Physics

Special Relativity – Part 2

[latexpage]

Review

From Special Relativity – Part 1, we know that the speed of light is constant in all inertial frames of reference

  • Michelson-Morely experiment (1887) showed that the wave-nature of light is not necessarily a mechanical wave causing a disturbance in a medium
    • In fact, the results of this experiment suggested that there is no medium through which light travels as a mechanical wave
  • However, it would be a few more decades until the wave-particle duality nature of light was explained in terms of electromagnetic waves

Some Newtonian Physics

Frame of reference (FoR) = an abstract coordinate system that is uniquely fixed (i.e., located and oriented) based on a set of physical reference points

Inertial frame of reference = a FoR in which objects that have a net force of zero acting on them are not accelerated (i.e., they are at rest or move at a cohnstant velocity without changing directions)

  • In other words, any object that “lives” inside an inertial FoR only accelerates when a physical force is applied to it
  • Also sometimes called a Galiliean reference frame
  • Importantly, Einstein’s theory of special relativity only applies for inertial FoRs traveling at a constant velocity relative to one another
    • A few decades later, Einstein developed general relativity, which applies to inertial reference frames that are accelerating relative to one another

Notes:

  • For simplicity, we will only consider 1D movement (specifically, movement in the horizontal direction)
  • As we do more advance physics, we shouldn’t necessarily think as time “driving” position (i.e., graphing an objects position as a function of time)
    • Although it may feel counterintuitive at first (in most contexts, we graph time on the horizontal axis), but in the context of these “path-time” and later spacetime diagrams, we are going to plot position on the horizontal axis and time on the vertical axis
    • Maybe changes in position are what drive time!

Constructing our Thought Experiment

Consider a scenario where an observer named Donald Trump is sitting in a spaceship that is drifting through space at a constant velocity

  • Furthermore, let’s define $S:(x,t)$ as an inertial reference frame in 1D space, whose $x=0$ coordinate corresponds with Trump’s position at any given time $t$
  • Thus, from Trump’s perspective, his position is constant at the $x=0 \mathrm{m}$ in the $S$-frame
  • Here is a Newtonian path-time diagram for Trump’s position in the $S$-frame (each $T_i$-th point represents Trump’s position in the $S$-frame at $t=i$ seconds)

    S-frame
    Figure 1: Newtonian path-time diagram of Trump’s position in $S:(x,t)$

Now, let’s introduce a second observer named Barack Obama who is sitting in a different spaceship that is traveling at half the speed of light in the positive $x$-direction

  • We can define a new inertial FoR called $S'(x’,t’)$ whose $x’=0$ coordinate corresponds with Obama’s position at any given time $t’$
    • Note: we can classify this as an inertial FoR because it is moving at a constant velocity $v=0.5c$ (where $c = 3 \cdot 10^8 \mathrm{m/s}$ is the speed of light) relative to another inertial FoR (i.e., the $S$-frame, which is defined by Trump’s position)
  • Thus, from Obama’s perspective, his position is constant at the $x’=0 \mathrm{m}$ in the $S’$-frame
  • Here is a Newtonian path-time diagram for Obama (each $O_i$-th point represents Obama’s position in the $S’$-frame at time $t’=i$ seconds)

    S'-frame
    Figure 2: Newtonian path-time diagram of Obama’s position in $S’:(x’,t’)$

Transforming Between Different Intertial FoRs

Aside: The Galilean Transformation

Newtonian/classical physics uses the Galilean transformation to transform between points of view (i.e., coordinate systems) for observers/objects “observing” the same event $E$ from different inertial FoRs

  • Put more simply, Newtonian/classical physics assumes the Galilean transformation accurately describes transforming between different coordinate systems for different intertial FoRs
  • For example, if we wanted to transform between the coordinates describing where/when an event occurs in the $S$-frame to coordinates in the $S’$-frame, we would use the following equations:

$\begin{matrix} t’ = t \\ x’ = x – vt  \end{matrix}$ where $v=$ the velocity of the $S’$-frame relative to the $S$-frame

The invariance of spatial distances is the most significant property in Galilean transformations (i.e., $\Delta x = \Delta x’$)

  • This property eventually breaks down for speeds near the speed of light

Galiliean physics also relies on the concept of absolute time

  • Absolute time = the time difference between two events $E_1$ and $E_2$ is the same in all inertial FoRs

Take-home-messages about the Galielean transformation:

  • Newtonian/classical physcis assumes space and time are absolute in all FoRs
  • However, for this to hold true, the speed of light should vary among inertial FoRs moving at different velocities relative to one another
  • As it turns out, the speed of light is always observed to be $c \approx 3\cdot 10^8 \mathrm{m/s}$
    • Consequently, Newtonian/classical physics “breaks down” for different inertial FoRs traveling at velocities $v \approx c$ relative to one another
    • Specifically, a unit length for both the time and space axes in one inertial FoR (e.g., $S:(x,t)$ or Trump’s experience of the passage of time and the distance of an event relative to himself) will differ from the unit lengths of these axes in another inertial FoR traveling at a different velocity relative to the first FoR (e.g., $S’:(x’,t’)$ or Obamas experience of the passage of time and the distance of an event relative to himself)

Also, I should probably note that the primary goal of this post is to show that the Galilean transformation does not hold for inertial FoRs traveling at or near the speed of light

Overlaying Path-Time Diagrams Using the Galilean Transformation

Assuming the Galilean transformation holds, let’s overlay the path-time diagram for Obama in the $S’$-frame onto Trump’s path-time diagram in the $S$-frame where Obama’s spaceship passes Trump’s spaceship at $t=0 \mathrm{s}$:

S-frame overlayed with S'-frame
Figure 3: The $S’$-frame overlayed on top of the $S$-frame
  • This path-time diagram shows Obama’s position changing relative to Trump’s, where Trump’s position is assumed to be constant
  • Since we are assuming the Galiliean transformation accurately transforms the $S$ and $S’$ coordinate systems, we drew the $t’$ axis so that $t’=t \, \, \forall t’$ and $x’=x$ at $t=0$

Similarly, Trumps path-time diagram overlayed on top of Obama’s would look something like this:

S'-frame overlayed with S-frame
Figure 4: The $S$-frame overlayed on top of the $S’$-frame

Let’s add some more obersvers/events to our thought experiment

Suppose there are two other spaceships, one containing George Bush and the other containing Bill Clinton

  • Assume the following $S$-frame coordinates for the spaceships
    • Bush’s spaceship: $(3\cdot 10^8 \mathrm{m}, 0 \mathrm{s})$
    • Clinton’s spaceship: $(6\cdot  10^8 \mathrm{m}, 0 \mathrm{s})$
  • Let’s also assume the velocity of these spaceships are the same as Obama’s
    • Thus, from Trump’s perspective (i.e., the $S$-frame), these spaceships are traveling at half the speed of light in the positive $x$-direction
    • From Obama’s perspective (i.e., the $S’$-frame), these spaceships remain at a fixed distance from his spaceship
  • Here is a path-time diagram of the $S’$-frame overlayed on top of the $S$-frame for these events:
    S-frame overlayed with S'-frame with Bush and Clinton
    Figure 5: The same path-time diagram depicted in Figure 3, but including Bush’s and Clinton’s path-time trajectories (green). Here, $B_i$ vertices denote Bush’s spaceship at time $t=t’=i$ seconds and $C_i$ vertices denote Clinton’s spaceship at time $t=t’=i$ seconds

     

Now, suppose Trump turns on a light located on the outersurface of his spaceship at $t=0$

  • If we draw a path-time diagram in the $S$-frame for the first photon emitted from this light, it would look something like this:
    S-frame with photon
    Figure 6: $S$-frame with a path-time trajectory for Trump and a path-time trajectory for the photon emitted at $t=0$

