Special Relativity

Description of motion from a special set of perspectives.

The Special Theory of Relativity

What is relativity?

Intro

The Special Theory of Relativity

What is relativity?

intro

How to change perspectives?

Relativity is concerned with the description of physical phenomena from different perspectives.

What physical quantities/laws are perspective dependent?

What physical quantities/laws are perspective independent?

The Special Theory of Relativity

What is relativity?

Relative and absolute quantities from a stationary perspective

Position is relative

Time is absolute

yet

Velocity is absolute

Acceleration is absolute

Displacement is absolute

= ((\red{x_2}\purple{-\Delta x})-(\red{x_1}\purple{-\Delta x}),...)

\vec{r}_1 = (x_1,y_1)

\vec{r}_2 = (x_2,y_2)

\vec{r}_1' = (x_1',y_1')

\vec{r}_2' = (x_2',y_2')

= (\red{x_1}\purple{-\Delta x},\red{y_1}\purple{-\Delta y})

= (\red{x_2}\purple{-\Delta x},\red{y_2}\purple{-\Delta y})

\Delta \vec{r} = \vec{r}_2 - \vec{r}_1

= (x_2-x_1,y_2-y_1)

\Delta \vec{r}' = \vec{r}_2' - \vec{r}_1'

= (x_2'-x_1',y_2'-y_1')

= (\red{x_2-x_1},\red{y_2-y_1})

\red{\vec{r}}\purple{\ne} \vec{r'}

\red{\Delta \vec{r}}\purple{=} \Delta \vec{r'}

The Special Theory of Relativity

What is relativity?

Relative and absolute quantities from an inertial perspective

K: spectator view

1: how it started

2: how it's going

K': TV view

The Special Theory of Relativity

What is relativity?

Relative and absolute quantities from an inertial perspective

Velocity is relative

Acceleration is absolute

Position is relative

Displacement is relative

The Special Theory of Relativity

What is relativity?

Relative and absolute quantities from an inertial perspective

\ \text{or is}\ c' = c-v\ ?\

\ \text{is}\ c' = c =\frac{1}{\sqrt{{\epsilon_0}{\mu_0}}} ?\

The Special Theory of Relativity

Meanwhile

Michelson (&Morley) 's Experiment to measure Ether drift

The Special Theory of Relativity

Meanwhile

Michelson's Experiment to measure Ether drift

Assuming ether drift as indicated on figure, and relative velocity given by Galilean transformation,

If you rotate the apparatus by 90 degrees, the presumed drift is now along the other arm of the interferometer.

implying a different value of the delay (corresponding to switching l1 and l2 in the above expression)

Michelson cleverly designed an interferometer, where the interference pattern depends on that time delay.

And he saw none!

we find that there will be a time difference between light traveling A-D-A and A-C-A given by:

i.e. there will be a shift in the interference pattern...

The Special Theory of Relativity

Meanwhile

Michelson's Experiment

Lorentz and FitzGerald, in an attempt to explain the null result, pointed out,

(in an ad-hoc way)

that it is possible to get

l_1 \rightarrow \ l_1 \sqrt{1-v^2/c^2}

That is to say, the null result would be justified if the length in the direction of the drift somehow contracted by a factor of

\sqrt{1-v^2/c^2}

The Special Theory of Relativity

Einstein's postulates

The laws of physics are the same in all intertial systems. There is no way to detect absolute motion, and no preferred intertial system exits.

Observers in all intertial systems measure the same value for the speed of light in vacuum.

Encompassing Electrodynamics

Einstein's Special Relativity

\blue{x'} = \frac{\red{x}-\green{v}\red{t}}{\sqrt{1-\frac{\green{v^2}}{c^2}}}

\blue{y'} = \red{y}

\blue{z'} = \red{z}

\blue{t'} = \frac{\red{t}-\frac{\green{v}\red{x}}{c^2}}{\sqrt{1-\frac{\green{v^2}}{c^2}}}

\blue{u'} = \frac{\red{u}-\green{v}\red{t}}{{1-\frac{\red{u}\green{v}}{c^2}}}

Lorentz Transformation

The Special Theory of Relativity

Lorentz Transform

Einstein's Special Relativity

\blue{x'} = \frac{\red{x}-\green{v}\red{t}}{\sqrt{1-\frac{\green{v^2}}{c^2}}}

\blue{x'} = \green{\gamma} (\red{x}-\green{v}\red{t})

