LOG#013. Spacetime.

spacetime

“(…)The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality(…)”

                                                                       – Hermann Minkowski, 1908

We have elucidated some amazing results from the Special Relavity (SR) postulates and more stuff is to come. Indeed, the own structure of Lorentz transformations is hinting some hidden symmetry between the concepts of space and time that are seen as independent from each other in Classical Mechanics. Einstein’s work on the principle of relativity and the electrodynamics of moving objects was showinga new symmetry of Nature, and it was pointed out by the old known Maxwell’s equations. Nobody realized it until Einstein’s papers were published.

Herman Minkowski was a professor of A. Einstein in Zurich. He was surprised by the fact Einstein were so deep in the problem of Electrodynamics. Then, Minkovski himself did a new advance on Einstein’s relativity and exposed his own works in 1908. Minkowski suggested that relativity meant that space and time are no longer independent entities, as in fact the Lorentz transformations show. Lorentz transformations mix space and time coordinates. Relativity was saying, according to Einstein, that space and time as absolute entities did not exist. Minkowski was a step further and suggested that relativity is just geometry of space and time merged together, a.k.a., that Lorentz transformations and relativity were… Physics in spacetime! Physics processes are then labelled by coordinates in “space” and “time”, or in “spacetime” for short. SR is just a set of rules or geometry handling with transformation between differente events in spacetime. Events in spacetime are labelled by some set of coordinates of space and time. If we restrict to the commonly known 3 dimensions of space and the single dimension of time we seem to observe, Minkowski exposed some mathematical framework to work in a 4-dimensional (D=3+1) spacetime. His tools can indeed be generalized to any arbitrary spacetime with D-dimension as well, but we will no go further in that direction today. We will only discuss 4D spacetime in this post.

Question: if time and space are relative, as Einstein suggested,…Does it mean that we have nothing “invariant” to study? Geometry in Minkowskian spacetime is the answer it. Fortunately, mathematicians in the 19th century had studied non-euclidean geometries and it was just rediscovered by Minkowski that non-euclidean geometries could fit the new theory of relativity.

Minkowski argued that events en spacetime E are given by four coordinate set of numbers:

    \[ \mathbb{X}=(ict,x,y,z)\]

where i=\sqrt{-1}. However, in modern language, physicists avoid the imaginary time (or equivalently, the so-called quaternionic formulation of SR) using a gadget called spacetime “metric”. Using the metric, you don’t need an imaginary time and you write:

    \[ \mathbb{X}=(ct,x,y,z)=(x^0,x^1,x^2,x^3)\]

The Minkovski spacetime was very helpful in the building of the relativistic theory of gravitation, a.k.a., general relativity in spite Einstein himself put critics on the spacetime formalism. Soon, he changed his view and he learned the Minkowki spacetime stuff.

Well, now, how can be the spacetime structure help in relativity? It is quite easy. It is true that time or space are not invariant by theirselves, as Lorentz transformations show but, it can be shown that, the “space-time” interval is invariant. What is a space-time interval? Easy! Take two events E_1, E_2 in spacetime. The squared spacetime interval between those events is defined as:

    \[ (\Delta S)^2=(\Delta x)^2+(\Delta y)^2+(\Delta z)^2-(\Delta ct)^2\]

or equivalenty, writing explicetly the coordinates of the two events in the S-frame E_1(ct_1,x_1,y_1,z_1) and E_2(ct_2,x_2,y_2,z_2), we get

    \[ \boxed{(\Delta S)^2=(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2-(ct_2-ct_1)^2=invariant}\]

In the S’-frame, by the other hand, we will have another spacetime interval:

    \[ \boxed{(\Delta S')^2=(x'_2-x'_1)^2+(y'_2-y'_1)^2+(z'_2-z'_1)^2-(ct'_2-ct'_1)^2=invariant}\]

If the S-frame and the S’-frame are related by Lorentz transformations, the invariant is the same. That is the key of spacetime! The invariant is called proper time, i.e., the time measured on the clock travelling attached to the reference frame, or, equivalently, the time measured by an observer in motion with the frame. Any other frame will not be “invariant” and it has to be Lorentz transformed in order to agree on clock measurements with any other observer in other different frame.