     

So far, everything looks alright, but if we try to overlay these sequence of events with the $S’$-frame and path-time trajectories for Bush and Clinton, we will see that things start to get contradictory with observations from nature (e.g., Michelson-Moorley experiment)

S-frame overlayed with S'-frame with Bush, Clinton, and Photon
Figure 7: The same path-time diagram depicted in Figure 5, but including a path-time trajectory for a photon emitted from the $S$-frame at $t=0 \mathrm{s}$ and $x=0 \mathrm{m}$
  • Consider the photon emitted from the flashlight Trump turned on at $t=0$ seconds (orange line in Figure 7)
    • Note: the velocity of the photon (that is, light) is $c=3\cdot 10^8 \mathrm{m/s}$
    • At $t=2 \mathrm{s}$, the photon will have traveled $6 \cdot 10^8 \mathrm{m}$ relative to Trump (i.e., in the $S$-frame)
    • Using the Galilean transformation and assuming the velocity of the $S’$-frame relative to the $S$-frame is $v=0.5c$, the position of the photon at $t=t’=2 \mathrm{s}$ is…

$x'(t’=2 \mathrm{s}) = x(t=2 \mathrm{s}) – v*2 \mathrm{s}$

$x'(t=2 \mathrm{s}) = 6 \cdot 10^8 \mathrm{m} – (0.5)(3 \cdot 10^8 \mathrm{m/s})(2 \mathrm{s})$

$x'(t=2 \mathrm{s}) = 6 \cdot 10^8 \mathrm{m} – 3 \cdot 10^8 \mathrm{m} = 3 \cdot 10^8 \mathrm{m}$

  • Thus, from Obama’s point of view (i.e., the $S’$-frame), the velocity of the photon relative to the $S’$-frame seems to be $1.5 \cdot 10^9 \mathrm{m/s}$ (i.e., half the speed of light) in the positive $x$-direction
    • However, we also know that in reality, regardless of which inertial frame of reference we are in, the speed of light is always the same (i.e., the speed of light should be $3 \cdot 10^8 \mathrm{m/s}$ in both the $S$ and $S’$-frames
    • Our classical/Newtonian/Galiliean understanding of the universe starts to break down!
      • Any time we make a prediction and it is not observed in nature, it means our conception of the universe is not complete (Note: it still isn’t complete today!)
      • We need a new way to conceptualize/visualize path-time diagrams

 

More content, editing, and diagrams coming soon! 😀 Also, I apologize if anyone takes offense to my version of Einstien’s thought experiement. I’m just trying to keep things interesting becuase this material can be a real mental “climb” if you know what I mean. Plus, I figure most people know who Trump, Obama, Bush, and Clinton are, so it applies to most people (I was going to use Game of Thrones characters, but not everyone knows who they are)

Personal remarks/thoughts:

  • Should we take into account the biological processes involved in the processing of time, space, and light sensation/perception?
  • 10/5/18 Update:
    • Photometry: the science of the measurement of light in terms of percieved brightness with respect to the human eye (wiki – photometry)
      • Note: this is distinct from radiometry – the science of measuring of radiant energy (which includes the light being measured in photometry)
      • Thought: Do the limitations/bounds of what an organism can/cant percieve have anything to do with different groups of spectral lines for a given element?

 

Sources

Physics

Waves in Physics

[latexpage]

General notes:

  • A hallmark property of waves is interference
  • In physics, waves are usually associated with the transmission of enegry between different points in space

Three Types of Waves

  1. Mechanical waves
  2. Electromagnetic waves
  3. Matter waves

Mechanical waves

Mechanical waves are propagations of a disturbance thorugh a material medium due to periodic motion of particles that comprise the medium from their mean positions

  • Existence of a medium is essential for propagation of a mechanical wave
    • Propagation occurs due to properties of the medium such as elasticity and inertia
  • Energy and momentum propagate via motion of particles in the medium BUT the overall medium remains in its original position
  • Two main types of mechanical waves:
    1. Transverse waves = the vibration of particles in the medium occurs perpendicular to the direction of wave propagation (e.g., vibration on a string)
    2. Longitudinal waves = the vibration of particles in the medium occurs parallel to the direction of wave propagation (e.g., oscillations in a spring, internal water waves, tsunamis, sound waves etc.)

