Reconciling Length Contraction and Time Dilation
Using Einstein's own description
First of all we should bear in mind that Length Contraction and Time Dilation were not terms that Einstein used but have become the accepted way of referring to what he described in:
Relativity: The Special and General Theory.* 1920.* Albert Einstein* (1879–1955).
Chapter XII: * The Behaviour of Measuring-Rods and Clocks in Motion.
(Note that Einstein used the superscript ' to denote the moving Frame of Reference (K') and all associated measurements).
Let us draw diagrams to see if we can picture what Einstein was saying in each of these two cases: the measuring-rod and the clock.
He wrote:
Originally Posted by Einstein
I PLACE a metre-rod in the x'-axis of k' in such a manner that one end (the beginning) coincides with the point x' = 0, whilst the other end (the end of the rod) coincides with the point x' = 1. What is the length of the metre-rod relatively to the system K? In order to learn this, we need only ask where the beginning of the rod and the end of the rod lie with respect to K at a particular time t of the system K
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the distance between the points being*
√1 – v²/c²
The rigid rod is thus shorter when in motion than when at rest.
Many will find it helpful to picture what happens to the said 1 metre rod and how well this matches Einstein's description and so I include the following diagram:
Fig. 1
This shows how the red 1 metre rod in Fig. 1 would be observed by an observer in that same K' system.
The Green 1 metre rod in Fig. 1 is that same rod rotated, as it would be seen by an observer in the stationary system K, when system K' is moving at 0.6c relative to system K.
When that green rotated rod is projected down onto the x axis of system K, as depicted by the blue rod it has the length:
√1 - v²/c² = 1/γ = 0.8
Which, being less than its original length, is quite reasonably referred to as Length Contraction and so gives us the formula:
x = x'√(1 - v²/c²) = x' / γ
And further, he continues to say:
And, indeed is required, in accordance with one being the reciprocal of the other.
What then if we draw a similar diagram for the t, t' axes in Fig. 2?
The two ordinates (t axes), t and t' of systems K and K', shown in blue and red respectively, with the addition of the rotated t' axis shown in green and how they are related.
Fig. 2
Einstein described this transformation thus:
So in our diagram we take the red bar and, multiply it by the Lorentz factor, to give us 1.25 units for the green bar on that K' rotated scale. A somewhat larger measurement, which, quite justifiably, has become known as Time Dilation.(red rod in Fig. 2 - lying along the t' axis). The first and fourth equations of the Lorentz transformation give for these two ticks:
t = 0
and*
As judged from K, the clock is moving with the velocity v; as judged from this reference-body, the time which elapses between two strokes of the clock is not one second, but*
seconds, i.e. a somewhat larger time. (green bar in Fig. 2 - lying along the rotated t' axis). ]As a consequence of its motion the clock goes more slowly than when at rest.
So we can see that Einstein was correct in what he wrote; but what do we conclude from it?
What lessons can we learn to resolve the mysteries, of Time dilation and Length contraction, that have so bedevilled students of Special Relativity?
Well the formula for time dilation that derives from the last equation above being:
t = γt'
leads into one of the puzzling facets of Special Relativity for if
Which flies in the face of Einstein's second postulate:
Which demands that if c = x/t in any Inertial Frame of Reference, then in system K', c = x'/t'The Principle of Invariant Light Speed – "... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body."
But don't worry all will be reconciled in the next section.
Summary
If the speed of light, c = x/t then how can one be dilated when the other is contracted?
Only if the same thing happens to each, therefore each must be both dilated and contracted.
Drawing it
This raises the question of how we draw it! The modern way is to use the ct axis of the stationary observer to provide the time scale on the moving objects ct’ axis.
WHY?
Because then we see that time is 'Dilated' on that scale - as SR predicts. - Rubbish!
Time dilation and length contraction are two terms for the same effect! 'dilation' and 'compression' mean the same?
YES THEY DO!
How? Well they are each taking a measurement from one FoR and comparing how it is seen by each of the observers. BUT those comparison's are not performed the same way round! One compares B to A and finds B is bigger - 'Dilation'; while the other compares A to B and finds A is smaller - 'Compression'!
