Eroica

2004-Mar-31, 01:56 PM

On another thread (http://www.badastronomy.com/phpBB/viewtopic.php?p=228155#228155) we were looking at the general equations for a ship that is travelling through space with constant jerk (jerk = rate of change of acceleration), and someone wondered what they would be like if relativistic considerations were included in the calculations. Here's how far I got.

There are two frames of reference: an inertial one that is at rest, and the ship itself. Inertial time and displacement are t and s respectively. The corresponding dilated time and contracted displacement as measured by the crew of the ship are t' and s'. At time t = 0, let us assume that s, v and a = 0. The ship's jerk = j, a constant.

The equations we derived are (all parameters relative to the inertial frame):

a = jt .... Equation 1

v = ½jt² .... Equation 2

s = (1/6)jt³ .... Equation 3

In order to calculate t' and s', I used the gamma factor = √[1-(v²/c²)]. If you plot the gamma factor against undilated time, the area under the graph is the corresponding dilated time. In other words:

t' = ∫γdt

Similarly:

s' = ∫γds

Let's take time first. Starting with Equation 2:

v = ½jt²

=> v² = ¼j²t^4

=> v²/c² = (j²t^4)/4c²

=> 1-(v²/c²) = (4c²-j²t^4)/(4c²)

=> γ = (1/2c)√[4c²-j²t^4]

=> t' = (1/2c)∫√[4c²-j²t^4]dt

Trouble is, this isn't a standard integral and I haven't been able to solve it yet. Any ideas?

As for s', it seems to be even more complex:

s = (1/6)jt³

=> ds = ½jt²dt

=> s' = (¼c)∫√[4c²-j²t^4]t²dt

with the same problem as before. ](*,)

[Edit: The last equation has an error. it should read:

s' = (j/4c)∫√[4c²-j²t^4]t²dt

Thank you, Gsquare, for highlighting this error.]

[Second Edit. I changed the notation to clarify things.]

There are two frames of reference: an inertial one that is at rest, and the ship itself. Inertial time and displacement are t and s respectively. The corresponding dilated time and contracted displacement as measured by the crew of the ship are t' and s'. At time t = 0, let us assume that s, v and a = 0. The ship's jerk = j, a constant.

The equations we derived are (all parameters relative to the inertial frame):

a = jt .... Equation 1

v = ½jt² .... Equation 2

s = (1/6)jt³ .... Equation 3

In order to calculate t' and s', I used the gamma factor = √[1-(v²/c²)]. If you plot the gamma factor against undilated time, the area under the graph is the corresponding dilated time. In other words:

t' = ∫γdt

Similarly:

s' = ∫γds

Let's take time first. Starting with Equation 2:

v = ½jt²

=> v² = ¼j²t^4

=> v²/c² = (j²t^4)/4c²

=> 1-(v²/c²) = (4c²-j²t^4)/(4c²)

=> γ = (1/2c)√[4c²-j²t^4]

=> t' = (1/2c)∫√[4c²-j²t^4]dt

Trouble is, this isn't a standard integral and I haven't been able to solve it yet. Any ideas?

As for s', it seems to be even more complex:

s = (1/6)jt³

=> ds = ½jt²dt

=> s' = (¼c)∫√[4c²-j²t^4]t²dt

with the same problem as before. ](*,)

[Edit: The last equation has an error. it should read:

s' = (j/4c)∫√[4c²-j²t^4]t²dt

Thank you, Gsquare, for highlighting this error.]

[Second Edit. I changed the notation to clarify things.]