Maths in Physics.
"The iron fist of the real, inside the velvet glove of airy mathematics."
- Gregory Benford on maths being a tool of the physicist.
Table Of Contents:
1. Special Relativity - The Lorentz Transformations.
2. Electromagnetism - Maxwell's Equations (Integral form).
Foreward.
In the rest of the Grail Diary, the mathematical content has been left to a bare minimum - we explored physics on a purely conceptual basis. However, Paul Dirac said: "Mathematics is only a tool" and as such it should be treated as one.
In this section, we will explore individual concepts from other chapters from a mathematical viewpoint. I strongly recommend reading the conceptual explanations found in other chapters before these mathematical ones.
Furthermore, the mathematical explanations found here are not essential reading - conceptually understanding physics is much more important than knowing the maths - and so the reader can feel free to skip over this section.
NOTE: When more complicated equations arise later in the chapter, the best thing for you to do is write them out yourself and follow the proof on a piece of paper because it is hard to visualise the maths when it is laid out without the use of divide-signs, indices or roots.
1. Special Relativity - The Lorentz Transformations.
As discussed in the chapter on special relativity, when objects move extremely fast (known as relativistic speeds), a range of strange phenomena occur. These phenomena include time dilation (time slows down for the moving object) and length contraction (the moving object becomes shorter in the direction of travel).
We discussed briefly that the effect of slowed time is proportional to the effect of shortened length but there is a much more elegant way of describing this known as the Lorentz transformation - named after the Dutch physicist Hendrik Antoon Lorentz.
The Principles.
All clocks have some element of a vibrating/oscillating part, this is the component that allows them to measure time. The easiest type of clock to prove the effects of time dilation with is a light clock.
A light clock is essentially two mirrors placed a certain distance apart. A photon is bounced between the two mirrors and the distance is set so that the time between the photon leaving one mirror and returning is 1 second.
This scenario is very hard to practically achieve because one light second is 299,792km so real light clocks use a shorter distance and a computer to determine how many bounces are equivalent to 1 second.
Suppose the light clock travels past us (perpendicular to our frame of reference in fancy physics-speak) and we observe it traveling sideways as seen in the diagram to the left. Because of it's shape, and Pythagoras' theorem, we can see that the distance travelled between the bounces by the photon has to increase if it is moving in this manner.
If we take dx as the horizontal distance between bounces and dy as the vertical distance between bounces, we can see that when the light clock is stationary (relative to us), dx becomes 0. So the total distance between bounces for the photon to move is just dy.
However, if the clock moves, the distance travelled between bounces (through Pythagoras theorem) becomes:
(dy^2 + dx^2)^1/2
This must be, by very definition, a longer distance than dy. This means when the clock is moving perpendicular to us as observers we can draw one of two very different conclusions:
1. The photon speeds up so that the light clock still measures one second correctly (correct co-ordinate time)
2. The photon travels the same speed and so the moving clock measures a slower second.
Conclusion 1 is not possible as it goes against the first principle of relativity - the speed of light is a constant to all observers. Therefore, it must be the case that conclusion 2 is correct and the clock has measured a slower value for a second when it moves. To examine this more closely:
Time Dilation Lorentz Transformation.
Let the proper length of the light clock be L and the proper time between ticks be T. A tick of the light clock is defined to be when the bouncing photon returns to the first mirror. Regardless of how many ticks make up a second we can say the following is always true:
EQUATION 1: 2L = c x T
However, this is only true for a stationary light clock. If we observe the light clock moving perpendicular to the observer, as pictured left, with a constant velocity v. The time between ticks of the moving clock is represented as t (the co-ordinate time).
We can therefore say that the clock moves a distance of vt between ticks and that the photon moves a total distance of ct (as it must move at the speed of light).
The distance travelled by the photon between the top and bottom mirror is:
[L^2 + (1/2vt)^2]^1/2
As v is a constant, the photon moves the same distance from the bottom to top mirror so the total distance travelled is:
2 x [L^2 + (1/2vt)^2]^1/2
This must be equal to ct (as both are expressions for distance traveled by the photon):
EQUATION 2: ct = 2 x [L^2 + (1/2vt)^2]^1/2
If we rewrite equation 1 into the form L = 1/2 cT and sub it into equation 2:
EQUATION 3: 1/2 ct = 2 x [(1/2 cT)^2 + (1/2vt)^2]^1/2
By rearranging equation 3 for T we get:
TIME DILATION: t = T / (1 - (v^2 / c^2))^1/2
This equation is incredibly useful because it gives a direct mathematical link between T (the proper time of the clock) and t (the co-ordinate time of the clock). The only thing we need to know to calculate how much time dilation occurs is the velocity of the object.