Thus, relativity unifies the classical notions of space and time into a wider and more general notion: spacetime. The speed of light is indeed the conversion factor between units of space and time. Sometimes, physicists use units in which the speed of light is set to the unity c=1. Common units can be recovered carefully reintroducing “c”. If we go from the interval to an infinitesimal variation of spacetime coordinates in our events, the proper time is defined as

    \[ dS^2\equiv -c^2d\tau^2=dx^2+dy^2+dz^2-c^2dt^2\]

or, from the S’-frame

    \[ dS'^2\equiv -c^2d\tau^2=dx'^2+dy'^2+dz'^2-c^2dt'^2\]

Please, note that, the speed of light is invariant as it should, accordingly to the SR postulates.

The proper time can be related to the usual time with some algebraic manipulations:

    \[ -c^2\left(\dfrac{d\tau}{dt}\right)^2=\left(\dfrac{dx}{dt}\right)^2+\left(\dfrac{dy}{dt}\right)^2+\left(\dfrac{dz}{dt}\right)^2-c^2\left(\dfrac{dt}{dt}\right)^2\]

so, knowing that the velocity in space (or 3d velocity) is indeed v=\left(\dfrac{dx}{dt},\dfrac{dy}{dt}\dfrac{dz}{dt}\right). we get

    \[ -c^2\left(\dfrac{d\tau}{dt}\right)^2=v^2-c^2=-c^2\left(1-\dfrac{v^2}{c^2}\right)=-\dfrac{c^2}{\gamma ^2}\]

where

    \[ \gamma =\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}\]

is the well know relativistic dilation factor we found in our previous studies. We have then obtained the relation between proper time and usual time, an important mathematical relationship that is very useful in order to simplify calculation in Minkowski spacetime (even if it is D-dimensional):

    \[ \boxed{\dfrac{d\tau}{dt}=\dfrac{1}{\gamma }\longleftrightarrow \dfrac{dt}{d\tau}=\gamma}\]

Minkovski ideas were proved useful and important for general relativity and the own mathematical framework of Special Relativity. The legacy of H. Minkowski is with us today yet:

1) The use of spacetime vectors. In the case of a D=4=3+1 spacetime, spacetime vectors are called 4-vectors and they represent events in the continuum spacetime geometry. This hyperbolic non-euclidean geometry was indeed studied by Gauss, Lobachevski and other bright mathematicians during the 19th century. The use of these vectors simplifies long calculations.

2) The use of Minkowski diagrams. Minkowski suggested that the causal structure of the geometrical spacetime was given by the structure of the “light-cones”. In \mathbb{R}^4, two hyperboloids (“hyperplanes”) can intersect in one point and it settles the physical events. The use of Minkowski diagrams is intuitive and it helps to visualize and solve physical problems too.

You can see a light-cone in Minkowski space above, fixing the causal structure of spacetime.

LOG#012. Michelson-Morley.

During the 19th century, the electromagnetic theory of Maxwell assumed that electromagnetic waves travelled in a medium called ether. The Michelson-Morley experiment was an experiment devoted to detect the ether. We can think about the electromagnetic waves like an analogue of waves in a medium (for instance, water). Look at this picture:

RiverMMex1

Imagine a boat sailing in this river. The river flows from the left side to the right side at constant velocity v relative to the banks. Now, the trick of this analogy. Imagine that the boat is light, i.e., the boat are our loved electromagnetic waves ( whatever they are, and it implies they propagate in some kind of elastic medium like the water in the river; this medium was called ether by the physicists during the 19th century). When the boat ( remember it is an electromagnetic field hidden in the analogue model) is sailing downstream (or upstream), its velocity with respect to the banks will be c+v downstream and c-v upstream. If you have to cover a distace l_1 from the points A to B and back to A, it will spend a time that can be easily computed:

    \[ t_1=T_1+T_2=\dfrac{l_1}{c+v}+\dfrac{l_1}{c-v}=\dfrac{2l_1 c}{c^2-v^2}=\dfrac{2l_1}{c}\dfrac{1}{1-\dfrac{v^2}{c^2}}\]

