Special Relativity: From Galileo to Einstein

The intent of this article (adapted from an original paper I wrote about a decade ago) is to show how Special Relativity can be arrived at by examining Galilean boosts along with the inherent assumptions and then removing invalid assumptions. During the process it becomes clear that there is a potential source of confusion associated with the standard expression of time dilation. Note that this is the “methodical” version; there is also a simplified version which (I hope) will make more sense to those who are less gifted mathematically.

1. Introduction

Galilean Relativity relies on a number of assumptions. A number of these assumptions have been shown to be invalid in the real world and Einstein’s Special Relativity, which dispenses with those assumptions, is a far superior conceptual framework. Einstein arrived at Special Relativity via “mind experiments” and brilliant reasoning. I intend to demonstrate an alternate method of arriving at Special Relativity.

2. Galilean Boost

A Galilean boost is the equation which allows x’ to be calculated in Figure 1 where S is nominally stationary and M is nominally in motion with a velocity v. The equation relies on the following assumptions:

– when t = 0 and t’ = 0, S and M are coincident

– S and M observe the same event

– S and M measure time and space identically

– S and the observed event (X) share a preferred frame

– information is transmitted instantaneously

The boost is given by:

x’ = x – vt

t’ = t

3. Information is not transmitted instantaneously

We know that it actually takes time for information about the event X to reach each of the observers.

This results in a modification to an assumption, such that event X takes place when S and M are coincident, and t = 0 and t’ = 0.  The extant assumptions are, therefore:

– the observed event takes place when S and M are coincident, and t = 0 and t’ = 0

– S and M observe the same event

– S and M measure time and space identically

– S and the observed event (X) share a preferred frame

The modified Galilean boosts are:

x’ = x – vt

ct’ = x’ = x – vt = ct – v.x/c

t’ = t – v.x/c²

4. No preferred frame

We referred to S as stationary, but this merely means that S is stationary relative to a nominated “observer” (conceptually the reader). Relative to M, S has a velocity of -v. There is no valid reason to assume that either of the perspectives (or “frames”) is preferred.

Note that the unprimed frame pertains to S and the primed frame pertains to M.  That is to say:

- in the unprimed frame, S is stationary and M has a velocity of v_s according to S and

- in the primed frame, M is stationary and S has a velocity of v_m according to M.

The magnitude of v_s and v_m are such that:

v_s = -v_m = v

The extant assumptions are:

– the observed event takes place when S and M are coincident, and t = 0 and t’ = 0

– S and M observe the same event

– S and M measure time and space identically

The third assumption cannot stand without a preferred frame, since x’_m > x’_s and x_m > x_s where v_s > 0.

5. S and M do not measure time and space identically

Figure 3 is modified slightly to reflect that v_s = -v_m = v.

The extant assumptions are:

– the observed event takes place when S and M are coincident, and t = 0 and t’ = 0
– S and M observe the same event.

According to each observer, the other utilises different, but consistent units of measurements – in the other’s own frame.  The unprimed frame pertains to S, so:

x_m = ย.x_s        (1)

The primed frame pertains to M, so:

x’_s = ย.x’_m        (2)

where ย is yet to be determined.  Taking first the perspective of S:

x’_s = x_s – vt’_s 

x_s = x’_s + vt’_s

but since x_m = ย.x_s and t’_s = x’_s/c

x_m = ย.x_s = ย.(x’_s + vx’_s/c) = ย. x’_s (1 + v/c)         (3)

Then taking the perspective of M:

x_m = x’_m - (-vt_m) = x’_m + vt_m

x’_m = x_m - vt_m         (4)

but since x’_s = ย.x’_m and t_m = x_m/c

x’_s = ย.x’_m = ย.(x_m - vx_m/c) = ย. x_m (1 - v/c)        (5)

Applying (5) to (3):

x_m = ย. x’_s (1 + v/c) = ย. [ย. x_m (1 - v/c)].(1 + v/c)

x_m =  ย². x_m (1 - v/c).(1 + v/c)

1 =  ย². (1 - v²/c²)

ย = 1 / (1 - v²/c²)^½          (6)

6. Lorentz boosts

Understanding a Galilean boost is relatively simple, since there is only one perspective. With a Lorentz boost, which we can derive from the information in the previous section, we have two perspectives to select from – that of S and that of M. Since we have already stated that S is stationary by virtue of being stationary relative to us, we will examine the situation from the perspective of S.

