Lorentz transformations (2nd year relativity)

Also, please clarify what the initial event is that you're trying to express with a spacetime point.In summary, the conversation discusses a light signal being sent from the origin of a system K to a specific point, and the question of when the signal is received. The first part is easily solved by dividing the distance by the speed of light. The second part involves using the Lorentz transformations to find the values in a different frame of reference. The formula used is more accurately written with a delta indicating the difference between two spacetime events. The incorrect attempts are using t=0 and plugging in the wrong values for the transformation equations.
  • #1
affans
11
0

Homework Statement



A light signal is sent from the origin of a system K at t = 0 to the point x = 1 m, y = 8 m, z = 13 m. a) At what time t is the signal received?
b) Find ( x', y', z', t' ) for the receipt of the signal in a frame K' that is moving along the x-axis of K at a speed of 0.6c.

Homework Equations



The lorentz transformations:

x' = [tex]\gamma * (x - vt)[/tex]
y' = y
z' = z


The Attempt at a Solution



Part a was easy. I got the right answer. I just took the length of the vector given by the co-ordinates and divided by the speed of light. The answer is [tex] 5 * 10^8 [/tex] I am having trouble with part b.

Ofcourse, y' and z' were easy to get. t' (i had 3 tries, and i used them all) so I lost a mark there. I have one try left on x'.

Using the equation, we first have to solve for [tex]\gamma[/tex]. Plugging the numbers into the equation for gamma:
[tex] \frac{1}{\sqrt{1-(v/c)^2}}\; \text{yields} \; 1.25[/tex]

Then using the lorentz transformation I have the following eqn:
x' = [tex]\gamma[/tex] (x - vt) . Plugging in the numbers yeilds
x' = 1.25(1 - 0.6 * c *(5E-8))

I get 10 as the answer. It is wrong. I also thought t = 0 could work since that's when the event happened. But the answer 1.25 is also wrong.

My third attempt yielded 11.25m however, I am scared to submit it. If anyone can please verify my number for me.
 
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  • #2
The Lorentz transformations are more accurately written
[tex]\Delta t' = \gamma(\Delta t - v\Delta x/c^2)[/tex]
[tex]\Delta x' = \gamma(\Delta x - v\Delta t)[/tex]
The [itex]\Delta[/itex] indicates that the numbers to be plugged in should be the difference between two spacetime events. So putting in t=0 is a mistake that you should not make again.

Can you show your calculations for the other two attempts?
 

Related to Lorentz transformations (2nd year relativity)

1. What are Lorentz transformations?

Lorentz transformations are a set of equations that describe how the measurements of space and time change between two reference frames that are moving relative to each other at constant velocity. They were developed by Hendrik Lorentz in the late 19th century and were later incorporated into Albert Einstein's theory of special relativity.

2. Why are Lorentz transformations important?

Lorentz transformations are important because they provide a mathematical framework for understanding the effects of time and space dilation, as well as length contraction, which are fundamental principles of special relativity. They also allow us to make accurate predictions and calculations in situations where objects are moving at speeds close to the speed of light.

3. How do Lorentz transformations differ from Galilean transformations?

Lorentz transformations differ from Galilean transformations in that they take into account the constancy of the speed of light, whereas Galilean transformations assume that the speed of light is infinite. This difference is what allows for the equations to accurately describe the effects of relativity at high speeds, where the assumptions made by Galileo and Newton's laws of motion break down.

4. What is the Lorentz factor?

The Lorentz factor, denoted by the symbol gamma (γ), is a term that appears in the equations of Lorentz transformations and is defined as 1/√(1 − v²/c²), where v is the relative velocity between the two reference frames and c is the speed of light. It is a key factor in calculating the effects of time dilation and length contraction.

5. Are Lorentz transformations only applicable to objects moving at speeds close to the speed of light?

No, Lorentz transformations can be applied to any situation involving two reference frames moving relative to each other at a constant velocity, regardless of the speed. However, the effects of time dilation and length contraction become more significant as the relative velocity approaches the speed of light, making these transformations particularly useful in the realm of high-speed physics.

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