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Paradoxes and paradigms in Physics

The Lorentz Transformation

Introduction
The Galilei Transformation
The Lorentz Transformation
The Lorentz Transformation of velocities
The mixed Lorentz Transformations
Space-time intervals
Time dilation
Length contraction
The Muon Paradox

 

Illustration from: "Mr. Tompkins in Wonderland or stories of c, G and h",
George Gamow

Introduction

The more "verbal-intensitive" text in c + c = c gives the first attempt to understand the Special Theory of Relativity but lacks a more precise treatment.
The only way to avoid the many traps and apparent paradoxes is to base the treatment on a more " mathematical-intensitive" presentation using the concept of "space-time events" and the transformation of these using the Lorentz Transformation.
Happily the Math is mostly simple algebraic manipulations of expressions and solving linear equations. Calculus can be efficient to apply in some cases, but central here.

The Galilei transformation

The inertial system S' is moving with velocity v relative to the inertial system S, v having constant size and direction.
(0,0) of S' is passing (0,0) of S at the time t = t' = 0.
Here and in the following v is assumed directed along the x-axis.

  - and the reverse:

x' = x - v t
y' = y
z' = z
t' = t

x = x' + v t'
y = y'
z = z'
t = t'

The Galilei Transformation assumes absolute simultaneity, that is, time t in S is equal to time t' in S' independent of position in space.

Transformation of velocities, from  vx in S  to  v'x in S', can be derived by substition of the reverse Galilei Transformation into

x  = vx t   becoming:   x'  + v t' = vx t'    or   x'  = (vx - v) t' = vx' t'   so that:   v'x = vx -  v

The y and z velocities simply becomes: v'y = vy and v'x = vx .

Alternatively the velocity transformations can be derived differentiating:

dx'/dt'  =  dx/dt  -  v   or   v'x = vx -  v

Newton's laws in classical mechanics are invariant under the Galilei Transformation,
Maxwell's equations in electro-magnetism are not.

The Lorentz transformation

Einstein's made two assumptions:

  1. The speed of light is absolute:
    The speed of light in vacuum is approximately c = 300.000 km/s and absolute in the sense that it is independent of both the speed of the emitter and the receiver, that is independent of the observer's choice of an inertial system.
  2. The principle of relativity is valid in all inertial systems.
    All laws of Physics are having the same form in all inertial systems.
    (Originally formulated by Galilei and Newton)

The consequence of these two assumptions will be that we must give up other 'traditional' assumptions:
Time and space are not absolute, but relative entities.
The Galilei-transformation must be substituted by another kind of transformation.

Based on these two assumptions (and a few more concerning linearity and symmetry) the Lorentz Transformation can be derived (*1) resulting in this:

With inertial systems S and S' as defined under the Galilei Transformation:

  - and the reverse:

(1a)  x'  =   γ (x  -  v/c ct)
         y'  =   y
         z'  =   z
(2a)  ct' =   γ (ct -  v/c x)

(1b)  x  =   γ (x'  +  v/c ct')
         y  =   y'
         z  =   z'
(2b)  ct =   γ (ct' +  v/c x')

- with γ ("gamma") defined as shown.

You can get the reverse transformations either in the original transformations interchanging variables with apostrophe (') with variables without apostrophe and vice versa combined with substitution of v with -v ,
or by solving for x, y, z, and t.

We will use the term "space-time event" having coordinates (t, x, y, z) in S and (t', x', y', z') in S'.

The Lorentz Transformations presented in this form, with time represented by ct, make the symmetry between space and time quite apparent. It also gives simple calculations using as units:

x in lightyears (cy), t in years (y), v as fraction of speed of light, v/c (dimensionless).

Example with numbers:

Let v/c = 0.6, that is γ = 1.25, x = 6 c y, t = 8 y. Then:

x'  =   γ (x  -  v/c ct) = 1.25 * (6 cy - 0.6 * 8 cy) = 1.5 cy,
ct' =   γ (ct -  v/c x)  =
1.25 * (8 cy - 0.6 * 6 cy) = 5.5 cy, that is, t' = 5.5 y.

We will sometimes skip units totally in calculations, like this:

x'  =   γ (x  -  v/c ct) = 1.25 * (6 - 0.6 * 8) = 1.5,
ct' =   γ (ct -  v/c x)  =
1.25 * (8 - 0.6 * 6) = 5.5.

The Lorentz transformation of velocities

Transformation of velocities, from  vx in S  to  v'x in S', can be derived by substition of the reverse Lorentz Transformation into x  = vx t , y  = vy t , and z  = vz t.   After some algebraic manipulations you get:

       

You can also find this differentiating, but in contrast to the Galilei Transformation dt' is different from dt. This also implies the non-trivial transformations of velocities along y and z.

The "mixed" Lorentz transformations

It can sometimes be convenient to express x and t' by x' and t, and to express x' and t by x and t':

From (1a):   x   =   (1/γ) x'  +  v/c ct
From (2b): ct'   =   (1/γ) ct  -  v/c x'

From (1b):  x'  =   (1/γ) x   -  v/c ct'
From (2a): ct   =   (1/γ) ct' +  v/c x

Space- and time intervals - and the invariant "space-time interval"

The Lorentz Transformation can also be formulated in the form of space- and time intervals, Δx = x2 - x1 meaning the increment of x and Δt = t2 - t1 the increment of t between two space-time events (x1 , t1 ) and (x2 , t2 ) :

  - and the reverse:

(delta1a)  Δx'  =   γ (Δx  -  v/c cΔt)
(delta2a) cΔt'  =   γ (cΔt -  v/c Δx)

(delta1b)  Δx  =   γ (Δx'  +  v/c cΔt')
(delta2b) cΔt  =   γ (cΔt' +  v/c Δx')

The "space-time interval" Δs is defined as:  Δs2 = (cΔt)2 - Δx2 - Δy2   - Δz2

The definition of Δs is analog with the distance formula in ordinary space, but with this strange change of sign between the space part and the time part.
(You will just as often find in the literature that the space-time interval is defined with plus (+) in front of the squares of space coordinates and minus (-) in front of the time coordinate. We prefer the form of the definition given here.)

