- #1
alc95
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Homework Statement
An observer on the bridge of a spaceship is undergoing a proper acceleration a, so that the
observer’s worldline expressed in terms of the coordinates of an inertial reference frame S
is given by
[tex]t(\tau)=(c/a)\sinh(a\tau/c)[/tex][tex]x(\tau)=(c^2/a)\sinh(a\tau/c)[/tex]
(a) Draw on a spacetime diagram for the reference frame S the worldline of the observer
for tau > 0.
At the instant tau = 0 a radio beacon is dropped off by the observer. At the instant of being
dropped off, the beacon is at rest relative to the observer. At this instant the beacon also
starts emitting at regular intervals (according to a clock on the beacon) a series of very short
pulses towards the observer.
Consider the signal pulse emitted at a time tE in the reference frame S.
(b) Draw on the spacetime diagram of part (a) the worldline of the beacon. Hence show
that this emission event has the spacetime cordinates [tex] (ct_E,x_E)=(ct_E,c^2/a)[/tex]
(c) Show that the spacetime displacement of this emission event with respect to the origin
of S is given by the four vector
[tex]
\textbf{s}_E=ct_E\textbf{e}_0+(c^2/a)\textbf{e}_1
[/tex]
(d) Draw on the spacetime diagram the world line of the pulse emitted at time tE:
Homework Equations
[tex]
(\cosh(x))^2-(\sinh(x))^2=1
[/tex]
A four vector is defined as:
[tex]
\textbf{s}=\Delta ct\textbf{e}_0+\Delta x\textbf{e}_1+\Delta y\textbf{e}_2+\Delta z\textbf{e}_3
[/tex]
The Attempt at a Solution
(a)
Rearranging the given equations:
[tex]\sinh(a\tau/c)=at/c[/tex][tex]\cosh(a\tau/c)=ax/c^2[/tex]
Substituting into the relation between sinh and cosh:
[tex](ax/c^2)^2-(at/c)^2=1[/tex][tex](a^2x^2-a^2c^2t^2)/c^4=1[/tex][tex]x^2=c^2t^2+c^4/a^2[/tex] (1)
(b)
when tau=0:
[tex]t(0)=0[/tex][tex]x(0)=c^2/a[/tex]
this part I'm not so sure of, but here is what I did:
using the equation of motion [tex]x_f=x_i+ut+(1/2)at^2[/tex][tex]x = -½ac^2t^2 + c^2/a[/tex]
rearranging:
[tex]ct=\sqrt{(c^2/a-x)/(a/2)}[/tex] (2)
I'm not quite sure of my solutions for (a) and (b). Assuming I have the correct equations (1) and (2), all that remains is to plot them on the same plot of ct vs x?
(c)
substituting the spacetime coordinates [tex](ct_E,c^2/a)[/tex] into the definition of a four-vector:
[tex]
\textbf{s}_E=(ct_E-0)\textbf{e}_0+(c^2/a-0)\textbf{e}_1+(0-0)\textbf{e}_2+(0-0)\textbf{e}_3=ct_E\textbf{e}_0+c^2/a\textbf{e}_1
[/tex]
(d)
For this part I'm assuming that I have to draw a straight line of slope 1 or -1, intersecting the wordline of the beacon at an arbitrary time coordinate t_E, such that it also intersects the wordline of the observer. Is this correct?
Any help would be appreciated :)