3 The inertial observer in Minkowski spacetime

The inertial observer O has zero acceleration and there is no cosmic expansion. Its velocity is defined by

u⁰=c , u¹=s , u²= u³= 0,

where $c:=\cosh\psi$ and $s:=\sinh \psi$, the rapidity $\psi$ being a constant. Its world line is $x ^{0}=c~\tau,~x^{1}=s~\tau,~x^{2}= x^{3}= 0$. A light-ray passing through x reaches O at proper time $\tau$ such that

$$(x^{0}-c~\tau)^{2}=(x^{1}-s~\tau)^{2}+ (x^{2})^{2} + (x^{3})^{2}.$$ (15)

For the arbitrary point x, it is convenient to define $x^{0}=\Delta ~\cosh \beta,x^{1}=\Delta ~ \sinh \beta$, so that the solutions of this equation give

$${\cal N}_{\varepsilon} \Delta ~\cosh (\psi-\beta)$$

$$+\varepsilon ~ \sqrt{\Delta ^{2} ~\sinh (\psi-\beta) ^{2}+(x^{2})^{2} + (x^{3})^{2}}$$ (16)

$$T =\Delta ~\cosh (\psi-\beta)=c~ x^{0} -s~x^{1},$$ (17)

=$$\sqrt{( s~ x^{0} -c~x^{1} )^{2}+ (x^{2})^{2} + (x^{3})^{2}},$$

$$\sqrt{\Delta ^{2} ~\sinh (\psi-\beta) ^{2}+(x^{2})^{2} + (x^{3})^{2}}$$ (18)

and N=1.

Figure 4: For the inertial observer in Minkowski spacetime, with arbitrary velocity (rapidity $\psi$), space is the hyperplane with inclination $\psi$. We have drawn a curve $R= C^{\rm te}$, in this plane.

The surface $\Sigma _{\tau }$ is the plane of equation $c~ x^{0} -s~x^{1}~= \tau$, inclined by $\psi$ with respect to the vertical, and thus orthogonal to the world line of the inertial observer. Thus, space is different for all inertial observers.

Finally, $u= c~ {\rm d} x^{0} -s~{\rm d}x^{1}$ and

$$\begin{eqnarray*}\vec{\tilde{b}}&=&\frac{1}{R}~\left[\left( s~ x^{0} -c~x^{1} \r... ...t.\\ &&+ \left. x^{2}~{\rm d}x^{2}+ x^{3}~{\rm d}x^{3} \right] \end{eqnarray*}$$

points towards the observer at time $\tau=T(x)$. In the space $\Sigma _{\tau }$ (Fig. 4), it is natural to define the coordinate y:=-s  x⁰ +c x¹ =y/c so that R²=y²+ (x²)² + (x³)² and the spatial metric $-{\rm d}s^{2}={\rm d}y^{2}+ ({\rm d}x^{2})^{2}+({\rm d}x^{3})^{2}$.

  
3.1 The Langevin observer in Minkowski spacetime

It is well known that the solution of the celebrated "Langevin's twin paradox'' lies in geometry. I define a Langevin observer as an inertial observer which is initially inertial, then suffers an instantaneous acceleration, and then is inertial again (Fig. 5). Such an observer is able to meet his twin, who remained always inertial, with a different lapse of proper time. Is is often quoted (see, e.g., Misner et al. 1973) that it is impossible to define space globally for such an observer. Here I show that the synchronicity prescription applies perfectly and provides an unambiguous definition of space for this observer. Thus I define the space-time trajectory of this observer as

$$x^{0}=\tau,x^{1}=x^{2}=x^{3}=0, \mbox{~~~~~for~~~~~}t<0,$$

$$x^{0}=c~\tau,x^{1}=s~\tau, x^{2}=x^{3}=0, \mbox{~~~~~for~~~~~}t>0,$$

with $c:=\cosh\psi$ and $s:=\sinh \psi$.

Figure 5: World's line and (cuts of) space at various moments for the Langevin observer. Its line cone is indicated by the dashed lines.