By the other hand, if the boat travels at right angles to the river ( remember it is playing the role of the ether), we can also calculate the time required by the boat to travel the distance l_2 from C to D and come back to C. The velocity in this case is calculated using the Pythagorean theorem (or doing the calculus by components) and it is equal to V=\sqrt{c^2-v^2}. In this way, the time will be then:

    \[ t_2=\dfrac{2l_2}{V}=\dfrac{2l_2}{\sqrt{c^2-v^2}}\]

The time difference between these two times is calculated as well in a straightforward way:

    \[ \Delta t = t_1-t_2=\dfrac{2l_1}{c}\left(\dfrac{1}{1-\dfrac{v^2}{c^2}}\right)-\dfrac{2l_2}{\sqrt{c^2-v^2}}\]

or equivalently, the fundamental formula for the Michelson-Morley light-dragging through the ether reads:

    \[ \boxed{\Delta t =\dfrac{2l_1}{c}\left(\dfrac{1}{1-\dfrac{v^2}{c^2}}\right) - \dfrac{2l_2}{c} \left( \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}\right)}\]

Michelson and Morley guessed an experimental set-up to measure the drag velocity v of light in the ether. From the hypherphysics project page,  hyperphysics.phy-astr.gsu.edu/hbase/hframe.html, we get this nice illustration of their device:

mmorexp

How is this related to the previous calculation? Well, it is pretty nice and simple. Michelson and Morley built what is called an stellar interferometer for light. Suppose that we write l_1=l_2=L in out main boxed equation above. L is the arm lenght of our stellar interferometer. Similar changes can be done in the formulae for t_1 (t' in the picture above) and t_2(t'' in the picture). In that case the time difference will be:

    \[ \Delta t =\dfrac{2L}{c}\left(\dfrac{1}{1-\dfrac{v^2}{c^2}}-\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}\right)\]

This mathematical expression is complicated. But if we suppose that the drag velocity is small compared with the speed of light, we can make an approximation ( the technical “magic words” would be Taylor series):

    \[ \Delta t\approx \dfrac{2L}{c}\left(1+\dfrac{v^2}{c^2}\right)- \dfrac{2L}{c}\left(1+\dfrac{v^2}{2c^2}\right)=\dfrac{2L}{c}\left(\dfrac{v^2}{2c^2}\right)=\dfrac{L}{c}\left(\dfrac{v^2}{c^2}\right)\]

As it is show in the figure as well, for a typical low drag velocity, the interferometer ( thanks to the wave character of light) could indeed measure tiny time separations searching for “moved fringes”. The fringe “shift” due to the rotation of the arms of the interferometer can be easily calculated. After a 90º rotation, the time diference \Delta t flips its sign, so the light should get a phase difference 2\Delta t. We know that the period of light is given by the relationship T=\lambda/c. Then, the fringe shift pattern that Michelson and Morley expected to obtain was:

    \[ \boxed{\Delta N=\dfrac{2\Delta t}{T}\approx \dfrac{2L}{c}\left(\dfrac{v}{c}\right)^2}\]

When Michelson did his first experiment in 1881, he had got L=1.2m and \lambda \sim 5 \cdot 10^{-7}m, and thus the expected fringe shift provided the minimum \Delta N \sim 1/20. The shift was not observed, and performing some improvements, Michelson and his collaborator Morley, in 1887, got L=11m and achieved a better ressolution in the fringe shift pattern. But, surprinsingly, there was no fringe shift in both cases!

In 1892, George Fitzgerald and H. Lorentz proposed (before A. Einstein true explanation with his relativity) that objects in the aether were contracted in the same fashion  (with the same formula) than we saw in relativity. In their conception, they indeed derived the Lorentz (or Lorentz-Fitzgerald) transformations but they intepreted (wrongly) in the context ot the electromagnetic ether theory. In this way, bodies were contrated in their motion through the aether and it could explain the null result of the Michelson-Morley experiment and other similar experiments. It changed radically when Einstein published his articles on relativity with the correct physical insight and consequences of their transformations, that Einstein himself derived from two simple principles, and more remarkably, neglecting the own existence of the ether!

