- #1

NoahsArk

Gold Member

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"Julius Ceasar was murdered on March 15 in the year 44 B.C. at the age of 55 approximately 2000 years ago. Is there some way we can use the laws of relativity to save his life?

Let Caesar's death be the reference event, labeled O: X

_{O }= 0, t

_{O}= 0. Event A is you reading this exercise. In the Earth frame the coordinates of Event A are X

_{A}= 0 light years, and t

_{A}= 2000 years. Simultaneous with event A in your frame, Starship Enterprise cruising the Andromeda galaxy sets off a firecracker: event B. The Enterprise moves along a straight line in space that connects it with Earth. Andromeda is 2 million light-years distant in our frame. Compared with this distance, you can neglect the orbit of the Earth around the Sun. Therefore, in our frame, event B has the coordinates X

_{B}= 2 x 10

^{6}light years, t

_{B}= 2000 years. Take Caesar's murder to be the reference event for the Enterprise too (X

^{1}

_{O}= 0, t

^{1}

_{O}).

a. How fast must the Enterprise be going in the Earth frame in order that Caesar's murder is happening NOW (that is, t

_{1}

_{B}) in the Enterprise rest frame? Under these circumstances is the Enterprise moving toward or away from Earth?

d. (I intentionally skipped parts b. and c. for now) Can the Enterprise firecracker explosion warn Caesar, thus changing the course of Earth history? Justify your answer."

For the answer to a. the book works out the answer this way:

"t

_{B}= vϒX

^{1}

_{B}

X

_{B}= ϒX

^{1}

_{B}

We do not yet know the value of X

^{1}

_{B}. Solve for v by diving the two sides of the first equation by the respective sides of the second equation. The unknown X

^{1}

_{B}drops out (along with ϒ), and we are left with v in terms of the known quantities t

_{B }and X

_{B}.

v = t

_{B }/X

_{B}= 2 x 10

^{3}years/ 2 x 10

^{6 }years = 10

^{-3}= .001

Since the velocity is a positive quantity, the Enterprise is moving away from Earth"

If first saw this problem about four months ago and have re-read it a few times since then, and I want to understand it as best as I can:

1) Is this problem saying that from the Enterprise frame, the exploding of the firecracker and Caesar's death have happened at the same time?

2) In the Earth frame, these events have happened 2,000 years apart, is the reason for the difference in time between these two events in the Enterprise frame the fact that, from the Enterprise frame, the location of Caesar being murdered compared with the firecracker's location is at the "front end" or "leading end" of the horizontal line between those two events? So Caesar's death happens much later (2000 years later) for someone in the Enterprise frame compared with someone in the Earth frame, and this is because of the leading clocks lag principle?

3) The leading clocks lag principle is built into the Lorentz transformations by the vϒX

^{1}term. While qualitatively I think I understand the leading clocks lag principle with illustrations of a beam of light needing to catch up to the front end of a rocket, I still don't quite understand quantitatively why the term vϒX

^{1}captures that principle.

4) With respect to question d., The book says it would be impossible to save Caesar's life because a signal connecting the two events would have to travel at infinite speed. Doesn't this depend on where the Enterprise is located at the moment the fire cracker explodes, and also how fast the Enterprise is moving? If we choose a little faster speed for the Enterprise, can't we create a situation where Caesar's death hasn't happened yet, and therefore the Enterprise can warn Caesar if the Enterprise is close enough to Earth?

5) Does it matter for this problem whether the firecracker exploded inside the Enterprise or outside of it?