> Suppose I is an inertial observer and A is an arbitrarily accelerated
> observer that constantly monitors the acceleration he feels as a
> function a(T) of his own proper time T (this is 'tau').
> At each time T, observer A can write the amount with which his
> velocity has changed during a small (infinitesimal) proper time
> interval dT during which the acceleration does not change as
> dV(T) = a(T) dT
> This change of velocity is to be regarded with respect to the
> instantaneously comoving inertial frame at time T.
> Observer I can parametrize the worldline of A with the same
> proper time T, so he will see infinitesimally consecutive velocities
> of A as v(T) and v(T+dT).
> Since v(T+dT) is the standard SR composition ('addition') of the
> velocities v(T) and dV(T), we can write (using c=1):
> v(T+dT) = [ v(T) + dV(T) ] / [ 1 + v(T) dV(T) ]
> which, using the expression for dV(T), becomes:
> v(T+dT) = [ v(T) + a(T) dT ] / [ 1 + v(T) a(T) dT ]
> To calculate dv(T)/dT we use
> [ v(T+dT) - v(T) ] / dT = a(T) [1-v^2(T)] / [ 1 + v(T) a(T) dT ]
> Take the limit dT --> 0
> dv(T)/dT = a(T) [ 1 - v^2(T) ]
> Rearrange:
> dv(T) / [ 1 - v^2(T) ] = a(T) dT
> Integrating between 0 and T
> argtanh(v(T)) - argtanh(v(0)) = Int{ 0 to T; a(T') dT' }
> and using the abbreviation
> A(T) = Int{ 0 to T; a(T') dT' },
> we get
> argtanh(v(T)) - argtanh(v(0)) = A(T)
> Inverting to
> [ v(T) - v(0) ] / [ 1 - v(T) v(0) ] = tanh(A(T))
> and isolating v(T) gives:
> v(T) = [ v(0) + tanh(A(T)) ] / [ 1 + v(0) tanh(A(T)) ]
> or simply, if v(0) = 0:
> v(T) = tanh(A(T))
> Use the standard SR Lorentz transformation (with c=1) between
> the frame I and the instantaneously comoving inertial frame of A
> at time T:
> dt = gamma(T) ( dT + v(T) dX )
> dx = gamma(T) ( dX + v(T) dT )
> Since we are working on the worldline of I where X=0 and thus
> dX=0, we get:
> dt/dT = gamma(T)
> = 1 / sqrt( 1-v^2(T) )
> = 1 / sqrt( 1-tanh^2(A(T)) )
> = cosh(A(T))
> and
> dx/dT = gamma(T) * v(T)
> = v(T) / sqrt( 1-v^2(T) )
> = tanh(A(T)) / sqrt( 1-tanh^2(A(T)) )
> = sinh(A(T))
> Integrate beteen 0 and T:
> x(T) = Int{ 0 to T; sinh(A(T')) dT' }
> t(T) = Int{ 0 to T; cosh(A(T')) dT' }
> If possible, eliminate T to find the equation of the worldline x(t)
> of the accelerated observer A in the frame of the intertial
> observer I:
> x(t) = ...
> If possible, invert the expression for t(T) to find the proper time
> of A as a function of the coordinate time t of I:
> T(t) = ...
> and use it to find the velocity v(t) as a function of coordinate time t:
> v(t) = tanh(A(T(t)))
> Summary:
> ========
> x and t = coordinates of object as seen by inertial frame.
> a(T) = felt proper acceleration as function of proper time T.
> A(T) = Int{ 0 to T; a(T') dT' }
> v(T) = tanh(A(T)) if v(0) = 0
> dt/dT = cosh(A(T))
> dx/dT = sinh(A(T))
> x(T) = Int{ 0 to T; sinh(A(T')) dT' }
> t(T) = Int{ 0 to T; cosh(A(T')) dT' }
> eliminate T to find worldline equation x(t)
> Example:
> ========
> The rocket with constant acceleration of the FAQ:
> Take
> a(T) = a = contant
> So
> A(T) = Int{ 0 to T; a(T') dT' }
> = a T
> So
> v(T) = tanh(A(T))
> = tanh(a T)
> and
> x(T) = Int{ 0 to T; sinh(a T') dT' }
> = 1/a * ( cosh(a T) - 1 )
> and
> t(T) = Int{ 0 to T; cosh(A(T')) dT' }
> = 1/a * sinh(a T)
> Eliminate T:
> (x+1/a)^2 - t^2 = 1/a^2
> giving the hyperbola
> x(t) = 1/a [ sqrt( 1 + (a t)^2 ) - 1 ]
> Also proper time as a function of coordinate time:
> T(t) = 1/a * argsinh(a t)
> so
> v(t) = tanh( a T(t) )
> = tanh( argsinh(a t) )
> = a t / sqrt[ 1 + (a t)^2 ]
> Re-introduce c and we find:
> v(T) = c tanh(aT/c)
> x(T) = c^2/a [ cosh(aT/c) - 1 ]
> t(T) = c/a sinh(aT/c)
> gamma(T) = cosh(aT/c)
> and as functions of t:
> T(t) = c/a argsinh(at/c)
> and
> x(t) = c^2/a sqrt( 1 + (at/c)^2 ) (the hyperbola)
> v(t) = at / sqrt( 1 + (at/c)^2 )
> gamma(t) = sqrt( 1 + (at/c)^2 )
> which are the equations of the FAQ entry.
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