Reference Frames

In order to understand frames of reference, we must understand the concept of spacetime. Spacetime refers to the way in which space and time together determine a set of coordinates in the same way that 2D graphs represent coordinates to represent data. Space and time cannot be considered separate when understanding special relativity. We must grasp that we move through 4 dimensions, the standard 3 Euclidean space dimensions and time.

Event: a set of spacetime coordinates defining a point in spacetime.

Reference Frame: a way to represent the position of an object in spacetime, often through the use of a Cartesian diagram with 3 axes that are orthogonal to one another and an origin. 

In order to explain certain effects in some reference frames, forces must be invented (like the centrifugal force). To avoid this, we use inertial frames of reference which is a frame that is not accelerated (all frames move at a constant velocity compared to one another), meaning that Newton's first law applies. Example: a spaceship with engines off in deep space that is unaffected by gravity. 

Postulates of Special Relativity

There are infinite inertial frames of reference in the universe. IB describes 2 that ways that we can move between them.

Translation: movement between two offset frames of references.

Boost: a change in coordinates of two frames of reference that are moving with constant velocity relative to another. This means that the distance between the origins of the frame of reference must be changing by a relative velocity v every second (t), allowing us to arrive at the familiar equation v x t to describe the distance between them. 

Galilean Transformations: equations that link reference frames by relative velocities.

Two Galilean Transformations that represent the relative state of two reference frames that were coincident at t = 0 are:

x' = x - vt, representing the relative location

u' = u - v, representing the relative velocity

While the above equations relate to a Newtonian view of the universe (space and time are fixed and absolute), Einstein arrived at different postulates. 

1. The laws of physics are the same in all inertial reference frames
2. The speed of light relative to any observer in an Inertial frame is not dependent on the motion of the source of light relative to the observer. The speed of light is absolute.

As a result, individuals that have a velocity relative to a stationary observer will experience time moving more quickly (think of the penlight and mirror thought experiment).

Time Dilation: time observed by an individual outside of the inertial frame of reference of an event will experience the time passing more slowly than the observer of the proper time interval of an event. 

Lorentz Transformation

Foundational to Lorentz transformations, the Lorentz factor represents how time, length, and other physical properties change due to motion. 

Calculating the Lorentz factor: γ = 1/sqrt(1 - v²/c²)

All Galilean Transformations will be multiplied by the Lorentz coefficient to determine the Lorentz Transformations that relate to special relativity. 

Spacetime Diagrams and Worldlines

View count: 60