Year 1 Sem 1/LC Classical Mechanics and Relativity 1/Lecture 08
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The Relativistic Doppler Effect
For an emitted frequency $f_0$, emitted by an object moving with velocity (in 1D) $u$ relative to an observer, the received frequency $f$ is given by:
For a relativistic speed $u$. Note that we cannot use the standard Doppler formula for relativistic speeds. Also note the lack of $\pm$, as we encode this into $u$. If the object moves towards the observer, $u$ is positive, and if the object is moving away $u$ is negative.
If $u$ is non-relativistic, we assume that $u/c \ll 1$:
And Taylor Series expanding:
Since $u/c$ is small, we ignore any quadratic, cubic etc terms of $u/c$, as these are very small.
Hence:
Since we assume $u/c \ll 1$:
This is the classical result that we’re familiar with, for non-relativistic speeds.
Lorentz Transformation
Say we have a reference frame $s^\prime$ which is moving along the $x$-direction relative to a static reference frame $s$.
An event in the $s$ frame has coordinates $(x, y, z, t)$ and the same event in the $s^\prime$ frame has coordinates $(x^\prime, y^\prime, z^\prime, t^\prime)$. We have the following transformations:
Noting that $y = y^\prime$, and $z = z^\prime$ as these are orthogonal to the direction of motion. We also have:
Please note that CMR1 does not include derivations of these equations (which collectively form the Lorentz Transformations), however for understanding’s sake I’ll include them here regardless.
Derivations
We want a transformation in time and space between a stationary frame $S$ and the moving frame $S^\prime$. We have three postulates to do this:
- Linearity: The transformation must be linear, i.e. a straight line in $S$ must map to a straight line in $S^\prime$.
- Standard rule for $c$: $c$ is invariant and has the same velocity in all frames, regardless of motion.
- Inverse symmetry: The inverse transformation ($S^\prime \to S$) is the same, but with $u \to -u$, as in $S^\prime$’s reference frame, it is static with $S$ moving with speed $-u$.
Deriving $x^\prime$:
We derive the transformation in position using length contraction:
Imagine a ruler at rest in the moving frame $S^\prime$. It has one end on the origin $O^\prime$ and the right end at some coordinate $x^\prime$. The ruler therefore has proper length $L_0 = x'$.
From the stationary frame $S$:
- The origin $O^\prime$ has moved distance $ut$ after some time $t$.
- The ruler is moving, so has been length contracted to the observer in $S$. The ruler now appears to have length $x^\prime / \gamma$.
The total coordinate as seen by $S$ is therefore:
And rearranging gives:
Deriving $t^\prime$:
We derive the transformation for time using the third postulate above:
If the transformation from $S \to S^\prime$ is:
Then the transformation from $S^\prime \to S$ is:
As the scenario is the same: $S'$ is moving with speed $u$ relative to $S$, if we instead consider $S^\prime$’s reference frame then it is static, and $S$ is moving in the opposite direction with the same magnitude of velocity, so the transformation must be the same with $u \to -u$.
Substituting the position transformation for $x^\prime$ into this:
Rearranging for $t^\prime$:
Using the identity $\gamma - 1/\gamma = \beta^2\gamma = \frac{u^2}{c^2}\gamma$:
Which gives us the time transformation.
Deiving $u_x^\prime$:
We find velocity in the moving frame as $dx' / dt'$.
The velocity in the moving frame $S^\prime$ is defined as $u^\prime_x = \frac{dx^\prime}{dt^\prime}$. Taking these derivatives
Substituting these into the definition of velocity:
Dividing through by $dt$:
And using $u_x = \frac{dx}{dt}$:
Dynamics
Kinematics is effectively looking at objects in motion. Dynamics is effectively “why” they move (Newton’s laws, static friction etc).
Newton’s Laws
We have three:
-
If there is no resultant force acting upon an object, there are two possibilities:
- The object was initially at rest, and stays at rest.
- The object moves at a constant speed with no acceleration.
- $\vec{F} = m \vec{a}$. Force and acceleration are proportional with a constant of proportionality $m$, the “inertial mass”. We call it inertial mass because this is theoretically distinct from gravitational mass, however all experiments give them as having them same value.
- The Reaction Principle. If a body $A$ is producing a force $\vec{F}$ on a body $B$, then $B$ acts back upon $A$ with a force of the same magnitude but the opposite direction “Every action has an equal and opposite reaction”.
Superposition Principle
If we have some body with $N$ forces acting upon it, the final resultant force that acts upon an object is a vector sum of these forces:
Example
Consider an object of mass $m$ hanging from the ceiling with an “ideal string”. This means that string is inextensible and is massless. The body is initially at rest.
Two forces act upon this body:
- The weight force due to local gravitational acceleration: $w = m g$.
- The force produced by the string (tension).
Since the body is at rest, the resultant force must be zero and:
Example II
Consider a body on a horizontal surface. We laterally pull the object with force $F$. We have these forces:
- Again a weight force: $w = mg$.
- The normal force produced by the table acting back upon the body iaw Newton’s Third Law.
- A frictional force acting in opposition to the direction of motion, $F_\text{fric}$
The frictional force is proportional to the Normal force with a coefficient depending on the materials used:
When moving an object there are two stages:
- Attempting to take an object from stationary to actually moving.
- Continuing the motion of the object once it’s moving (this is easier).
We therefore have multiple coefficients of friction. Here we consider the coefficient of static friction $\mu_s$ and the coefficient of kinetic friction $\mu_k$. There is also the coefficient of rolling friction $\mu_r$ seen in labs. Generally, $\mu_s > \mu_k$.