The Michelson-Morley is likely one of the null experiments most famous in the history of Physics. Indeed, it advanced the rising of the special theory of relativity and, in perspective, it was saying that the ether hypothesis was not necessary for the electromagnetic field and its waves to exist and propagate “in vacuum”.

Of course, from the modern viewpoint, relativity moved the question of the ether into another…Nobody questions today how can light travel at speed of light in vaccum always at the same speed, independly from the source. Nobody questions that light can self-sustain their own oscillations in vacuum in space and time everywhere in the known Universe. Simply, we do know that it can and it does…Light and vacuum are entangled somehow. Indeed, the questions about the nature and properties of the ether has shifted now into the question what the vacuum is...That is, now we ask about what are the properties and nature of the vacuum and its properties. But that is another story that advanced the rising of the theory and methods of Quantum Physics and Quantum Field Theory. Too far for this modest post today!

Finally, I would like to remark that similar experiments to the Michelson-Morley experiment have been performed. Modern experiments are based on the concept of optical resonators and they have a set-up like this:

michelson-morley-experiment

Every result of Michelson-Morley type experiments has been null. The ether as the 19th century physicists imagined does not exist.

LOG#011. Relativistic accelerations.

TransAccelFrameSR

Imagine the S’-frame moves at constant velocity (see the frames above this line):

    \[ \mathbf{v}=(v,0,0)\]

relative to the S-frame. In the S’-frame an object moves with acceleration

    \[ \mathbf{a}'=(a'_x,a'_y,a'_z)=\left(\dfrac{du'_x}{dt'},\dfrac{du'_y}{dt'},\dfrac{du'_z}{dt'}\right)\]

QUESTION: What is the acceleration in the S-frame?

Of course it has to be something like this

    \[ \mathbf{a}=(a_x,a_y,a_z)=\left(\dfrac{du_x}{dt},\dfrac{du_y}{dt},\dfrac{du_z}{dt}\right)\]

In order to get the relationship between both accelerations (and frames) we have to use the addition law of velocities from the previous post. Without loss of generality, we will use the rule (I) and we leave the general case as an exercise for the eager reader. We obtain:

    \[ a_x=\dfrac{du_x}{dt}=\dfrac{\dfrac{du_x}{dt'}}{\dfrac{dt}{dt'}}=\dfrac{\dfrac{d}{dt'}\left[\dfrac{u'_x+v}{1+\dfrac{u'_xv}{c^2}}\right]}{\dfrac{1}{c}\dfrac{d}{dt'}\left[\gamma(ct'+\beta x')\right]}=\dfrac{\dfrac{du'_x}{dt'}\left(1+\dfrac{u'_xv}{c^2}\right)-(u'_x+v)\dfrac{v}{c^2}\dfrac{du'_x}{dt'}}{\left(1+\dfrac{u'_x v}{c^2}\right)^2 \gamma \left(\dfrac{dt'}{dt'}+\dfrac{1}{c}\beta \dfrac{dx'}{dt'}\right)}\]

Therefore,

    \[ a_x=\dfrac{\left(1-\dfrac{v^2}{c^2}\right)a'_x}{\left(1+\dfrac{u'_x v}{c^2}\right)^2 \gamma \left(1+\dfrac{u'_x v}{c^2}\right)}=\dfrac{1}{\gamma^3\left(1+\dfrac{u'_x v}{c^2}\right)^3}a'_x\]

Thus we get the first transformed component transformation for the acceleration:

    \[ \boxed{a_x=\dfrac{a'_x}{\gamma^3\left(1+\dfrac{u'_x v}{c^2}\right)^3}}\]

