski is an extension of Euclid's geometry for making measurements in three dimensions of space. An example, of Euclid, is measuring the height, length and breadth of a box-room from one of its corners. Minkowski treated time as a fourth dimension of space and integrated it into a four-dimensional Euclidean space-time geometry.

The advantage of this new mathematical formalism was that all observers' different space and time measurements could be re-stated in a formula, known as 'the Interval', a constant spatio-temporal measurement, which is the same for all observers. This reflects the fact that, tho the observers were using their own co-ordinate systems, they were all seeing the same event.

( The qualification needs to be made that the Interval only applies to observers moving in 'uniform relative motion'. To remove this arbitrary restriction, Einstein had to come up with his 'principle of equivalence' of acceleration to gravity, which was the basis of a general theory of relativity. )

The only mathematical difference between the three spatial dimensions and the one temporal dimension is that time units are multiplied by the mysterious square root of minus one ( given the symbol i ).

We may see solid shapes changing in time. We may imagine our three dimensional world in a historical perspective. However, the three spatial dimensions are essentially the same in character and we need only one of them to show how the time dimension relates to spatial measurement. One space co-ordinate, x, and one time co-ordinate, t, can be represented, diagrammatically on a two dimensional page.

So, we are relating the spatio-temporal measurements of two observers, working in two different places and times, and therefore with two different co-ordinate systems: the x and t system is distinguished from an x' and t' system, by the latter's indices. Of course, both systems have two other dimensions of space to consider. But for simplicity's sake, we've assumed their measurements dont differ in that respect. In other words, their space measures on the y and z dimensions are equal, or y = y' and z = z', comparing the two observers' co-ordinate systems.

The Interval derives the Lorentz transformation for time, and for space, in much the same sort of way, partly with the use of trigonometry. The Lorentz transformation can be expressed as an equation of one observer's time, t', or the other observer's time, t. Likewise, for their respective distance measurements, x and x'.

The Interval points to a space-time co-ordinate position, that is both x + ict and x' + ict' These four values are essentially the Lorentz transformations, we've just been speaking of. Here, they form two different sets of co-ordinate axes, that are the two observers' frames of reference.

Using trigonometry, the Interval allows observers' varying space and time measures to be drawn as the varying opposite and adjacent sides, of a right-angled triangle, with respect to one of the other two angles. For instance, a more acute angle makes the opposite side, of the triangle, relatively short. These opposite and adjacent sides will vary according to the rotation of the hypoteneuse, as the constant radius of a circle.

Any position on the circle boundary, that the radius points to, is a given space-time position, perhaps marking some mutually observed physical event. Any number of observers can choose different ninety degree co-ordinate systems of space and time, on the circle, given that their axes all share the same origin at the centre of the circle.

With these different anglings of their rectangular co-ordinates, they can all measure any one given space-time position on the circumference. Their respective axes will all show different space-lengths and different time-lengths in relation to that common sighting. In effect, each co-ordinate system is like a triangle with varying opposite and ajacent sides for varying distances and times.

But all these different triangles have the same hypoteneuse. By Pythagoras' theorem, their varying lengths, on different space and time axes, add up to the same unitary space-time measurement, given by the length and direction of their shared hypoteneuse, as a constant radius pointing out any given space-time event on the circle circumference.

This space-time radius vector is 'the Interval'. ( A vector means a length or magnitude in a pointed direction. ) The Interval applies not only in one dimension of space and one dimension of time. For example, making measurements in two dimensions of space, as well as one dimension of time, the radius could vector-in on a given three-dimensional space-time position, represented as a point on a sphere, instead of a circle.

( Pythagoras' theorem, for finding the Interval as hypoteneuse, extends to apply to three or more dimensions. )

For the full four-dimensional space-time treatment, the Interval would, presumably, be a radius 4-vector pointing out a position on a four-dimensional hyper-sphere, whatever that is -- something beyond the imagination of our three-dimensional experience, perhaps.

The Interval: diagram explained.

Having explained the Interval in words, simple equations will give a more precise idea, with the help of the following diagram.

Diagram of the Interval The Interval is given by the radius vector OP, which is the hypoteneuse of triangulations based on both co-ordinate systems. Thereby, the point, P, is located by both systems, the two sets of co-ordinate positions, x + ict and x' + ict', being equal or differently representing the same position.

The diagram shows, schematically, the essential difference between two observers measuring, in space and time units, the same event at position, P. Special relativity recognises local ruler or 'rod' lengths and local clock rates. These correspond to the different co-ordinate systems the observers are working with.