Electromagnetic waves

Electromagnetic waves = periodic distortions in electric and magnetic fields

  • Two components: an electric component and a magnetic component
  • Initiated when charged particles (e.g., electrons) begin vibrating due to various forces acting on them
    • The vibration of these charged particles then results in the emission of energy called electromagnetic radiation
  • Importantly, electromagnetic waves do NOT require a medium to travel through
    • As far as we know, this property is unique to electromagnetic waves
  • Other properties:
    • Travel at the speed of light ($3 \cdot 10^8 \mathrm{m/s}$) in a vacuum
    • Can be polarized
    • Transverse in nature
      • Propagate out from their source (i.e., the vibrating particles)
      • Oscillations of waves occur perpendicular to the direction they are propagating/traveling
        • Furthermore, in the case of electromagnetic waves, oscillations in the magnetic component occur in a direction perpendicular to oscillations in the electric componet
    • No medium required
    • All EM waves have momentum (thus, they have kinetic energy)

Matter waves

Matter waves (or, de Broglie waves) = depict the wave-like properties of all matter

  • Assumes wave-particle duality for all matter
  • Frequency of these waves depends on their kinetic energy
  • Momentum is not directly (or, inversely) proportional to position of the wave

Miscellaneously-classed waves

Surface waves (or, Rayleigh waves) = can have mechanical or electromagntic nature

Standing waves = a wave that remains constant

 

Chemistry, Mathematics, Physics

Special Relativity – Part 1

[latexpage]

Luminiferous ether

Mid-ninteenth century: mainstream science believed light particles were (mechanical) waves traveling through a medium called the luminiferous ether

Mid-ninteenth century definition of a wave:

  • Waves = a disturbance traveling through a medium 
    • Note: this is essentially what our current definition of a mechanical wave is today
    • Medium = the material or substance through which a wave is traveling
  • Example 1: a dewdrop falling into a pond (Figure 1)
    water dropplet
    Figure 1
    • Here, the medium is the water in the pond
    • The initial disburbance is the dewdrop landing in the water
      • The water particles initially disturbed by the dewdrop further disturbs the position of surrounding water particles
      • This disturbance is further propagated throughout the medium (i.e., the pond)
  • Example 2: sound waves from clapping
    • Medium = air
    • Initial disturbance = compression of air molecules
      • Compressed air molecules causes them to collide with one another and generate sound waves
  • Note: both of these examples are mechanical waves

We knew from research such as Young’s double-slit experiment (1801) that light has wave-like properties

  • Specifically, it showed one of the hallmark signs of wave behavior: interference
  • Some unaswered questions of the mid-19th century
    • How could they define light in terms of its wave-like properties?
      • Note: they were trying to define light in terms of mehcanical waves (they didn’t know about electromagntic waves back then)
    • They theorized that light (e.g., light traveling from the Earth to the Sun) could be explained as a disturbance propagating through a medium
      • People called this medium the luminiferous ether
    • Big question: does the luminferous ether exist?