For the transverse components, say a_y ( an analogue symmetrical calculation provides a_z), we calculate

    \[ a_y=\dfrac{du_y}{dt}=\dfrac{\dfrac{du_y}{dt'}}{\dfrac{dt}{dt'}}=\dfrac{\dfrac{d}{dt'}\left[\dfrac{u'_y}{\gamma \left(1+\dfrac{u'_x v}{c^2}\right)}\right]}{\dfrac{1}{c}\dfrac{d}{dt'}\left[(ct'+\beta x')\right]}=\dfrac{\dfrac{du'_y}{dt'}\gamma \left(1+\dfrac{u'_x v}{c^2}\right)-u'_y \gamma \dfrac{v}{c^2}\dfrac{du'_x}{dt'}}{\gamma^2\left(1+\dfrac{u'_x v}{c^2}\right)^2 \gamma \left(\dfrac{dt'}{dt'}+\dfrac{1}{c}\beta\dfrac{dx'}{dt'}\right)}\]

Some basic algebra manipulations allow us to get:

    \[ a_y=\dfrac{\gamma \left(1+\dfrac{u'_x v}{c^2}\right)a'_y-\gamma\dfrac{u'_y v}{c^2}a'_x}{\gamma^3\left(1+\dfrac{u'_x v}{c^2}\right)^2\left(1+\dfrac{u'_x v}{c^2}\right)}=\dfrac{a'_y}{\gamma^2 \left(1+\dfrac{u'_x v}{c^2}\right)^2}-\dfrac{\dfrac{u'_y v}{c^2}a'_x}{\gamma^2\left(1+\dfrac{u'_x v}{c^2}\right)^3}\]

Thus, the complete transformation of accelerations between frames from S’ to S(and from S to S’) are given by the following tables:

    \[ \boxed{\mbox{Transf.of acc.in SR:} S'\rightarrow S \begin{cases}a_x=\dfrac{a'_x}{\gamma^3\left(1+\dfrac{u'_x v}{c^2}\right)^3}\\ \; \\ a_y=\dfrac{a'_y}{\gamma^2 \left(1+\dfrac{u'_x v}{c^2}\right)^2}-\dfrac{\dfrac{u'_y v}{c^2}a'_x}{\gamma^2\left(1+\dfrac{u'_x v}{c^2}\right)^3}\\ \; \\ a_z=\dfrac{a'_z}{\gamma^2 \left(1+\dfrac{u'_x v}{c^2}\right)^2}-\dfrac{\dfrac{u'_z v}{c^2}a'_x}{\gamma^2\left(1+\dfrac{u'_x v}{c^2}\right)^3}\end{cases}}\]

    \[ \boxed{\mbox{Transf.of acc.in SR:} S\rightarrow S' \begin{cases}a'_x=\dfrac{a_x}{\gamma^3\left(1-\dfrac{u_x v}{c^2}\right)^3}\\ \; \\ a'_y=\dfrac{a_y}{\gamma^2 \left(1-\dfrac{u_x v}{c^2}\right)^2}+\dfrac{\dfrac{u_y v}{c^2}a_x}{\gamma^2\left(1-\dfrac{u_x v}{c^2}\right)^3}\\ \; \\ a'_z=\dfrac{a_z}{\gamma^2 \left(1-\dfrac{u_x v}{c^2}\right)^2}+\dfrac{\dfrac{u_z v}{c^2}a_x}{\gamma^2\left(1-\dfrac{u_x v}{c^2}\right)^3}\end{cases}}\]

where to obtain the second table we used the usual trick to map primed variables to unprimed variables and to map v into -v.

One important comment is necessary in order to understand the above transformations. One could argue that the mathematical description of accelerated motions is beyond the scope of the special theory of relativity since it is a theory of inertial reference frames. THAT IDEA IS WRONG! Special Relativity (SR) IS based rather on the concept that every reference frame used for the calculations IS an inertial reference frame. It has NOTHING TO DO with accelerations between frames, and as we have showed you here, they can be calculated in the framework of SR!

In conclusion: the description of accelerated motions relative to inertial reference frames makes sense in SR and IS NOT subject to any limitation (unless, of course, you change/modify/extend the basic ideas and/or postulates of SR).

LOG#010. Relativistic velocities.

