Question

The Federation space cruiser Execrable is floating in Federation territory at rest rel-

ative to the border of Klingon space, which is 6.0 min away in the +x direction.

Suddenly, a Klingon warship flies past the cruiser in the direction of the border at

speed B = 3/5. Call this event A and let it define time zero in both the Klingon and

cruiser reference frames. At tB = 5.0 min according to the cruiser clock, the Klingons

emit a parting phasor blast (event B) that travels at the speed of light back to the

cruiser. The phasor blast hits the cruiser and disables it (event C), and shortly later

(according to cruiser radar measurements) the Klingons cross the border into Klingon

territory (event D).

a. Draw a 2-system space-time diagram of the situation, taking the cruiser to de_ne

the stationary frame and the Klingon to define the moving frame. Draw and

label the world-lines of the cruiser, the Klingon territory boundry, the Klingon

warship, and the phasor blast. Draw and label events A,B,C,D as points on

your diagram.

b. When does the phasor blast hit, and when do the Klingons pass into their own territory, according to clocks in the cruiser's frame? Read these events directly

from the diagram.

c. The Klingon-Federation Treaty states that it is illegal for a Klingon ship in Fed-

eration territory to damage Federation property. When the case comes up in

Interstellar court, the Klingons claim that they are within the letter of the law,

since according to measurements made in their reference frame, the damage to

the Execrable occurred after they crossed back into Klingon territory-hence

they were not in Federation territory at the time. Did event C really happen

after event D in the Klingon's frame? Check your answer with the Lorentz Transformation equations.

Answer 1

Solution

Part a)

We need to start to calculate the value of so:

In this problem,

Therefore, replacing this data into equation (1) we have:

Solving we obtain:

Now, if

The slope of the t' axis is:

Now, the scale of the t'-axis is obtained using:

Isolating :

This means that, for we have that:

Therefore, the clock of klingon warship is 4 min. Now with all this data we can build the 2-system space-time diagram of the situation:

Part b)

When does the phasor blast hit according to clocks in the cruiser's frame?

Based on the diagram, the phasor blast hits at t=8 min according to clocks in the cruiser's frame.

When do the Klingons pass into their own territory according to clocks in the cruiser's frame?

Based on the diagram, Klingons pass into their own territory at t=10 min according to clocks in the cruiser's frame.

Part c)

The Lorentz transformations equations are defined as:

Now, for event C the clock of Cruiser give:

Replacing this data into equation (3):

solving we obtain:

Simillarly, for event D the clock of Cruiser give:

Replacing this data into equation (3) we have:

solving we obtain:

Therefore, from the Klingon's frame, the event c occurs after event d