\blue{x'} = \green{\gamma} (\red{x}-\green{\beta}\ c\red{t})

Expressing the relative speed of the perspectives as a fraction of the speed of light:

\green{\beta}=\frac{\green{v}}{c}\;,

and defining the relativistic factor:

\green{\gamma}=\frac{1}{\sqrt{1-\green{\beta^2}}}

\blue{t'} = \frac{\red{t}-\frac{\green{v}\red{x}}{c^2}}{\sqrt{1-\frac{\green{v^2}}{c^2}}}

c \blue{t'} = \frac{c \red{t}-\frac{\green{v}\red{x}}{c}}{\sqrt{1-\frac{\green{v^2}}{c^2}}}

c \blue{t'} = \green{\gamma} (c \red{t}-\green{\beta}\red{x})

Then the Lorentz Transformation can be written as:

The Special Theory of Relativity

Lorentz Transform - note 1

Einstein's Special Relativity

\blue{x'} = \green{\gamma} (\red{x}-\green{\beta}\ \red{ct})

\green{\beta}=\frac{\green{v}}{c}\;,

\green{\gamma}=\frac{1}{\sqrt{1-\green{\beta^2}}}

\blue{ct'} = \green{\gamma} (\red{ct}-\green{\beta}\ \red{x})

The Lorentz Transformation

\red{(x,ct)} \rightarrow \blue{(x',ct')}

In our universe, motion

mixes

space and time!

\blue{\left(\begin{matrix}x'\\ ct'\end{matrix}\right)} = \green{\gamma} {\left(\begin{matrix}1&-\green{\beta}\\ -\green{\beta}&1\end{matrix}\right)} \red{\left(\begin{matrix}x\\ ct\end{matrix}\right)}

The Special Theory of Relativity

Lorentz Transform - note 2

Einstein's Special Relativity

\blue{x'} = \green{\gamma} (\red{x}-\green{\beta}\ \red{ct})

\green{\beta}=\frac{\green{v}}{c}\;,

\green{\gamma}=\frac{1}{\sqrt{1-\green{\beta^2}}}

\blue{ct'} = \green{\gamma} (\red{ct}-\green{\beta}\ \red{x})

The Lorentz Transformation

\red{(x,ct)} \rightarrow \blue{(x',ct')}

\red{x} = \green{\gamma} (\blue{x'}+\green{\beta}\ \blue{ct'})

\green{\beta}=\frac{\green{v}}{c}\;,

\green{\gamma}=\frac{1}{\sqrt{1-\green{\beta^2}}}

\red{ct} = \green{\gamma} ( \blue{ct'}+\green{\beta}\ \blue{x'})

The inverse Lorentz Transformation

\red{(x,ct)} \leftarrow \blue{(x',ct')}

The Special Theory of Relativity

Lorentz Transform - note 3

Einstein's Special Relativity

\blue{x'} = \green{\gamma} (\red{x}-\green{\beta}\ \red{ct})

\green{\beta}=\frac{\green{v}}{c}\;,

\green{\gamma}=\frac{1}{\sqrt{1-\green{\beta^2}}}

\blue{ct'} = \green{\gamma} (\red{ct}-\green{\beta}\ \red{x})

The Lorentz Transformation

\red{(x,ct)} \rightarrow \blue{(x',ct')}

Taylor expansion of the relativistic factor at small relative velocities

The Special Theory of Relativity

Lorentz Transform - note 3

Einstein's Special Relativity

\blue{x'} = \green{\gamma} (\red{x}-\green{\beta}\ \red{ct})

\green{\beta}=\frac{\green{v}}{c}\;,

\green{\gamma}=\frac{1}{\sqrt{1-\green{\beta^2}}}

\blue{ct'} = \green{\gamma} (\red{ct}-\green{\beta}\ \red{x})

The Lorentz Transformation

\red{(x,ct)} \rightarrow \blue{(x',ct')}

At small velocities:

\green{\beta}={\green{v}}/{c} \approx 0

\green{\gamma}=\frac{1}{\sqrt{1-\green{\beta^2}}} \approx 1

\blue{x'} = \green{\gamma} (\red{x}-\green{\beta}\ \red{ct}) \approx \red{x}-\green{v}\ \red{t}

\blue{ct'} = \green{\gamma} (\red{ct}-\green{\beta}\ \red{x}) \approx \red{ct}

The Lorentz Transformation

reduces to

the Galilean Transformation.