On the diagram, the two observers each have a pair of ninety-degree axes. One observer measures distance and time, in units x and t ( or rather, ict, where t is just multiplied by c, or constant speed of light, and by i, sometimes described, in mathematics, as an operator. Just as the plus sign, +, means the operation of adding, so the i sign, as it happens, can mean the operation: turn thru ninety degrees ).

The other observer measures distance and time, in his units, x' and t', on his x' and ict' co-ordinates. But the diagram shows the only real difference between this co-ordinate system, and the other, mathematically, is that it is a turn, thru angle, a, away from the x and ict axes system. ( Of the four angles, a, on the diagram, I refer to the angle a, at the origin, O, and marked by the up-wards curling arrow, to show angle, a, is the angle of tilt between the ( x, ict ) right-angled co-ordinate system and the indexed co-ordinate system. The triangle for this particular angle, a, is color-coded yellow in the diagram. )

Position, P, where the event is, the observers are both measuring, is equally well located by one set of axes as another: x' + ict' marks the spot as well as x + ict. There is no privileged or prior frame of reference. The frames are 'only relative' to each other. This would be true for any number of observers, with their own co-ordinate frames, at varying angles, about origin, O, from each other.

But they would all have in common what these two observers have. They would share the same hypoteneuse, OP. This is 'the Interval', a constant space-time measurement, that all observers ( in constant relative motion ) can agree is the same measurement of the event at P. Yet, the observers arrive at the Interval from their different measurements of space and time, treated by each observer as separate measurements on a separate space axis and a separate time axis.

It is in the context of the Interval, that special relativity is about an integrated concept of 'space-time'.

The value of the Interval can be found from any co-od. system. Taking the ( x,t ) system, and using triangle OPJ: As shown on the diagram, the length of OJ is x = ut. The length of JP is ict. By Pythagoras' theorem, the length of the hypoteneuse, OP, which is the Interval, is found from: ( OP )² = x² + ( ict )².

So, the Interval equals ( t²( u² - c² )) = t( u² - c² )

Taking the other observer's system, ( x',t' ), the Interval can be found by the same procedure, this time using triangle OPM. The diagram shows that, as this other right-angled triangle has the same hypoteneuse, OP, the Interval, as such, is obviously the same value.

In the indexed co-ods., the Interval equals t'( u'² - c² ).

Therefore, t( u² - c² ) = t'( u'² - c² ).

( Usually, text books leave the Interval in terms of the hypoteneuse squared, without taking the square root. )

The Interval: deriving the Lorentz transformations.

The Interval may derive the Lorentz transformations of one observer's distance, or time, measurements, into those of another.

The Lorentz transformation of distance, x, into distance x', is closely related to that for time ( which was given in the preceding web page ):

x' = ( x - vt )/( 1 - v²/c² )

This can be rendered:

x' = x.(c/( c² - v² )) + ict.( iv/( ( c² - v² )).

The diagram of the Interval also provides an equation for x':

x' = x.cos(a) + ict.sin(a).

The length, x', is the line OM, which the diagram marks as on the x' axis. OM can be split into lengths OL and LM. By trigonometry, OL is the length, x.cos(a) ( derived with angle, a, in triangle OLJ ), and LM is the length, ict.sin(a), ( derived with angle, a, in triangle PJK ).

( The former angle, a, is color-coded yellow, and the latter angle, a, is color-coded green, on the diagram. )

The Lorentz transformation can be derived from the diagram of the Interval, by equating the above two equations for x'. This implies that the factors of their x terms must be equal: namely, c/( c² - v² ) = cos(a). Likewise, the factors of their ict terms must be equal, or: iv/( c² - v² ) = sin(a).

To realise these equalities, certain simple conditions are assumed. An event, say a flash of light, happens where one of the observers actually is. So, he measures zero distance from the event. Or, x' = 0 ( assuming this is the observer with the indexed co-od. system ).

The other observer has been moving away from him, at a relative velocity, v, so that, by time, t, when the event happens, she is a distance, away, of x = vt. ( Normally, the velocities are u' = x'/t' and u = x/t. But in this instance, u' is zero and v, their relative velocity, can be set equal to u. Tho, v is generally not a simple difference between observers' velocities, when an observer's velocity approaches significantly close to light speed -- this being when relativistic physics revises the common sense assumptions of classical physics. )

Substituting x' = 0 and x = vt, in the Interval diagram's equation for x':

x' = 0 = vt.cos(a) + ict.sin(a).

Or, iv.cos(a) = c.sin(a).