According to S, the distance between the event X and the point at which M observes that event is x’_s = x_s – vt’_s.  The Lorentz boost doesn’t seek to tell us this, it seeks to tell us what that distance is in terms of what M observes.  To arrive at the relevant equation, we make use of (2), (4) and (6):

x’_s = ย.x’_m

x’_s = ย.(x_m - vt_m) = (x_m - vt_m) / (1 - v²/c²)^½          (7)

and since x’_s = ct’_s and x_m = ct_m so that t_m  = x_m / c:

x’_s=ct’_s=(x_m - vt_m) / (1 - v² / c²)^½=(ct_m - vx_m/c) / (1 - v²/c²)^½

t’_s = (t_m - vx_m/c²) / (1 - v²/c²)^½         (8)

It is trivial to show that a similar equation can be derived from the perspective of M, in terms of what S observes, so long as one keeps in mind that v_s = -v_m = v.

7. Finding x and t in the other frame

Rather confusingly, a different priming convention applies for denoting relativistic effects on space and time.  When considering relativistic effects, a primed value refers to a measurement within a frame in relative motion, as made by an observer from within that frame in relative motion – in terms of units which pertain to the nominally stationary frame.

Consider x_s and x_m:

– x_s is a measurement made by observer S within the S frame which, according to observer M, is in motion

– what is x_s in terms of x_m?

This is equivalent to the question what is x’ in terms of x?  We apply (7) to (1):

x_m = ย.x_s

x_s = x_m / ย = x_m . (1 - v² / c²)^½

                                

→ x’ = x . (1 - v² / c²)^½         (9)

 and since x_s = ct_s and x_m = ct_m:

x_s =  ct_s = ct_m . (1 - v² / c²)^½

t_s = t_m . (1 - v² / c²)^½

→ t’ = t . (1 - v² / c²)^½          (10)

The same logic can be applied to x’_s and x’_m and the same result achieved.

8. On the result achieved by standard derivations

Standard derivations achieve a different result, viz:

L' = L . (1 - v² / c²)^½          (11)

Δt' = Δt / (1 - v² / c²)^½         (12)

This is because, in the standard derivations, Δx’ corresponds with x’ as expressed in the boost while Δt’ refers to something quite different, something that is conceptually inverse to t’ as expressed in the boost.  Conceptually, the term Δt’ refers a period measured in a frame in motion (relative to the observer).  In other words, Δt’ is equivalent to Δt_m rather than Δt_s:

It must, however, be noted that a key assumption in this derivation is that if x = ct then x’ = ct’, and by extension that if Δx = Δct then Δx’ = Δct’, so that the use of the prime notation is consistent.  If the use of the prime notation used in the length contraction and time dilation equations is intended to be consistent, then there is a problem with this assumption as used explicitly to transition from (8) to (9).  Therefore it behoves me to present a derivation which does not rely on this assumption.

9. Finding t in the other frame by other means

Recall that according to each observer, the other utilises different, but consistent units of measurements – in the other’s own frame.  The unprimed frame pertains to S, so:

t_m = ท .t_s

ท = t_m / t_s        (13)

The primed frame pertains to M, so:

t’_s = ท .t’_m

ท = t’_s / t’_m         (14)

where ท is yet to be determined.  Multiplying (13) by (14):

ท . ท = (t’_s / t’_m).(t_m / t_s)

Rearranging:

ท² = (t’_s / t_s).(t_m / t’_m)        (15)

According to S, M moves towards the event at velocity v while information about the event moves towards both S and M at c, so:

t_s / t’_s = c/(c - v) = 1 / (1 - v/c)       (16)

According to M, s moves away from the event at velocity v while information about the event moves towards both S and M at c, so:

t’_m / t_m = c / (c + v) = 1 / (1 + v/c)          (17)

Applying (16) and (17) to (15):

ท² =(t’_s / t_s).(t_m / t’_m)=[1 / (1 - v/c)].[1 / (1 + v/c)]           

ท² = 1 / (1 - v²/c²)

ท = 1 / (1 - v²/c²)^½ = ย        (18)

Applying (18) to (13):

t_s = t_m / ท = t_m . (1 - v²/ c²)^½

Since t_s is a measurement made by observer S within the S frame which, according to observer M, is in motion, then:

t_s = t_m . (1 - v²/c²)^½

t' = t . (1 - v²/c²)^½      (10)

As described in Section 8, this corresponds to the time dilation equation (where t_m = Δt' and t_s = Δt), so:

Δt = Δt' . (1 - v²/c²)^½
Δt' = Δt / (1 - v²/c²)^½      (12)

10. Conclusion (as modified)

The equations for Special Relativity can be derived mathematically by means of systematically removing invalid assumptions associated with Galilean relativity.