The light cone of the observer at t=0, ${\cal L}^{0}$, divides the space-time into three parts I, II and III (see Fig. 5) corresponding to the past, future and spatially related regions to the acceleration point. Studying the light-rays from [to] the point x to [from] the observer gives the functions:

• Region I
  

  $${\cal N}^{\varepsilon} (x)=x^{0}+\varepsilon \sqrt{(x^{1})^{2} +(x^{2})^{2}+(x^{3})^{2}},$$ (19)

  
  
  
  

  $$T(x)=t, R (x) = \sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}.$$ (20)

  
  

• Region II
  

  $${\cal N}^{\varepsilon}(x) c~x^{0}-s~x^{1} +\frac{\varepsilon}{c}$$ = (21)

  $$\times \sqrt{(c~x^{0}-s~x^{1})^{2} {-}(x^{0})^{2}{+}(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}$$

  $$c~x^{0}-s~x^{1} +\frac{\varepsilon}{c}$$ =

  $$\times \sqrt{( s~ x^{0} -c~x^{1} )^{2}+ (x^{2})^{2} + (x^{3})^{2}},$$

  
  
  
  
  T (x)= c x⁰-s x¹,
  
  
  
  

  $$\sqrt{(c~x^{0}-s~x^{1})^{2} {-}(x^{0})^{2}{+}(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}$$ R(x) =

  $$\sqrt{( s~x^{0}-c~x^{1})^{2} + (x^{2})^{2}+(x^{3})^{2}}.$$ = (22)

  
  

• Region III
  

  $${\cal N}^{+}(x)$$ = c x⁰-s x¹

  $$+ \sqrt{( s~x^{0}-c~x^{1})^{2} +(x^{2})^{2}+(x^{3})^{2}},$$ (23)

  
  
  
  
  $${\cal N}^{-}(x)=x^{0} -\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}},$$
  
  
  
  

  2T (x) = x⁰+c x⁰-s x¹

  $$+\sqrt{(s~x^{0}-c~x^{1})^{2} +(x^{2})^{2}+(x^{3})^{2}}$$

  $$-\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}$$ (24)

  
  
  and
  

  2R (x) = -x⁰+c x⁰-s x¹

  $$+\sqrt{(s~x^{0}-c~x^{1})^{2} +(x^{2})^{2}+(x^{3})^{2}}$$

  $$+\sqrt{(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}}.$$ (25)

  
  

The surface $\Sigma _{\tau }$ of equation $T(x)=\tau$ defines space for the observer at proper time $\tau$.

Let us consider its projection in the (x⁰,x¹) plane (Fig. 5):

• Region I
  

  $$x^{0}=\tau.$$ (26)

  
  
  This is a straight horizontal line, where R (x) = x¹.
• Region II
  

  $$c~x^{0}-s~x^{1}=\tau$$ (27)

  
  
  or
  

  $$\tau/\cosh \psi= x^{0}-\tanh \psi~x^{1}.$$ (28)

  
  
  This is a line inclined of $\psi$ with respect to the vertical, and thus orthogonal to the world line of the observer in that region. In this line,
  
  $$\begin{eqnarray*}R(x)&=& \sqrt{( c~x^{0}-s~x^{1})^{2} -(x^{0})^{2}+(x^{1})^{2} }\\ &=& c~x^{1}-s~x^{0}. \end{eqnarray*}$$
  
  

• Region III
  

  $$2\tau$$ = x⁰+c x⁰-s x¹

  $$+\sqrt{(c~x^{0}-s~x^{1})^{2} -(x^{0})^{2}+(x^{1})^{2} } - x^{1}$$

  = x⁰+c x⁰-s x¹ +c x¹-s x⁰ - x¹, (29)

  
  
  or
  

  $$\frac{\tau~\exp(\psi/2)}{\cosh(\psi/2)} =- x^{0}+ x^{1}~\tanh (\psi/2).$$ (30)

  
  
  This is a line inclined of $\psi/2$ with respect to the vertical, i.e., at equal hyperbolic angle $\psi/2$ of the two previous lines (Fig. 5). In this line,
  

  2R (x) = -x⁰+c x⁰-s x¹

  $$+\sqrt{(c~x^{0}-s~x^{1})^{2}-(x^{0})^{2}+(x^{1})^{2} } + x^{1}$$

  = - x⁰+c x⁰-s x¹ +c x¹-s x⁰ + x¹. (31)

  
  

For the observer at an arbitrary moment, space is made of a plane circle S$^{\rm I}$ [or S$^{\rm II}$] up to the light cone ${\cal L}^{0}$ and is continued by a composite surface S $^{\rm III}$ beyond. Except at the single moment when the observer experiences the instantaneous acceleration, space is not flat, nor homogeneous.

This is the simplest example where our prescription differs from the other ones. As it is well known, it is impossible to extend the Fermi coordinates outside the conical regions. And no homogeneous hypersurfaces would be convenient. Thus, in this simple case, our prescription is the only one providing a reference frame associated to the observer valid in the whole space-time, to extend the validity of his proper time, and to consider unambiguous synchronicity procedures (a similar conclusion has been reached by Dolby & Gully 2001).