Length Contraction Lorentz Transformation.
We return to our light clocks from the time dilation formula, the photon bounce between the mirrors in exactly the same way. However, rather than having a stationary vertical clock we will imagine a moving horizontal clock as pictured left.
We will examine three events:
- The photon bounces off the left-side mirror.
- The photon bounces off the right-side mirror.
- The photon bounces off the left-side mirror again.
We will measure the time of each event as t1, t2 and t3 respectively in such a way that t3 occurs after t2 which in turn occurs after t1.
In order to find the Lorentz transformation for length contraction, we need to find two expressions for t3 - t1 (the co-ordinate time difference between ticks of the clock):
Method 1 to Find t3 - t1.
Our horizontal clock is moving at a constant velocity, v and has proper length L and co-ordinate length l.
This means that between t1 and t2, the clock moves a distance of v (t2 - t1). Similarly, as it travels at the speed of light, c, the photon travels a distance of c (t2 - t1). Because the light pulse also travels the length of the clock:
c ( t2 - t1) = l + v (t2 - t1)
By rearranging this formula for t2:
EQUATION 1: t2 = l + [(c - v) t1] / c - v
We can apply the same logic to events 2 and 3. This time however, because the photon now moves against the direction of the clock, we use:
c ( t3 - t2) = l - v (t3 - t2) (same as above formula but using - not +)
Once again, we rearrange the formula for t2:
EQUATION 2: t2 = [(c + v) t3] - l / c + v
By setting equation 1 and 2 equal to each other (they both express t2) and cross-multiplying we can produce:
l(c + v) + (c^2 - v^2) t1 = (c^2 - v^2) t3 - l(c - v)
Notice how the bold brackets above are produced from a perfect square expansion!
By spotting that both sides contain the bold bracket (the only difference being the expanding constant of t1 or t3), we can put them both on one side:
l(c + v) + l(c - v) = (c^2 - v^2) (t3 - t1)
Expanding the left side and then collecting like terms before rearranging for t3 - t1 gives:
t3 - t1 = 2lc / c^2 - v^2
By dividing every term by c^2, we can write this equation as:
EQUATION 3: t3 - t1 = (2l / C) / (1 - v^2 / c^2)
Method 2 to Find t3 - t1.
So far we have only used co-ordinate time, t1, t2 and t3 are all measures of time from our external reference point. We will use T1, T2 and T3 to measure the proper time of these events.
By using the equation generated from time dilation (found at the end of method 1), we can see that:
EQUATION 4: t3 - t1 = T3 - T1 / (1 - (v^2 / c^2))^1/2
The above equation is just the time dilation occurring between the proper time and coordinate time of events t1 and t3. We found it by simply substituting into the time dilation formula.
Because T3 - T1 is just a fancy way of saying the time between ticks, we can return to equation 1 from the time dilation proof (remember L is the proper length of the stick) which shows that:
EQUATION 5: T3 - T1 = 2L / c
Therefore, by subbing equation 5 into equation 4, we can generate:
EQUATION 6: t3 - t1 = (2L / c) / (1 - (v^2 / c^2))^1/2
An extremely interesting note (and one which caused an unbelievable amount of syntax checking on my part) is that method 1 generates an answer only involving co-ordinate length: l. On the other hand, method 2 generates an answer which exclusively uses proper length: L.
Equation 3 and equation 6 are incredibly mathematically similar, despite one using L and one l. The only difference is that the (1 - v^2 / c^2) part of equation 6 is square rooted, whereas in equation 3 it is not. I would recommend the reader physically write out both equations on paper to compare this similarity as it is incredibly elegant.
This is therefore the perfect opportunity to examine the relationship between the proper length of an object and its co-ordinate length. The easiest way to do this is to set the right sides of equation 3 and 6 equal to each other (remember they both define t3 - t1):
EQUATION 7: (2l / C) / (1 - v^2 / c^2) = (2L / c) / (1 - (v^2 / c^2))^1/2
To simplify this algebra we will make a substitution: we will define the variable x as:
x = 1 - v^2 / c^2
By subbing this into equation 7 and multiplying through by x^1/2 we produce:
(2l / c x^1/2) / x = 2L / c x
Collecting and simplifying like terms allows us to get:
l / L = x / x^1/2
x / x^1/2 must equal x^1/2 and so we can rearrange for l:
l = L x^1/2
Remembering the x substitution:
LENGTH CONTRACTION: l = L (1 - v^2 / c^2)^1/2
This is our second clever equation of special relativity as it shows the mathematical relationship between the coordinate length of the clock, l, and the proper length of the clock, L.
As it is not some strange property of the clock that makes this equation work it is therefore a model of coordinate space against proper space.
