Lorentz Transforms + Relativity of Simultaneity

[Dakar]

Hi,

I'm studying special relativity this semester and stumbled onto a couple of problems. These have nothing to do with my homework or anything, they're just a couple of things I thought up. Since I have little contact with my teacher (it's an online course), it'd probably be a good idea to put them here. I'm sorry if this is the wrong forum, but again this isn't homework but questions on the topic.

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Person A sets up, in the lab frame, a string of collinear firecrackers evenly spaced one meter apart. Person B is in the rocket frame, traveling parallel to the firecrackers at speed .5*c. A sets off the firecrackers so they all go off the same time, her frame. To B, the distances every two consecutive firecrackers are lorentz contracted to 1*sqrt(1 - .5^2) .866 meters. So after the first flash of light reaches him, the next one should arrive after 1/(1 - .5) 1.73 seconds. Is this correct? If not, what did I accidentally add or leave out? Does this demonstrate the relativity of simultaneity?

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We'll define A to be gamma. We have a lab frame (with coordinates x
and t), and a rocket frame (with x' and t'). In the lab frame, there
are two points, x1 and x2. The distance between them is dx. In the
rocket frame, the two points are at locations A(x1 - vt) and A(x2 -
vt). Therefore, in the rocket frame dx' is A*dx.

Now, we place a bar between those two points of length dx. It Lorentz
contracts to length dx/A as observed in rocket frame. So the space
increased to A*dx and length of the bar in that space decreased to
1/A * dx. Is this all correct, or am I missing something here?

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Thanks!

Dakar

 

[Histspec]

Person A sets up, in the lab frame, a string of collinear firecrackers evenly spaced one meter apart. Person B is in the rocket frame, traveling parallel to the firecrackers at speed .5*c. A sets off the firecrackers so they all go off the same time, her frame. To B, the distances every two consecutive firecrackers are lorentz contracted to 1*sqrt(1 - .5^2) .866 meters. So after the first flash of light reaches him, the next one should arrive after 1/(1 - .5) 1.73 seconds. Is this correct? If not, what did I accidentally add or leave out? Does this demonstrate the relativity of simultaneity?

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Well, you have to add the relativity of simultaneity, i.e. the time transformation [tex]\gamma(t-vx/c^2)[/tex].

For example, we have 2 firecrackers starting simultaneously at the coordinates x1=0, t1=0 and x2=1m, t2=0.

What are the coordinates (both space and time) of those two events in the rocketframe?

Regards,

 

[Entropia]

ai

Hi Dakarai,

Your understanding of the Lorentz transforms and relativity of simultaneity seems to be correct. In the first scenario, you have correctly calculated the time and distance intervals between the firecrackers as observed by person B in the rocket frame. This does demonstrate the relativity of simultaneity, as the events that are simultaneous in one frame (A setting off the firecrackers) are not simultaneous in the other frame (B observing the firecrackers going off at different times). This is a key concept in special relativity and shows that the perception of time and space is relative to the observer's frame of reference.

In the second scenario, your understanding of the Lorentz contraction is also correct. As you mentioned, the bar in the lab frame has a length of dx, but in the rocket frame it is Lorentz contracted to a length of 1/A * dx. This shows that the length of an object is also relative to the observer's frame of reference and is affected by their relative velocity.

Overall, it seems like you have a good understanding of these concepts. Keep up the good work! If you have any further questions or need clarification on anything, don't hesitate to ask. Good luck with your studies!