Intro to Relativity

Speed of Light

raymo — Fri, 12 Apr 2019 19:19:39 +0000

Maxwell’s theory of electromagnetism contains 4 equations not required by IB.

Definition: An invariant quantity or property of a system remains unchanged as the system undergoes transformations.

One of the important conclusions of Maxwell’s theory of electromagnetism is that the speed of light is invariant in vacuum. This also means that the value for c is independent of any relative motion between source and observer.

The table below summarizes the understandings of 4 renowned physicists on addition close to the speed of light:

Galileo & Newton	Maxwell & Einstein
0.5c_source + c_light = 1.5c_light	sum of 0.5c_source and c_light gives c_light
0.9c_source + c_light = 1.9c_light	sum of 0.9c_source and c_light gives c_light
-0.7c_source + c_light = 0.3c_light	sum of -0.7c_source and c_light gives c_light

Galileo and Newton

raymo — Fri, 12 Apr 2019 16:40:40 +0000

Spacetime coordinates in one IRF can be used to calculate spacetime coordinates in another IRF, when given the relative velocity of IRF2 to IRF1.

S’ is moving to the right at speed v with respect to S

One can use the vector displacement method to write Galilean coordinate transformation equations, simplified to one dimension and uniform relative velocity. For example, for two different coordinate systems $S$ and $S^{\prime}$ , the vector displacement equation would be $x_1={x_{1}}^\prime+v{\Delta}t$ .

Note: There is no movement in the y-axis, the offset is only for clarity.

In this diagram, $x$ and $x^{\prime}$ denote position in coordinate systems $S$ and $S^{\prime}$ , and the measured distance (length along the x-axis) can be denoted with the equation ${\Delta}x=x_2-x_1$ . Time in the systems is denoted as $t$ and $t^{\prime}$ , and an interval of time can be denoted as ${\Delta}t=t_2-t_1$ .

Galilean stated that time is consistent, and $t=t^{\prime}$ , ${\Delta}t={\Delta}t^{\prime}$ for any reference frame.

When looking at relative motion, the Galilean law of velocity addition is $u=u^{\prime}+v$ . Note that all these quantities are vectors!

In Galilean transformations, acceleration is invariant. However, in Einstein’s relativity, acceleration is not invariant.

Newton’s postulates of space and time state that absolute space and time are independent from reference frames. However, Newton describes motion as always being relative.

Definition: Postulates are basic assumptions without proof used to make rules for scientific theories, that are neither right nor wrong.

Newton’s postulates also state that IRFs are identical concerning the laws of physics, and there is no experimental evidence to claim that any IRF is truly at rest, and measurements of space and time are made independent of any given frame of reference.

Galileo, Newton and Einstein agree all uniform motion is relative.

Reference Frames

raymo — Fri, 12 Apr 2019 16:29:10 +0000

To understand relativity, one must first understand reference frames.

All measurements in physics are taken from the origin. 3D Cartesian coordinate space has $x, y, z$ axes at right angles to each other.

Any event can be represented by observer with x, y, z, t (time)

There are many possible reference frames, and all measurements are made from a frame of reference.

Reference frames

S

and

S^{\prime}

relate to each other by

x^{\prime}=x-{\Delta}x

Inertial reference frames (IRF) are defined as having no acceleration. Though Earth is not a true IRF, its acceleration is so small that it can be ignored.

Galileo and Einstein agree that all IRFs and their laws of physics are identical.

Reference frames can differ from the origin via rotation or translation. Alternatively, one frame of reference can be moving relative to another.

Various different reference frames. In the bottom right diagram,

S^{\prime}

is moving at a speed

v

with respect to

S