Then, sin(a) = ( iv/c )cos(a).

But, by Pythagoras' theorem, ( sin(a))² + ( cos(a))² = 1.

Substituting:

(( iv/c )cos(a))² + ( cos(a))² = 1.

So, ( cos(a))²( (iv/c)² + 1 ) = 1.

Then, cos(a) = 1/( 1 - v²/c² ) = c/( c² - v² ).

And substituting this equivalent for cos(a), to get an equivalent for sin(a):

c²/( c² - v² ) + ( sin(a))² = 1.

Then, sin(a) = iv/( c² - v² )

These are the two equivalent values we were looking for, to make the Interval diagram equation for x' equal to the Lorentz transformation for x'.

The identities for cos(a) and sin(a) can be substituted in the trigonometric formula, sin(b+a) = sin(b).cos(a) + cos(b).sin(a), where angle, b, is shown in the Interval diagram. Then:

ict/t( u² - c² ) = ( ict'/ t' ( u'² - c² )).( c/( c² - v² ) + ( u't'/t'( u'² - c² )).( iv/( c² - v² )).

This equation transforms into:

( u'² - c² )/( u² - c² ) = ( 1 + u'.v/c² )/F = t/t', where F = ( c² - v² )/c, the contraction factor, in the Lorentz transformation of observers' times, given in the middle term's equation to the right side term, t/t'.

Whereas, t/t', equated to the left side term, gives the Interval:

t'( u'² - c² ) = t( u² - c² ).

The Lorentz transformation of time could have been deduced in a like way to the Lorentz transformation of distance. From the Interval diagram, we could deduce that

ict = x'.sin(a) + ict'.cos(a).

This result derives from the top-right angle, a ( color-coded orange in the diagram ) as the angle in the x'.sin(a) term.

The angle, a, color-coded blue, derives the ict'.cos(a) term. I havent marked the exact triangles for the calculations, that these angles belong to ( not wanting to clutter the diagram ).

The Lorentz transformation of time would also be split into two terms, that are factors of x' and t'. Then, the two factors of each equation, for t, could be respectively equated, by the sort of simplifying assumptions made, above, for the case of observers' spatial measurements.

At the end of the preceding web page, the calculation for the Michelson-Morley experiment was shown as a restricted form of the Interval. How would this look on the Interval diagram?

On our diagram, x = ut is the length OJ. But in our Interval version of the Michelson-Morley calculation, u = 0. So, x = 0, and the length OJ shrinks to nothing. Instead of measuring a rectangular co-ordinate position, to mark the event at point, P, this frame of reference consists only of one line, the time axis, ict, that points directly to P, and is, therefore, also that frame's Interval measurement.

Michelson-Morley's ( x', t' ) system ( which measured the putative greatest ether drag on light speed ) has both space and time components greater than zero. It resembles the tilted rectangular system in the diagram. But its hypoteneuse, by which that system measures the Interval, will coincide with the vertical temporal axis, in t ( of which the putative least-ether-drag system solely consists ).

The Interval: Phase-different frames of reference.

The Interval is a rotatable radius ( tho, the diagram doesnt include a circle, showing the origin, O, at its centre, and the Interval's particular position, given at P, as on its circumference ). No mathematician, I still couldnt help notice that such a radius vector draws out waves, in either time or space, or both. Such wave equations have independent variables of up to three dimensions in space and one independent variable for time, on the other side of the equation. Both sides act on a dependent variable of wave displacement. The Interval seems rather like this classical wave displacement, acted upon by four separate components of space and time, but summarising them mathematically as a unified space-time measurement.

The Interval is a constant for any given space-time event but becomes a variable once one considers it moving thru a circle or sphere of events ( or 4-D space-time hyper-sphere of events ).

The Interval was marked at P, on the diagram, by two observers' co-ordinate systems. But, in the process, it was also marked by two different kinds of co-ordinate system: rectangular or rectilinear co-ordinates and polar co-ordinates.

The Interval was described as a space-time vector, giving both length ( including length of time ) and direction. In our two-dimensional example, any point on a plane, or sheet of paper, could be indicated by this means.

The same can be done with complex numbers. These are an extension and completion of the number system, with rules of counting somewhat different from traditional numbers. Complex numbers are taught to young children as part of the 'new maths' in schools.

The two observers each described the position, P, by a complex number: x + ict or x' + ict' ( which may be set equal to z and z', respectively. ) This meant that, in this instance, the two complex numbers were equal. These complex numbers are in the system of rectilinear co-ordinates. This works like a net. Depending on how fine the net is, you can mark any position, on a stretch of ground it covers, by which node or knot in the net lies over it.