In our daily experience, we live in a “non-relativistic” world with a very high degree of accuracy. Thus, if you see a train departing from you ( you are at rest relative to it) with speed V (in the positive direction of the x-axis), you  move with relative velocity V-u respect to the train if you run in pursuit of it with speed u, or maybe you can also run with relative speed V+u if you run away from it in the opposite direction of motion.

However, light behavior is diferent to material bodies. Light, a.k.a. electromagnetic waves, is weird.  I hope you have realized it from previous posts. We will see what happen with an SR analogue gedanken experiment of the previously mentioned “non-relativistic” train(S’-frame)-track(S-frame) experiment and that we have seen lot of times in our ordinary experience. We will discover that velocities close to the speed of light add in a different way, but we recover the classical result ( like the above) in the limit of low velocities ( or equivalently, in the limit c\rightarrow \infty).

Problem to be solved:

In the S’-frame, an object (or particle) moves at constant velocity \vec{v}=\mathbf{v}=(v,0,0) relative to the S-frame. In the S’-frame, the object/particle moves with velocity

    \[ \vec{u}\,'=\mathbf{u}'=(u'_x,u'_y,u'_z)=\left( \dfrac{dx'}{dt'},\dfrac{dy'}{dt'},\dfrac{dz'}{dt'}\right)\]

The question is: what is the velocity in the S-frame

    \[ \vec{u}=\mathbf{u}=(u_x,u_y,u_z)=\left( \dfrac{dx}{dt},\dfrac{dy}{dt},\dfrac{dz}{dt}\right)\]

CAUTION ( important note): v is constant (a relative velocity from one frame into another). u is not constant, since it is a vector describing the motion of a particle in some frame.

From the definiton of velocity, and the Lorentz transformation for a parallel motion, we have

    \[ u_x=\dfrac{dx}{dt}=\dfrac{\dfrac{dx}{dt'}}{\dfrac{dt}{dt'}}=\dfrac{\dfrac{d}{dt'}\left[\gamma (x'+\beta ct')\right]}{\dfrac{d}{dt'}\left[\gamma (t'+\frac{\beta }{c}x')\right]}=\dfrac{\dfrac{dx'}{dt'}+\beta c \dfrac{dt'}{dt'}}{\dfrac{dt'}{dt'}+\dfrac{\beta}{c}\dfrac{dx'}{dt'}}\]

and thus we get the addition law of velocities in the direction of motion

    \[ \boxed{u_x=\dfrac{u'_x+v}{1+\dfrac{u'_x v}{c^2}}}\]

We can also calculate the transformation of the transverse components to the velocity in the sense of motion. We only calculate the component u_y since the remaining one would be identical but labelled with other letter(the z-component indeed):

    \[ u_y=\dfrac{dy}{dt}=\dfrac{\dfrac{dy}{dt'}}{\dfrac{dt}{dt'}}=\dfrac{\dfrac{dy'}{dt'}}{\gamma \left( 1+\dfrac{u'_x v}{c^2}\right)}\]

    \[ \boxed{u_y=\dfrac{u'_y}{\gamma \left( 1+\dfrac{u'_x v}{c^2}\right)}}\]

Therefore, the transverse velocity also changes in that way! There is an alternative deduction of the above formula using space and time coordinates. We will proceed in two important cases only.

The first case is when the motion happens with parallel relative velocity. Suppose two inertial frames S and S’. S is moving relative to S’ with velocity V along the X-axis. Moreover, suppose an object that is moving parallel to OX, with velocity v. Imagine two “frozen pictures” of the object at two different times according to S, e.g., fix two times t_1 and t_2. The two events have coordinates of space and time given by

    \[ E_1(t_1,x_1,y_1,z_1)\]

and 

    \[ E_2(t_2,x_2,y_2,z_2)\]

But we do know that t_2=t_1+\Delta t and so,

    \[ (t_2,x_2,y_2,z_2)=(t_1+\Delta t,x_1+v\Delta t,y_1,z_1)\]