Mass - Energy Equivalence Lorentz Transformation.
Conclusion.
Time Dilation.
The relationship between proper time, T, and co-ordinate time, t, of any moving object, traveling at constant velocity, v, can be mathematically modelled through the equation:
t = T / (1 - (v^2 / c^2))^1/2
Length Contraction.
The relationship between proper length, L, and co-ordinate length, l, of any moving object, traveling at constant velocity, v, can be mathematically modelled through the equation:
l = L (1 - (v^2 / c^2))^1/2
Mass - Kinetic Energy Equivalence.
The relationship between proper mass, M, and co-ordinate mass, m, of any moving object, traveling at constant velocity, v, can be mathematically modelled through the equation:
m = M / (1 - (v^2 / c^2))^1/2
The Lorentz Factor.
Both equations contain the constant term: (1 - (v^2 / c^2))^1/2, we will define it here as the Lorentz Factor. The symbol used is γ
The only difference between the equations therefore is that to find co-ordinate time, you divide proper time by γ but to find co-ordinate length you multiply proper length by co-ordinate length.
2. Electromagnetism - Maxwell's Equations (Integral form).
In the electromagnetism, we discussed Maxwell's equations for particle-like points in space. We assumed the objects placed inside the electric and magnetic fields had no dimensions - that is to say they didn't have a size or shape.
However, in the real world, all objects have dimensions and so we need to make an adjustment to Maxwell's equations so they can more accurately describe objects in real life. The form these equations take are called the Integral form of Maxwell's Equations:
Law 1: Gauss' Law of Electric Fields.
Vexing Vectors.
→n - The
unit normal vector. It has a magnitude of 1 and acts perpendicularly away from
the surface of the object as seen on the left.
→E - A vector that describes the motion of a charged particle inside an electric field. Conceptually, this is a mathematical way to define the vector field (see the electromagnetism chapter) of an electric field.
→A . →B - The dot-product of two vectors is equal to |→A | |→B | cosθ where θ is the angle between the two. It is defined as the magnitude of A in the direction of B.
Therefore we can say that →E . →n is the amount of the electric field that acts in the direction of n (that is to say perpendicularly away from the surface of the shape).
Incandescent Integrals.
∯…..da - Closed surface integral:
A surface integral (represented by ∬.... da is the summation of an infinite number of scalars contained within infinitesimal areas on a surface).
For example if you are spreading paint on a curved wall, the surface integral would be adding together the volume of paint on each cm^2 of the wall. If you made the area of each section smaller (tending it towards an infinitesimally small area), the total number of sections on the wall would get larger (and tend towards infinity).
This tending towards infinity causes the estimate of total volume of paint on the wall to get closer and closer to the actual value. In mathematics, surface integrals sum together an infinite number of infinitesimally-small areas to gain the true value of whatever scalar is being measured on the surface - in this case it would be total volume of paint used on the wall.
Left Side.
The observant reader at this point leaps from their chair to point out that Gauss' law uses the symbol ∯ not ∬. ∯ shows a closed surface integral.
This means it can only be applied to closed objects. Closed objects are ones in which you can't reach the interior without going through it. We can say that a sphere or a doughnut is a closed surface but a bowl is not.
The surface integral of the dot product between an electric field and the unit normal vector is so helpful in physics, we give it a unique name: electric flux. This gives us a much more concise way of defining the left hand side of Gauss' first law: the electric flux of an area in space.
Now for the fun bit:
∯ →E .→n da (the electric flux through a closed shape) can take any positive or negative value. A positive flux shows that more field lines leave the object than converge into the object. The reverse is true for a negative value of flux.
THIS MEANS FLUX IS JUST A NEW WAY TO SAY DIVERGENCE FOR AN OBJECT WITH A SURFACE AREA.
However, as all the electric flux entering the surface of the object must eventually come out, ∯→E .→n da must always be 0, the only way for the total electric flux of an object to be non-0 is if the flux appears or disappears inside the object.
The mysterious creation and destruction of flux is not as complicated as it may sound. It is produced/destroyed by the electric charge within the object as shown in the right side of Gauss' first law:
Right Side.
The right side of Gauss' Law shows us that the electric flux (through a closed surface) of an object is equal to qE / ε0.
qE - short for q enclosed, it is the total electric charge enclosed by the object. It is important to note here that qE being zero does not mean that the object contains no charge. If I filled a balloon with 50% 1- ions and 50% 1+ ions, the total qE is still 0 as the charges cancel each other out.
ε0 – the permittivity of free space, sometimes called the electric constant. The permittivity of a substance is it's resistance to an electric field passing through it and so the electric constant is how much free space (a vacuum) resists having an electric field run through it. It has a constant value of 8.854 x 10^-12 Nm^-1.
