# Deriving Linear Transformations - Special Relativity

• erok81
In summary, the conversation is about linear transformations and reference frames, specifically in the context of a stationary observer and another object moving at the speed of light. The boundary conditions, transformations and final results are discussed, with the speaker expressing a desire to understand the method for deriving these results rather than just following notes. They also ask what class typically covers linear transformations.

#### erok81

I wasn't sure if this counted as intro physics. Feel free to move if I have it in the wrong place.

## Homework Statement

In class we learned some linear transformations where we have a stationary observer and another moving near the speed of light.

Describing the reference frames:
s' -> x'=u't'
s-> x=ut

Boundary conditions:
x=x'=t=t'=0

Therefore at the origin of x we have x=ut=0, x'=u't', x=-vt'

Using these for the transformations:
x'=Ax+Bt
t'=Cx+Dt

## Homework Equations

Final transforms:

$$x'=\gamma(x-vt)$$

$$t'=\gamma(\frac{-vx}{c^2}+t)$$

$$x=\gamma(x'+vt')$$

$$t=\gamma(\frac{vx'}{c^2}+t')$$

## The Attempt at a Solution

So I can solve these as long as I have my notes that I can follow so I know what to solve for and when. My professor ended up solving for the A,B,C,D first then subbing in gamma etc to arrive at the final results.

My questions is: I'd love to be able to derive these on my own. My problem I don't know what I am trying to solve for or the point of what I am solving for. Point may not be the best word - I understand the end result and know how it is used, I just don't understand the steps/method to get there. (the algebra I get, not the method.

Would any more information help? I'm not sure if what I am saying makes total sense. :)

Ok...I finally figured this out and can do it.

I don't think I've really done linear transformations. What class are these usually taught in?

## 1. How is special relativity related to linear transformations?

Special relativity is a theory in physics that describes how objects behave at high speeds, close to the speed of light. Linear transformations are mathematical operations that describe how coordinates in one reference frame relate to coordinates in another reference frame. Special relativity uses linear transformations to explain how the measurements of space and time change for observers in different reference frames.

## 2. What is a Lorentz transformation?

A Lorentz transformation is a type of linear transformation that is used in special relativity. It describes how the measurements of space and time change for observers in different reference frames that are moving at constant speeds relative to each other. It was first proposed by Hendrik Lorentz in 1904 and was later incorporated into Albert Einstein's theory of special relativity.

## 3. How are the equations of special relativity derived using linear transformations?

The equations of special relativity, such as the time dilation and length contraction equations, can be derived using linear transformations. This involves using the Lorentz transformation to transform the coordinates of an event from one reference frame to another. By equating the transformed coordinates to the original coordinates, the equations of special relativity can be derived.

## 4. Can linear transformations be used to describe the effects of gravity on space and time?

No, linear transformations cannot fully describe the effects of gravity on space and time. This is because gravity is described by the theory of general relativity, which uses a more complex mathematical framework called curved spacetime. While linear transformations are still used in the theory of general relativity, they cannot fully capture the effects of gravity like they do in special relativity.

## 5. Are there any practical applications of understanding linear transformations in special relativity?

Yes, understanding linear transformations in special relativity has many practical applications. For example, it is crucial for accurately predicting the behavior of particles in particle accelerators, GPS technology, and other high-speed technologies. It also helps us understand the behavior of objects at high speeds, such as in space travel or high-speed transportation systems.