Luminiferous ether = medium through which light (supposedly) propogates

  • One major goal of mainstream science back then was to detect/validate the existence of this medium
  •  Note: if there is a luminiferous ether, the Earth must be traveling fast relative to it
    • Not only is the Earth rotating on its own axis, but is also treaveling along an elliptical orbit around the Sun at $\approx 30 \mathrm{km/s}$
    • Moreover, the Sun is estimated to orbit around the center of the galaxy at $\approx 200 \mathrm{km/s}$
    • As far as our galaxy is concerned, we don’t really know what it’s doing, we just know its moving
      • Most scientists theorize our galaxy is rotating around a black hole
    • Take-home-message: if the luminiferous ether exists, Earth’s position should be constantly changing relative to it
      • Reasoning behing this:
        • The odds of us being stationary relative to such a medium are essentially zero
        • We should either be moving relative to the ether or the ether should be moving relative to us
          • Thus, we should be able to detect some sort of “ether wind” or the “current” associated with the luminiferous ether
  • Aside: waves propagate faster in the direction which current is moving
    • Example: a dew drop falling in a stream with a current flowing (Figure 2)
      water dropplet 2
      Figure 2
      • Here, the medium is the water in the stream and the initial disturbance is the dewdrop falling into the stream
      • Propagation of medium distortion (i.e., the waves/ripples in the stream) will occur more quickly in the direction of current (i.e., movment to the left)

 

The Michelson-Morely Experiment:

Experiment Background

Assuming there did exist a luminiferous ether, let $\overrightarrow{s}$ be the velocity of the its ether wind

  • From our dicussion on wave propogation speed and currents, light that is propagated in the same direction as $\overrightarrow{s}$ show travel at a faster velocity than light propagated in the $-\overrightarrow{s}$ direction
  • For a while, no one could figure out how to test this because the tools/technology did not yet exist that could detect velocities near the speed of light (thus, any differences they would have expected to find were inmeasureable)

Eventually Michelson and Morely designed an experiment that was able to work around this issue using wave interference

  • Recall: interference is a hallmark behavior of waves
  • Instead of attempting to measure the speed of light emitted in different directions, they split light into two different directions, recombined them, and observed the interference patterns
    • They reasoned, that if light emitted in different directions traveled at different speeds, then different interference patterns would result
    • However, this isn’t what happened!!
  • No matter how they oriented the apparatus (no matter the time of day and/or year), they always observed the same interference pattern
    • Conclusion: the luminiferous ether doesn’t seem to affect light waves ⇒ breakdown of the idea behind a luminiferous ether and/or an “absolute” inertial frame of reference through which light traveles
  • Titled one of “the most famous failed experiments”
    • Note: there were other experiments besides this one at the time that were also causing people to question the existence of a luminiferous ether

As it turns out, no matter the reference frame, light always travels at a constant speed!

 

Sources

Physics

Gravity

[latexpage]

Background

Gravity is fundamental to everything humans have experienced in (written) history

  • Newton was the first person recorded in modern history to question why objects always fall towards the ground
    • Used gravity to explain this phenomena
    • Lead to the development of classical mechanics (later updated by Einstein)
    • Note: classical mechanics applies to macroscopic objects that are traveling far slower than the speed of light

Importantly, we currently do not have a very good understanding of what is causing gravity

  • Question: what is mass and why do bodies of mass “gravitate” towards each other?
    • Answer: we dont know
    • Comments/random thoughts: this is kind of like the opposite of diffusion (diffusion = the tendency for particles to move from areas of high concentration gradients – i.e. – areas with a lot of particles, to areas of low concentration gradients – i.e. – areas with few particles)
      • Like diffusion, gravity seems to occur spontaneously, so maybe it is the result of something that can be described as “energetically favorable”
      • Q: is there a “randomness” component to gravity like there is for the diffusion of particles?