The Special Theory of Relativity

Lorentz Transform - summary of the story so far

Einstein's Special Relativity

\blue{x'} = \green{\gamma} (\red{x}-\green{\beta}\ \red{ct})

\green{\beta}=\frac{\green{v}}{c}\;,

\green{\gamma}=\frac{1}{\sqrt{1-\green{\beta^2}}}

\blue{ct'} = \green{\gamma} (\red{ct}-\green{\beta}\ \red{x})

The Lorentz Transformation

\red{(x,ct)} \rightarrow \blue{(x',ct')}

Different inertial perspectives

\red{K}, \blue{K'}, \gray{K''}, ...

will assign to the same event

a different set of Spacetime coordinates

\red{e_1=(x_1, y_1 , z_1,ct_1)},

\text{event 1}

\blue{e_1'=(x_1',y_1',z_1',ct_1')},

\gray{e_1''=(x_1'',y_1'',z_1'',ct_1'')}, ...

To transform between these sets, you need to know the relative velocity of the frames.

The Special Theory of Relativity

Einstein's Special Relativity

Example 1

What are the spacetime coordinates of these events in an inertial system K' that is moving at 25% the speed of light in the +x direction relative to K.

The Special Theory of Relativity

Einstein's Special Relativity

Example 1 - solution

What are the spacetime coordinates of these events in an inertial system K' that is moving at 25% the speed of light in the +x direction relative to K.

The Special Theory of Relativity

Einstein's Special Relativity

The relativity of simultaneity

Suppose that you would assign the same time to two distinct events happening at different places from your perspective.

According to a perspective moving relative to yours, would the two events be assigned the same time as well?

t_1 = t_2

x_1 \ne x_2

Suppose that you would assign the same time to two distinct events happening at different places from your perspective.

t_1' = t_2' ?

According to a perspective moving relative to yours, would the two events be assigned the same time as well?

The Special Theory of Relativity

Einstein's Special Relativity

Example - 2

The Special Theory of Relativity

Einstein's Special Relativity

Example 2 - Solution

The Special Theory of Relativity

Einstein's Special Relativity

Example 3

The Special Theory of Relativity

Einstein's Special Relativity

Example 3 - Solution

The Special Theory of Relativity

Einstein's Special Relativity

Invariant quantities

If space is relative and time is relative, is there any spacetime quantity that is invariant under the Lorentz transformation?

The spacetime interval

\Delta s^2 = \Delta x^2 - c^2\Delta t^2

Lightlike separation

Space-like

separation

timelike

separation

\Delta s^2 = 0

\Delta s^2 < 0

\Delta s^2 > 0

No causal connection!

The Special Theory of Relativity

Einstein's Special Relativity

Example - Invariant quantities

What are the spacetime coordinates of these events in an inertial system K' that is moving at 25% the speed of light in the +x direction relative to K.

Calculate the spacetime intervals between events 1 and 2 in K and K'

Is it possible that event 2 caused event 1 ?

The Special Theory of Relativity

Einstein's Special Relativity

Example - Invariant quantities

What are the spacetime coordinates of these events in an inertial system K' that is moving at 25% the speed of light in the +x direction relative to K.

Calculate the spacetime intervals between events 1 and 2 in K and K'

Is it possible that event 2 caused event 1 ?

No Way! Since the space time interval is positive, the two events are spacelike separated, which implies that for the signal to travel between the two events it would have had to travel faster than light!

The Special Theory of Relativity

Visual representation of spacetime

Spacetime as a map

From any given perspective, events unfold in a 4-D spacetime, where a new version of 3-D space manifests at every point in time -- like a 4-D flip book.

Spacetime is a description of relationships between entities and events, constructed from sensory (and extended-sensory) data mainly communicated via electromagnetic waves (i.e. photons) and travelling vibrations (i.e. phonons.)

