Relativity. Spacetime question. finding times

[ttttrigg3r]

Homework Statement

Imagine that your boss is on the Earth-Pluto shuttle which travels
at a constant velocity of 0.60 straight between Earth and Pluto a
distance of 5.0 h in an inertial frame attached to the Sun. An
hour into the flight (according to your boss’s watch) your boss
sends a laser message to you on Earth, asking you to send a
wake-up call appropriately timed so that your boss can catch a
1-h nap (as measured on your boss’s watch). You immediately
reply with the wake-up call and an apology that the call is late,
claiming in your defense that the laws of physics prevented a
timely response.

How long has your boss slept (according to your boss’s watch)
by the time your message is received at the shuttle?

Homework Equations

s^2=t^2-d^2
s=spacetime, t=time, d=distance

The Attempt at a Solution

I am utterly stuck from the beginning. My first lead is that an hour into the flight means 1h as measured by the boss. Is that 1 hour considered space time or is that just "t" and you have to calculate the space time? Does the velocity of the boss, .6, have any play? Can anyone give me some helps/hints?
edit: I also know that the messages has velocity=c which is 1 in both the Sun's frame and the boss's frame. (Home and other)

 

[BruceW]

The convention is being used that c=1. (Unless maybe they made a mistake). It is a quite commonly used convention when people need to do relativistic calculations. are you familiar with this?

For example, they say a distance of 5 hours, they literally mean that the distance (in natural units), is equal to 5 hours. We know that in natural units, c=1, so to get back to SI units, you just need to multiply or divide by c such that you end up with units of distance. Give it a try, maybe practice with natural units a bit. http://en.wikipedia.org/wiki/Natural_units

 

[oli4]

Hi ttttrigg3r (not easy to write
Your boss' clock runs slow by a factor of 1.25
therefore when he sends the message, as far as you are concerned he has already traveled 1.25*u (u being his speed)
the message gets to you at the speed of light, so it will take 1.25*u/c=1.25*0.6=0.75 that you add to the already 1.25.
It means you get the message 2h after he left.
you send now a message right away which will have to catch up with the ship.
Can you calculate how long it will take (for you) to reach the ship ?
you divide that then by 1.25 and indeed, it will be too late for the 1h nap.

 

[ttttrigg3r]

Hi ttttrigg3r (not easy to write
Your boss' clock runs slow by a factor of 1.25
therefore when he sends the message, as far as you are concerned he has already traveled 1.25*u (u being his speed)
the message gets to you at the speed of light, so it will take 1.25*u/c=1.25*0.6=0.75 that you add to the already 1.25.
It means you get the message 2h after he left.
you send now a message right away which will have to catch up with the ship.
Can you calculate how long it will take (for you) to reach the ship ?
you divide that then by 1.25 and indeed, it will be too late for the 1h nap.

First question. Why a factor of 1.25? I understand that c=1 for all frames. I see what you're doing but I do not understand the reasoning behind your calculations.
I'm in the chapters with the metric eq. of space time: s^2=t^2-d^2 ,so I'm certain it has to do with calculating spacetime.

As for C=1 and SR units I've studied that. The messages are light speed, 1, therefore they travel 5 light-hours (distance) in 5 hours (time).

 

[oli4]

Hi trigger, 1.25 is just the result of applying the time dilation formula using the speed of the ship.
\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}
for the given speed (0.6*c), that gives you a time dilation factor of 1.25, that is, when your boss clock moves one hour forward, your clock moves one hour and 15 minutes forward.

 

[ttttrigg3r]

Hi trigger, 1.25 is just the result of applying the time dilation formula using the speed of the ship.
\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}
for the given speed (0.6*c), that gives you a time dilation factor of 1.25, that is, when your boss clock moves one hour forward, your clock moves one hour and 15 minutes forward.

Ok let me get this straight. I know that as velocity approaches the speed of light, time slows down for the person traveling that velocity. The Boss's lock measures 1h, but a clock at the sun observing his travel would actually measure 1.25h. So according to your reference frame, He would send a message back at 1.25h instead of 1h.
1.25h *.6= .75h traveled. This means that the message (him to you) has to travel .75h at the speed of c=1 to get back to you. So according to you, by the time the message gets to you, he is already at t= .75h+1.25h=2h.
So this 2h, is this 2h that he spent time traveled? or 2 light-hours traveled.

so using x=2+.6t and y=t, setting x=y (when your message,y, and his path,x, cross)
2+.6t=t, t=5h according to your watch.
According to his watch, his time would read t=5h/1.25 = 4h.
According to the boss, he sent out a message at t=1h and gets one back at t=4h, he would have been asleep for 3 hours by the time your message gets there.
Did I do everything right?
Does anyone know how to do it using the metric equation ?

 

[oli4]

Hi, when you get the laser message sent by your boss, your clock says 2 hours passed since he left (and somehow magically was already at constant velocity of 0.6c)
At this moment, he is at 2u (I call u his velocity) away from you.
you then send a laser message which goes at c, and must catch him going at ut.
so you want to solve 2u+ut=ct.
that will tell you how long it takes your message (according to you) to reach your boss (3h).
You then have to translate that into how much time for his clock, which is more than 1h.

 

[ttttrigg3r]

Hi, when you get the laser message sent by your boss, your clock says 2 hours passed since he left (and somehow magically was already at constant velocity of 0.6c)
At this moment, he is at 2u (I call u his velocity) away from you.
you then send a laser message which goes at c, and must catch him going at ut.
so you want to solve 2u+ut=ct.
that will tell you how long it takes your message (according to you) to reach your boss (3h).
You then have to translate that into how much time for his clock, which is more than 1h.

I'm a little lost here without units attached to numbers.
When you say 2u, is that 2h*.6 or does 2u have another meaning?

 

[oli4]

yes, sorry, it is 2h*u, it doesn't really matter, if it was 2 minutes, 2 seconds, 2 days, whatever, it would work the same, as long as you keep using the same unit of time all the time.

 

[ttttrigg3r]

yes, sorry, it is 2h*u, it doesn't really matter, if it was 2 minutes, 2 seconds, 2 days, whatever, it would work the same, as long as you keep using the same unit of time all the time.

I got it thanks a bunch.

 

[oli4]

Great, you are welcome, so did you find how much you boss overslept ?

 

[ttttrigg3r]

I got 2.4h as according to his clock. He slept for a total of 1.4 hours so your message can never reach him in time.

 

[oli4]

2.4h is the time (according to his clock) to reach him
but you also have to add the time it took for you to get his original message.
your answer took 2.4h, but your are answering already with a delay.
(remember that he sends the message at what for him is 1h of travel, but you get it at 2h since the travel started according to your clock), since your boss when he sends the message says "wake me up in 1h", you have to start counting at this moment.
So that would be 3+2, 5h for your clock -> 4h for his clock -> he overslept 3h.