Although we still dont exactly know why gravity exists, we are pretty good at describing how gravity behaves (in the context of systems that follow principles of classical mechanics)

Newton’s Law of Gravitation

  • Generally, gravity is defined as the attractive force $\overrightarrow{F}_g$ between two objects with positive nonzero masses $m_1$ and $m_2$ whose centers of mass are separated by a distance vector $\overrightarrow{r}$
    • Note: Newton’s law is stated in the context of particles
  • Equation for the gravitational force between two objects:

$\left\| \overrightarrow{F}_g \right\| = G \frac{m_1 m_2}{\left\| \overrightarrow{r} \right\|}$

  • Variables:
    • $\left\| \overrightarrow{F}_g \right\| = $ the magnitude of the gravitational force between object 1 and object 2 (SI units: Newton)
    • $G =$ the graviational constant $ \approx 6.67 \cdot 10^{-11} \mathrm{N (\frac{m}{kg})^2}$
    • $m_i$ = the mass of object $i$ (SI units: $\mathrm{kg}$)
    • $\left\| \overrightarrow{r} \right\| =$ the length of the vector describing the distance between the two objects (SI units: $\mathrm{m}$)
  • Also, assume $m_1$ is located at a point $P$ and $m_2$ is located at a point $Q$

Sources:

Mathematics, Physics

Kinematics

[latexpage]

*Note: for the purposes of this article, variables that represent a column vector may either be denoted using bolded letters (e.g., $\mathbf{a}$) or using over-head arrow notation (e.g., $\overrightarrow{a}$)

Linear Motion

 

Variables:

  • $\mathbf{x} =$  position vector (SI unit: $\textup{m}$)

$\Delta \mathbf{x} = \mathbf{x}_f – \mathbf{x}_i$

  • $\mathbf{v} =$ velocity vector (SI unit: $\textup{m/s}$)

$\mathbf{v} = \frac{\Delta \mathbf{x}}{\Delta t} = \mathbf{\dot{x}}$

  • $\mathbf{a} =$ acceleration vector (SI unit: $\textup{m} / \textup{s}^2 $)

$\mathbf{a} = \frac{\Delta \mathbf{v}}{\Delta t} =\mathbf{\dot{v}}= \mathbf{\ddot{x}}$

Kinematic formula (1D motion with constant acceleration):

$v(t)=v_0 + at$

$\Delta x (t) = (\frac{v(t)+v_0}{2})t$

$\Delta x (t) = v_0t + \frac{1}{2}at^2$

$v^2 (t) = v_0^2 + 2a \Delta x(t)$

Rotational Motion

Variables:

  • $\mathbf{\theta} =$  angle vector (SI unit: $\textup{rad}$)

$\Delta \mathbf{\theta} = \mathbf{\theta}_f – \mathbf{\theta}_i$

  • $\mathbf{\omega} =$  angular velocity vector (SI unit: $\textup{rad/s}$)

$\mathbf{\omega} = \frac{\Delta \mathbf{\theta}}{\Delta t} = \mathbf{\dot{\theta}}$

  • $\mathbf{\alpha} =$  angular acceleration vector (SI unit: $\textup{rad} / \textup{s}^2 $)

$\mathbf{\alpha} = \frac{\Delta \mathbf{\omega}}{\Delta t} =\mathbf{\dot{\omega}}= \mathbf{\ddot{\theta}}$

Kinematic formula (uniform circular motion)

$\omega (t)=\omega_0 + \alpha t$

$\Delta \theta (t)= (\frac{\omega (t)+\omega_0}{2})t$

$\Delta \theta (t) = \omega_0t + \frac{1}{2} \alpha t^2$

$\omega^2(t) = v_0^2 + 2\alpha \Delta \theta (t)$

Relations between linear and rotational variables

Let $\mathbf{x} = \mathbf{r} = x\mathbf{\hat{\imath}}+ y\mathbf{\hat{\jmath}}$, $r = \left|| \mathbf{r} \right||$, and $\omega = \left|| \overrightarrow{\omega} \right||$

$x = r \cos{\theta}$

$y = r \sin{\theta}$

$r^2 = x^2 + y^2$

$\theta = \tan^{-1}{(y/x)}$

$\mathbf{v} = \overrightarrow{\omega} \times \mathbf{r}$

$\mathbf{a} = \mathbf{\overrightarrow{\alpha}} \times \mathbf{r} – \omega^2 \mathbf{r} $

Sources