Neither Δt nor Δx are invariant under a Lorentz Transformation, but by inserting the Lorentz Transformation in the definition of space-time interval it can be shown after some algebraic manipulations that the space-time interval is invariant under a Lorentz Transformation:

Δs2 = (Δs')2    or     (cΔt)2 - Δx2 - Δy2   - Δz2   =   (cΔt')2 - Δx' 2 - Δy' 2   - Δz' 2

or with just one space coordinate x:      (cΔt)2 - Δx2   =   (cΔt')2 - Δx' 2

In the special case where   Δs2 = (Δs')2 = 0   the two space-time events (t1 , x1 ) and (t2 , x2 ) can be connected by a signal with the speed of light,

or, with just one space coordinate x:     c = Δx/Δt = Δx'/Δt'.
In this case the space-time interval is called "lightlike".

If Δs2 > 0  the two space-time events can be connected by a signal with speed lower than the speed of light, the absolute value of Δx/Δt becomes less than c.
In this case it can be proved that it is possible to transform to an inertial system for which Δx = 0 (and Δt ≠ 0).
When calculating Δs2 and Δs in this system we get: Δs2 = (cΔt)2 or Δs = (+/-)cΔt.   As Δx = 0, the two events correspond to an object at rest observed in a period Δt , which means that Δt is the proper time interval between the events. So Δs represents this proper time interval (as calculated in any transformed system, because Δs is invariant) .
In this case the space-time interval is therefore called "timelike".

If Δs2 < 0   the two space-time events cannot be connected by any signal - the required speed, the absolute value of Δx/Δt, would be greater than the speed of light which is not possible. Δs becomes an imaginary number.
In this case it can be proved that it is possible to transform to an inertial system for which Δt = 0 (and Δx ≠ 0).
When calculating Δs2 and Δs in this system we get: Δs2 = Δx2 or Δs = (+/-)Δx.   As Δt = 0, the two events are simultaneous. So Δs represents the space interval for two simultaneuos events (as calculated in any transformed system, because Δs is invariant) .
In this case the space-time interval is therefore called "spacelike".

Timelike inertial systems can never be transformed to spacelike inertial systems and vice versa.

See Sartori 1996, part 5.3, "Relativity and causality", references at The Twin Paradox revisited

Time dilation

Two space-time events occur in S' at the same point in space at two different times:
(t1', x') and (t2', x').

As seen from S the two events occur at two different positions at two different times:
(t1 , x1 ) and (t2 , x2 ) .

According to the Lorentz Transformation in the form of space-time intervals with Δx' = 0
we get:

(delta2b)   cΔt  =   γ (cΔt' +  v/c Δx') = γ (cΔt')    or:   Δt  =   γ Δt'.

The time interval Δt' for the two events in the same space point as seen from S' is called
"the proper time".
As γ > 1 we have:     Δt  > Δt'.  As seen from S the time is dilated in relation to as seen from S'.

Length contraction

Two space-time events occur in S' at the same time at two space points:
(t', x1') and (t', x2').

As seen from S the two events occur at two different positions at two different times:
(t1 , x1 ) and (t2 , x2 ).

According to the Lorentz Transformation in the form of space-time intervals with Δt' = 0
we get:

(delta1b)   Δx  =   γ (Δx'  +  v/c cΔt') = γ Δx'    or:   Δx  =   γ Δx'.

If we assume that x1 and x2 mark the coordinates of the endpoints of a stick at rest in S, then Δx = x2 - x1 can be interpreted as a measure of the length of the stick in S, "the rest-length" (independent of the time-parts t1 and t2).

In relation to any other inertial system, for example S' with speed v in relation to S, the stick is moving backward with that speed, and to get a reasonable concept of the length of the moving stick as seen from S' we must read the x'-coordinates of the endpoints of the stick simultaneously, that is at the same time t', and we get from above:

Δx' = x2' - x1' = (1/γ) Δx  <   Δx.

We call this "length contraction" as the "moving length" of the stick becomes smaller than "the rest length".

Remark:
There is always a lot of tradition how to describe these matters. Concerning length contraction the most often used way is to let the stick be at rest in S' (having the speed v in relation to S in the direction of the x-axis as here). The choice here of letting the stick be at rest in S makes the defining situation and result look "symmetric" compared to the time dilation case

The Muon Paradox

Applied to the example with the muon, "The Muon Paradox" the defining enteties become:

v = 0.999 c.   γ = 22.4.  Muon's proper lifetime: Δt'  =  2.2 10-6 s.  

From this and the Lorentz Transformation we get, assuming that the muon is at rest in S':

As seen from S':  Δt'  =  2.2 10-6 s.    Δx' = 0.
As seen from S:   Δt  =   γ Δt' = 49.2 10-6 s.    Δx = v Δt = 14.7 km.

If we from the muon point of view see a stick at rest in S of length 10 km pass backwards at a speed of 0.999 c, the muon will interpret the moving length of this stick (the distance down to us as observers) as: 
L'  =   1/γ 10 km = 447 m 

and in the muon's proper lifetime this stick can travel the distance:
0.999 c 2.2 10-6 s = 659 m > 447 m.

References:

(*1) http://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations

See more: Paradoxes i fysik - The Twin Paradox revisited

Opdateret 2-07-2016 , TM

 
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