Only perspectives that are at rest w.r.t. each other share a common spacetime.

The Special Theory of Relativity

Visual representation of spacetime

Minkowski spacetime diagrams

Minkowski diagrams are two-dimensional graphs that depict events as happening in a universe consisting of one space dimension and one time dimension.

The "time" axis has units of length (to match the space units.)

The world line of an entity travelling with a constant velocity is a straight line.

The scale is typically taken such that the motion of light is represented by worldlines that are at

\pm 45^\circ

Events are represented by points in the spacetime diagram e=(x,ct).

The trajectories of entities in a spacetime diagram are called world lines.

From your perspective, you are always at x=0. i.e. your world line in your own frame is the time axis.

The Special Theory of Relativity

Visual representation of spacetime

Minkowski spacetime diagrams

Draw the world lines representing the shown scenarios.

The Special Theory of Relativity

What stories are told by the world lines illustrated in the spacetime diagrams ?

Visual representation of spacetime

Minkowski spacetime diagrams

The Special Theory of Relativity

What you detect "now" is a collage of signals that were emitted at different times in the past!

Visual representation of spacetime

What is the composition of the "present"?

The Special Theory of Relativity

What you detect "now" is a collage of signals that were emitted at different times in the past!

Visual representation of spacetime

What is the composition of the "present"?

The Special Theory of Relativity

Only events within the light cone have the potential to influence your present or be influenced by it.

The Light-cones

Visual representation of spacetime

The Special Theory of Relativity

Visual representation of spacetime

The Special Theory of Relativity

Einstein's Special Relativity

Visualizing the Lorentz Transformation

link to notebook

K-frame

K'-frame

\vec{v}= +0.25 c\ \hat{i}

\blue{x'} = \green{\gamma} (\red{x}-\green{\beta}\ \red{ct})

\green{\beta}=\frac{\green{v}}{c}\;,

\green{\gamma}=\frac{1}{\sqrt{1-\green{\beta^2}}}

\blue{ct'} = \green{\gamma} (\red{ct}-\green{\beta}\ \red{x})

The Lorentz Transformation

\red{(x,ct)} \rightarrow \blue{(x',ct')}

The Special Theory of Relativity

Einstein's Special Relativity

Visualizing the Lorentz Transformation

link to notebook

In all of the examples on this slide, shown are the events in K and all the K'(s) overlaid on the same spacetime diagram.

\beta = 0:0.1:0.9

Example 3:

Combining

examples 1 & 2

Example 1:

What happens to simultaneous events in K as increases?

\beta

Example 2:

What happens to events at the same location in K as increases?

\beta

The Special Theory of Relativity

Einstein's Special Relativity

Visualizing the Lorentz Transformation

The Special Theory of Relativity

Einstein's Special Relativity

Time Dilation

Link to time dilation cell in notebook

Suppose that there are two events (a and b), which from the K-perspective happen at the same position, but are separated by a time interval T0. Question is, what time, T', separates the same two events from a moving perspective K'?

cT' = ct_b'-ct_a'

= \gamma \left(ct_b- \beta x_b\right)- \gamma \left(ct_a- \beta x_a \right)

= \gamma \left(ct_b- ct_a \right)

= \gamma\ cT_0

\ge cT_0

i.e. The shortest observed time between any two events is the "Proper Time," which is measured in a frame where they happen at the same location!

The Special Theory of Relativity

Einstein's Special Relativity

Length Contraction

The Special Theory of Relativity

Einstein's Special Relativity

Time Dilation & Length Contraction

The Special Theory of Relativity

Einstein's Special Relativity

Relativity of relative velocity

The Special Theory of Relativity

Einstein's Special Relativity

Transformation of velocities

What does motion look like from the different perspectives?

link to notebook

The Special Theory of Relativity

Einstein's Special Relativity

Transformation of velocities

Suppose that we are colliding two particles, A and B, whose speeds in the lab frame are 70% and 50% the speed of light, respectively. Describe the motion of the two particles from their respective perspectives. (i.e. calculate their speeds from A's perspective, and then from B's perspective.)