Back to Our diagram.
We are drawing the path of the moving object as the stationary observer sees it. When we draw the second Frame of Reference over the first, we have to use the scales as seen by that same stationary observer. (After all the diagram is from her perspective.)
The time passing, as measured within each frame, passes at the same rate - 1st Postulate - so if we draw them 1mm per unit on the first FoR we must draw them 1mm per unit for the second one too.
Measurements within each Inertial Frame of Reference will be identical in scale as each is stationary to an observer within that Frame of Reference. This is what Einstein's first Postulate is all about.
It is when measurements are transformed, that is the system of coordinates is rotated, as a function of the relative velocity, that things become interesting.
For moving objects, measurements of time and distance are transformed according to the Lorentz Transformation Equations as a function of their relative velocity. In accordance with the Lorentz Factor γ where:
and as both these transformations are applications of the Lorentz Factor, they are reciprocal, and so the Absolute Measurements (i.e. magnitude x quantity) will be unchanged.
Consider Fig. 2. The 1 second time in red (system K')rotates and dilates to the green length γ (= 1.25 seconds) when it is viewed as a moving frame. But that 1.25 seconds contracts to 1 second (1/γ) when viewed from system K, a stationary frame; (so the time in K equals the time in K'; complying with the first postulate.)
In the same way the one metre rod in K would have become γ units, = 1.25 metres in the rotated/transformed/moving system K'; and it would have then contracted back to 1 metre from the stationary observer's perspective. Remember that Einstein was placing a 1 meter rod in the moving system so, viewed from the stationary system, it contracted to 80cm.
It is all relative, reciprocal and only the observed measurements are affected.
So both time and distance are dilated and contracted, it all depends which transformation one is making.
So:
t' = γt,
t' = t/γ,
x' = γx,
x' = x/γ
are all correct! Depending on whether we are referring to the unit size or it quantity; and whether we are comparing A to B, or B to A! We must be careful to use the appropriate formulae for the calculation we are making.
This also explains the problem with the c = x'/t' for x' is referring to the unit size, while t', is referring to the number of time units; and if we change them to both refer to the same property, all is right with the world once more.
And it is worth repeating that in all cases (unit-size) x (unit-quantity) will give the same Absolute (total) Magnitude or Duration whether measuring a stationary object or a moving one.
This is not to say that the measurement of time doesn't dilate. It does. We know that because we measure it and use it in GPS calculations and innumerable experiments. Yet what we don't do is to measure the size of the units we are measuring.
Time dilates, more time passes [b] but the units of time decreases by the same factor!
And the same goes for the length. The 'metre rod' does indeed shorten, the metre unit becomes smaller, contracts, but the number of such units increase again by the same factor!
That is how a moving object can change its measured length a different amount for every observer travelling at a different relative velocity without any conflict.
I will say again, take notice of that one important phrase that is at the heart of so much:
"As a consequence of its motion …"
It is only when the system K' is observed from the system K that that extra translational vector is included in the measurements. We see this in Figs 1 & 2 where it is shown in green, rotated relative to the stationary system views; due to that rotation we see its measurements increased in quantity in comparison to the two local systems K, and K' where that relative motion is excluded. It is also where we see the measurements, projected from the rotated moving scale onto the vertical stationary scales, and the consequent contraction of those units.
It is, as was stated by Einstein himself, only when the body/Frame of Reference is moving relative to the observer, that Lorentz Transformations occur.
In Systems K, K' there are only the measurements made within each Reference Frame by the associated local observers; it is only when one is viewed by a moving observer that the appropriate vector representing that movement has to be added using the Lorentz Transformations, Transforming the Proper measurements to Coordinate measurements.
Summary
Length Contraction and Time Dilation are reconciled only when we appreciate that it is the way the measurements are calculated that changes; that all measurements of a moving Frame of Reference are both Length Contracted and Time Dilated, while the Absolute magnitude or duration remains the same.
Let me say that the repetition in this section is deliberate and I make no apology for that. It is a thing that I learned from the writing of Agatha Christie. Repetition drives the point home especially when read at speed; and these points are fundamental to understanding the Theory.