As ε0 is a positive constant, the only thing that determines whether the electric flux through an object is negative or positive is the enclosed charge of the object.
If qE is negative, then electric flux must be negative. This makes the object a source.
If qE is positive, then electric flux must be negative. This makes the object a sink.
Therefore, we can say that flux only appears at objects with a positive enclosed charge and disappears at objects with negative qE.
Voila! We have mathematically reinforced the entirety of pole-line diagrams and vector fields in electric fields. Before understanding Gauss first law, you had to blindly accept the seemingly-magical idea that electric field lines left positive charges and were sucked in by negative ones. Gauss' law of electric fields shows us that this magic is simply the enclosed charge of the objects in question.
The last quirk of Gauss first law is that electric flux is directly proportional to the magnitude of the enclosed charge. This can be immediately seen by re-arranging the equation into the form:
(∯ →E . →n da) x ε0 = qE.
The constant of proportionality between electric flux and enclosed charge is the permittivity of free space (how much a vacuum resists the presence of an electric field). So doubling the charge of the object from 1C to 2C, causes the amount of electric flux leaving it to double.
Conclusion.
To sum-up (integral pun not intended) Gauss' law of electric fields:
"The electric flux (divergence across the surface area of an object) as an electric field passes through any closed surface is proportional to the charge contained within the object. This electric flux depends on both the magnitude and sign of the enclosed charge and the constant of proportionality is the permittivity of free space."
Law 2: Gauss' Law of Magnetic Fields.
The left side of Gauss' second equation is almost the same as the first equation and the same logic can be applied to both.
However in this case we take the
dot-product of the magnetic field and the unit
normal vector passing through a closed object, rather than the electric field.
∯→B . →n da is called magnetic flux and is measured in Tesla (T) - 1T is defined as the field intensity generating one Newton (N) of force per Ampere (A) of current per meter of conductor.
This sounds reasonable but it turns out to be the most ridiculous unit of all time and an experimental nightmare. It turns out that 1 Tesla of flux is (to use the technical term) one hell of a lot. For perspective, the flux of the Earth's magnetic field is a measly 22,000 nanoTesla (0.000022T).
Using the same logic we applied in law 1: in order for magnetic flux to be non-zero there would have to be something inside the object that created or destroyed magnetic flux. This would be the equivalent of qE in Gauss' law of electric fields.
However, there is no such equivalent for magnetic fields as a magnetic monopole is never possible (see the electromagnetism chapter for conceptual proof of this).
Therefore, with nothing to create/dispel magnetic flux, any magnetic flux that enters an object must leave it sooner or later and so ∯→B . →n da must be 0.
Law 3: Faraday's Law.
If we look at the right side of Faraday's law first, we can see that ∬→B . →n da is almost identical to the left side of Gauss' second law. However, it represents the magnetic flux acting through the surface of an open object not a closed one.
The crucial difference is that the open-surface-integral doesn't have to be zero. Picture magnetic flux lines that pass through the open shape and loop back on themselves - as seen on the left. We can therefore say that the magnetic flux entering an open object does not always leave it and so ∬→B .→n da does not always equal 0 as ∯→B .→n da does.
Changing Time.
The right side of Faraday's law describes the effects of varying time on the magnetic flux through an open surface. This changing time is shown by d/dt which is the derivative of a function respect to time. In the context of Faraday's law it means the rate of change of magnetic flux.
Therefore, the best way to think about the entire right side of Faraday's law is to read it as "The rate of change of magnetic flux through an open object."
What does a changing magnetic flux over time actually mean though? There are three ways for our open object to have a changing magnetic flux:
- Change the strength of the magnetic field (changing the number of lines flowing through it) over time.
- Tilt the surface over time to change how much of the field is actually used in the dot-product between it and the normal unit vector.
- Change the surface area over time.
The faster any of these three things
happen, the greater the change in magnetic flux over time is – which is
represented in the right side of Faraday's law.
Left Side - Circulating Electric Fields.
The electric field vector in the left side of Faraday's Law is drastically different to the one in Maxwell's first law. That's because it is produced through magnetic induction as oppose to surrounding a point charge.
Electric fields produced by induction (by a changing magnetic field) always have 0 divergence and so loop back on themselves without points of origination or termination. These types of fields are sometimes called circulating electric fields.
Despite this, circulating electric fields still cause charged particles within them to act in exactly the same way as normal electric fields.
Path Integrals.
Mathematically there is also a difference in how we treat circulating electric fields which is shown by the letter c before the integral:
c∮→E . →dl is an example of a path integral whereas ∯→B . →n da (from Gauss) is an example of a surface integral.
Law 4: The Maxwell-Ampere Law.