The Special Theory of Relativity

Einstein's Special Relativity

Transformation of velocities

The Special Theory of Relativity

Einstein's Special Relativity

Transformation of velocities

How does (perpendicular) motion look like from the different perspectives?

link to notebook

The Special Theory of Relativity

Einstein's Special Relativity

Transformation of velocities

How does motion look like from the different perspectives?

link to notebook

The Special Theory of Relativity

Einstein's Special Relativity

Relativistic momentum and energy

The Special Theory of Relativity

Einstein's Special Relativity

Relativistic momentum

\vec{p} \ne m \vec{u}

With this thought experiment, you can show that, for objects traveling with speeds comparable to c,

We need a relativistic expression for momentum that is conserved at high speeds.

The Special Theory of Relativity

Einstein's Special Relativity

Relativistic momentum

\vec{p} = \gamma_u m \vec{u}

\gamma_u=\frac{1}{\sqrt{1-u^2/c^2}}

u^2=u_x^2+u_y^2+u_z^2

where

The relativistic momentum takes the form:

The Special Theory of Relativity

Einstein's Special Relativity

Relativistic Kinetic Energy

K = (\gamma_u-1) m c^2

\gamma_u=\frac{1}{\sqrt{1-u^2/c^2}}

u^2=u_x^2+u_y^2+u_z^2

where

The Special Theory of Relativity

Einstein's Special Relativity

Relativistic Energy

Example

Suppose you want to accelerate an electron from rest on a metal plate to strike another metal plate at 9E7 m/s.

What potential difference should you place across the plates?

(compare the classical and relativistic results)

The Special Theory of Relativity

Einstein's Special Relativity

Relativistic Energy

Example

Suppose you want to accelerate an electron from rest on a metal plate to strike another metal plate at 9E7 m/s.

What potential difference should you place across the plates?

(compare the classical and relativistic results)

The Special Theory of Relativity

Einstein's Special Relativity

Relativistic Energy

K = \gamma_u m c^2 - m c^2

K = (\gamma_u-1) m c^2

Total

Energy

\gamma_u m c^2 = K + m c^2

Kinetic

Energy

Rest

Energy

E\quad = K\quad + E_0

The Special Theory of Relativity

Einstein's Special Relativity

Relativistic Energy

Example

Suppose an electron is traveling at 0.5C in the lab frame. In units of eV, what is its rest energy, kinetic energy, and total energy, (a) in the lab frame, and (b) in its own frame?

The Special Theory of Relativity

Einstein's Special Relativity

Relativistic Energy

Example

Suppose an electron is traveling at 0.5C in the lab frame. In units of eV, what is its rest energy, kinetic energy, and total energy, (a) in the lab frame, and (b) in its own frame?

The Special Theory of Relativity

Einstein's Special Relativity

Energy and momentum

What is the relationship between the relativistic momentum and energy?

The Special Theory of Relativity

Einstein's Special Relativity

Relativistic momentum and energy

The Special Theory of Relativity

Einstein's Special Relativity

Energy and momentum

For a massless particle, its momentum and energy are related by:

p=\frac{E}{c}

The Special Theory of Relativity

Einstein's Special Relativity

Energy and momentum

Example

Suppose a particle accelerator accelerates a proton to the point that its total relativistic energy in the lab frame is 3 times its rest energy. Determine the following for the proton:

a) Its rest energy.

b) Its relativistic energy in the lab frame (in MeV)

c) Its relativistic Kinetic energy.

d) Its speed (as a fraction of the speed of light.)

e) The magnitude of its relativistic momentum in the lab frame.

f) Verify that, in the lab frame,

E^2=p^2c^2+m^2c^4

The Special Theory of Relativity

Einstein's Special Relativity

Energy and momentum

Example

Suppose a particle accelerator accelerates a proton to the point that its total relativistic energy in the lab frame is 3 times its rest energy. Determine the following for the proton:

a) Its rest energy.

b) Its relativistic energy in the lab frame (in MeV)

c) Its relativistic Kinetic energy.

d) Its speed (as a fraction of the speed of light.)

e) The magnitude of its relativistic momentum in the lab frame.

f) Verify that, in the lab frame,

E^2=p^2c^2+m^2c^4