Special Relativity and Lorentz Transformation

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Questions and Answers

What is the significance of the constant γ in the Lorentz transformation?

It ensures the transformation is linear.

It transforms space coordinates alone.

It represents the speed of light in vacuum.

It maintains the constancy of the speed of light across all frames. (correct)

In the context of the Lorentz transformation, if two events occur at the same location but are separated by a time interval, how is this perceived in a moving frame?

The time interval will appear shorter.

The time interval will appear longer. (correct)

There will be no perceived time interval.

They will appear at different locations.

If a spaceship moves at a velocity of v=0.8c, by how much does the time interval appear to stretch for events that are 4 seconds apart in the Earth frame?

2 seconds

8 seconds

6.67 seconds (correct)

4 seconds

What does the Lorentz transformation equations describe?

The change of time and space coordinates in different reference frames. Signup and view all the answers

During a light pulse's movement along the x-axis, what characteristic remains consistent across different reference frames?

Its speed. Signup and view all the answers

When spacecraft F' sends a radio signal lasting 1.2 seconds according to its clock, how would the communication officer on spacecraft F measure this interval?

Shorter than 1.2 seconds due to time dilation. Signup and view all the answers

Why is it necessary to modify the Galilean transformations for relativistic speeds?

To ensure the constancy of light speed. Signup and view all the answers

In the provided context, time dilation implies what effect on the perception of time in different frames?

Time appears slower in the moving frame. Signup and view all the answers

What is the main purpose of the Lorentz transformation?

To relate the space and time coordinates of observers moving at a constant velocity. Signup and view all the answers

Which of the following is NOT one of the assumptions in special relativity?

Principle of Gravity in all frames. Signup and view all the answers

How does the Lorentz factor (γ) affect time and space coordinates?

It modifies time and space to reflect constant light speed. Signup and view all the answers

What does the Galilean transformation fail to address in relation to the speed of light?

It does not maintain the constancy of the speed of light. Signup and view all the answers

At what relative velocity is Rico moving compared to Rica in this scenario?

0.8c Signup and view all the answers

Which relationship best describes the coordinates in the Lorentz transformation?

Linear relationship incorporating the Lorentz factor. Signup and view all the answers

In the context of the Lorentz transformation, what happens to time as the relative velocity approaches the speed of light?

Time dilation occurs, leading to slower passage of time. Signup and view all the answers

What condition must be satisfied for the Lorentz transformation to be applicable?

Observers must be moving at a constant velocity relative to each other. Signup and view all the answers

Study Notes

Lorentz Transformation

Relates space and time coordinates of two observers moving at a constant velocity relative to each other.
Introduced by Albert Einstein in 1905.
Fundamental to the theory of special relativity.

Assumptions of Special Relativity

Principle of Relativity: The laws of physics are the same in all inertial frames of reference.
Invariance of the Speed of Light: The speed of light in a vacuum, c, is the same for all observers, regardless of their motion relative to the light source.

Galilean Transformation

Describes the relationship between coordinates in two reference frames moving at a relative velocity v.
Not compliant with the constancy of the speed of light.

Lorentz Factor (γ)

A factor used to modify the Galilean transformation to ensure the speed of light remains constant in all frames.
Introduced by Hendrik Lorentz.
Depends on the relative velocity v and the speed of light c.
Defined as: γ = 1/√(1 - (v^2/c^2))

Deriving the Lorentz Transformation

Two events occur at points (x,t) and (x',t') in two frames of reference.
Postulate the constancy of the speed of light.
Modify the Galilean transformation to ensure the speed of light is constant.
Propose a general linear transformation for space and time:
- x' = γ(x - vt)
- t' = γ(t - vx/c^2)

Determining γ

For a light pulse moving along the x-axis, the speed of light is the same in both frames.
x = ct and x' = ct'
Equate these equations and solve for γ, resulting in the same equation as above: γ = 1/√(1 - (v^2/c^2)).

Final Lorentz Transformation Equations

x' = γ(x - vt)
t' = γ(t - vx/c^2)
y' = y
z' = z
Describe how space and time change between reference frames moving relative to each other at a constant velocity v.
Ensure the speed of light remains constant in all inertial frames.

Sample Problem 1: Time Dilation

Spaceship moves at v = 0.8c relative to Earth.
Two events occur at the same location in Earth's frame, separated by 4 seconds.
Calculate the time interval between these events as observed from the spaceship's frame.

Solution 1

Apply the time dilation equation t' = γt, where γ = 1/√(1 - (v^2/c^2)) and t = 4 seconds.
Time interval between the two events as observed from the spaceship is approximately 6.67 seconds.
Demonstrates time dilation: time passes more slowly in the spaceship's frame compared to Earth's frame.

Sample Problem 2

Spacecraft F' is at rest and heading towards Alpha Centauri.
Spacecraft F passes F' at a relative speed of c/2.
F' sends a radio signal lasting 1.2 seconds according to its clock.
Use the Lorentz transformation to find the signal's time interval measured by F's communication officer.

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Description

This quiz explores the Lorentz transformation, its assumptions, and the Galilean transformation. It delves into key principles introduced by Einstein and Lorentz that are fundamental to the understanding of special relativity. Test your knowledge of relativity concepts and their implications for physics.