What does the S’-frame observe? Using the Lorentz boost with speed -V, we get

    \[ (t'_1,x'_1,y'_1,z'_1) = (\gamma (V)(t_1+Vx_1/c^2),\gamma (V) (x_1+Vt_1),y_1,z_1)\]

and

    \[ (t'_2,x'_2,y'_2,z'_2)=(\gamma (V)(t_2+Vx_2/c^2),\gamma (V)(x_2+Vt_2),y_2,z_2)\]

Therefore, the velocity of the object according to the S’-frame will be:

    \[ v'=\dfrac{\Delta x'}{ \Delta t'}=\dfrac{x'_2-x'_1}{t'_2-t'_1}=\dfrac{x_2+Vt_2-(x_1+Vt_1)}{t_2+(V/c^2)x_2-(t_1+(V/c^2)x_1)}\]

and in this way, we obtain, dividing by t_2-t_1

    \[ v'=\dfrac{\dfrac{(x_2-x_1)+V(t_2-t_1)}{(t_2-t_1)}}{\dfrac{(t_2-t_1)+(V/c^2)(x_2-x_1)}{(t_2-t_1)}}\]

or equivalently

    \[ \boxed{v'=\dfrac{v+V}{1+\dfrac{V \cdot v}{c^2}}}\]

as before.

The second case is when in the velocities between the frames are orthogonal (perpendicular) can be also calculated in this way. Suppose that some object is moving in the orthogonal ( perpendicular) direction to the OX axis. For instance, we can suppose it moves along the y-axis (OY axis) with velocity v measured in the S-frame. We proceed in the same fashion that the previous calculation. We take two “imaginary pictures” of the body at t_1 and t_2=t_1+\Delta t. We write the coordinates of space and time of this object as

    \[ E_1(t_1,x_1,y_1,z_1)\]

and

    \[ E_2(t_2,x_2,y_2,z_2)\]

But we do know that t_2=t_1+\Delta t and so,

    \[ (t_2,x_2,y_2,z_2)=(t_1+\Delta t,x_1,y_1+v\Delta t,z_1)\]

We make the corresponding Lorentz boost on those coordinates

    \[ (t'_1,x'_1,y'_1,z'_1) = (\gamma (V)(t_1+Vx_1/c^2),\gamma (V) (x_1+Vt_1),y_1,z_1)\]

    \[ (t'_2,x'_2,y'_2,z'_2)=(\gamma (V)(t_2+Vx_2/c^2),\gamma (V)(x_2+Vt_2),y_2,z_2)\]

and now, the two components of this motion in the S’-frame will be given by ( note than our two set of coordinates have x_2=x_1 and y_2\neq y_1 in this particular case):

    \[ v'_{x'}=\dfrac{\Delta x'}{\Delta t'}=\dfrac{x_2+Vt_2-(x_2+Vt_1)}{t_2+(V/c^2)x_2-(t_1+(V/c^2)x_1)}=V\]

    \[ v'_{y'}=\dfrac{\Delta y'}{\Delta t'}=\dfrac{y'_2-y'_1}{\gamma (V)(t_1+(V/c^2)x_2-(t_1+(V/c^2)x_1))}=\dfrac{\dfrac{(y_2-y_1)}{(t_2-t_1)}}{\gamma (V)}=\dfrac{v}{\gamma (V)}\]

and so, in summary, in the orthogonal relative motion we have

    \[ \boxed{ v'_{x'}=V \;\; v'_{y'}=\dfrac{v}{\gamma (V)}}\]

Indeed, these two cases are particular cases of the general transformations we got before.

The complete transformation of the velocity components and their inverses (obtained with the simple rule of mapping v into -v, and primed variables into unprimed variables)  can be summarized  by these formulae:

    \[ \mbox{SR: Adding velocity(I)}\begin{cases}u_x=\dfrac{u'_x+v}{1+\dfrac{u'_x v}{c^2}} \; \; u_y=\dfrac{u'_y}{\gamma \left( 1+\dfrac{u'_x v}{c^2}\right)}\; \; u_z=\dfrac{u'_z}{\gamma \left( 1+\dfrac{u'_x v}{c^2}\right)}\\ \; \\ u'_x=\dfrac{u_x-v}{1-\dfrac{u_x v}{c^2}} \; \; u'_y=\dfrac{u_y}{\gamma \left( 1-\dfrac{u_x v}{c^2}\right)}\; \; u'_z=\dfrac{u_z}{\gamma \left( 1-\dfrac{u_x v}{c^2}\right)}\\ \; \\ \mathbf{u}=(u_x,u_y,u_z)\;\; \mathbf{u}'=(u'_x,u'_y,u'_z)\;\; \gamma =\dfrac{1}{\sqrt{1-\beta ^2}}\;\; \beta =\dfrac{v}{c}\end{cases}\]