But the Interval diagram also used another kind of co-ordinate system, because the two rectangular co-ordinate systems were related to each other, by sharing the same origin, and having their axes turned at a given angle to each other. They also shared the same hypoteneuse, a radius vector, which was the Interval itself.

A co-ordinate system which measures, in this case a plane, by sweeping out an angle, in a circle, with a radius, is called a polar co-ordinate system. The origin is the 'pole'.

We can also express z and z' in polar co-ordinates. From the Interval diagram, let the radius, from the origin, O, pointing to position, P, be called r.

Then, cos(b+a) = x/r.

i.sin(b+a) = ict/r.

z = x + ict = r( cos(b+a) + i.sin(b+a))

Thus, z is expressed both in rectangular and polar co-ordinates.

Similarly for z':

cos(b) = x'/r.

i.sin(b) = ict'/r.

And z' = x' + ict' = r( cos(b) + i.sin(b)).

A well-known result says the trigonometrical terms can be replaced by an equivalent term, using the exponent ( usually written 'e' or 'exp' ).

( So, complex number, z', equals radius, r, times the exponent, e, to the power of an index, which is the operator, i, times angle, b.
For, z, the only difference is that the angle in the index is not angle, b, but angle, b+a. )

These various expressions are standard text book solutions for z or z' as a wave form, in a so-called wave equation. Z can be imagined as the point, P, in the Interval diagram, moving around the circle, whose circumference it is on. This circular motion can be charted, horizontally on a graph, as a wave motion. The horizontal axis can represent the lapse of time or distance.

The ( positive part of the ) graph's vertical axis represents the height of the wave. Its full height or crest is called the amplitude. It is the same length as the radius. When the radius reaches the northern vertical position, on the Interval diagram, this corresponds to a crest of the wave. From there, for every full circle, the radius sweeps out, the graph of the wave reaches another crest.

Our Interval diagram only shows the positive quadrant of the circle co-ordinates. But when the radius reaches the southern vertical position, where the ict-axis has changed to -ict on the circle, this corresponds to a troff of the wave being drawn out on the graph.

This is why a solution of the wave equation is typically given as a function of: x ± ct. ( Whether the wave equation has a complex solution, containing i, depends on certain conditions, which qualify the shape of the wave. These are taken into account in the solution of a wave equation as a form of differential equation. )

Both z and z' mark an event, P. Presumably, the two observers could both track P, if it were to describe a circle in space-time, on the Interval diagram. In doing so, the two observers would trace out their respective z and z' wave forms on the graph. But these wave forms would not be synchronised, because there is a difference of angle, a, between their co-ordinate systems.

This difference of angle is called the phase. A difference of phase, between physical waves, is responsible for interference effects, such as the patterns made by colliding water waves.

On the Interval diagram, an angle, a ( color-coded yellow ) shows the two observers' spatial axes, x and x', out of phase to that degree. Consequently, their temporal axes, ict and ict', are also out of phase by angle, a ( color-coded green ).

In general, this situation applies to the Interval as a four-dimensional space-time. So, you could characterise different observers, of relativistic effects, as having co-ordinate systems, with space-time phase differences from each other.

It is only a phase they are going thru.

References.

For the Michelson-Morley experiment, I relied on:

Sir James Jeans, The New Background Of Science. Scientific Book Club, 1945.

Isaac Asimov, Asimov's Guide To Science, Vol. 1, The Physical Sciences. Penguin 1972. The appendix also contains a simple derivation of E = mc².

Arthur Beiser, Concepts Of Modern Physics. McGraw-Hill 1973. Chapter one.

For the Lorentz transformations, I cannot remember the first book that interested me in them, but I have read appreciatively, on the way:

Milton Rothman, The Laws Of Physics. Penguin 1963.

Allan M Munn, From Nought To Relativity. Allen and Unwin 1974.

Paul Davies, Space And Time In The Modern Universe. Cambridge University Press 1977. This carefully explains away the twin 'paradox'. Davies has written a good many popular physics books since then, most of which Ive read.

For the Minkowski Interval, I relied on:

George E Owen, Fundamentals of Scientific Mathematics. The John Hopkins Press 1961. However, I didnt follow the text's following of Lorentz's use of matrices to arrive, formally, at his transformations, but tried to make do with trigonometry.

Richard Lung.

Special relativity ( part three ): The Lorentz transformations.
An arithmetic example of the loss of simultaneous time between observers in relative motion comparable to light speed.

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