Of course, these transformations are valid in the case of a parallel relative motion between S and S’. What are the transformations in the case of non-parallel motion? Suppose that

    \[ \vec{\beta} =\dfrac{\mathbf{v}}{c}=(\beta_x,\beta_y,\beta_z)\]

and \mathbf{u}=(u_x,u_y,u_z), \; \mathbf{u}'=(u'_x,u'_y,u'_z) as before. Then, using the most general Lorentz transformations

    \[ \mathbf{u}'=\dfrac{d\mathbf{r}'}{dt'}=\dfrac{d\mathbf{r}+(\gamma - 1)\dfrac{\left(\vec{\beta}\cdot \mathbf{u}\right)\vec{\beta}}{\beta^2}-\gamma \vec{\beta}cdt}{\gamma dt - \dfrac{1}{c}\gamma \vec{\beta}\cdot d\mathbf{r}}\]

then, using the same trick as above

    \[ \mathbf{u}'=\dfrac{d\mathbf{r}'}{dt'}=\dfrac{\dfrac{d\mathbf{r}'}{dt}}{\dfrac{dt'}{dt}}\]

    \[ \mathbf{u}'=\dfrac{\dfrac{1}{\gamma}\mathbf{u}+\left(1-\dfrac{1}{\gamma}\right)\dfrac{\left(\vec{\beta}\cdot \mathbf{u}\right)\vec{\beta}}{\beta^2}-\vec{\beta} c}{1-\dfrac{1}{c}\vec{\beta} \cdot \mathbf{u}}\]

We have got the following transformations (we apply the same recipe to obtain the inverse transformations, also included in the box below):

    \[ \mbox{SR: Adding velocity(II)}\begin{cases} \mathbf{u}'=\dfrac{1}{1-\dfrac{\mathbf{u}\cdot\mathbf{v}}{c^2}}\left[\dfrac{\mathbf{u}}{\gamma}+\left[\left(1-\dfrac{1}{\gamma}\right)\dfrac{\mathbf{u}\cdot\mathbf{v}}{v^2}-1\right]\mathbf{v}\right]\\ \;\\ \mathbf{u}=\dfrac{1}{1+\dfrac{\mathbf{u}'\cdot\mathbf{v}}{c^2}}\left[\dfrac{\mathbf{u}'}{\gamma}-\left[-\left(1-\dfrac{1}{\gamma}\right)\dfrac{\mathbf{u}'\cdot\mathbf{v}}{v^2}-1\right]\mathbf{v}\right]\end{cases}\]

We observe that these equations are a non-linear addition of velocities. Equivalently, they can be rewritten as follows after some elementary algebra using a mathematical structure called gyrovector (or gyrovector addition):

    \[ \mbox{gyrovector law}\begin{cases}\mathbf{u}'\equiv -\mathbf{v}\biguplus_{REL}\mathbf{u}=\dfrac{1}{1-\dfrac{\mathbf{u}\cdot \mathbf{v}}{c^2}}\left[-\mathbf{v}+\dfrac{\mathbf{u}}{\gamma_{\mathbf{v}}}+\dfrac{1}{c^2}\left(\dfrac{\gamma_{\mathbf{v}}}{\gamma_{\mathbf{v}}+1}\right)\left(\mathbf{v}\cdot \mathbf{u}\right)\mathbf{v}\right]\\ \; \\ \mathbf{u}\equiv \mathbf{v}\biguplus_{REL}\mathbf{u}'=\dfrac{1}{1+\dfrac{\mathbf{u}'\cdot \mathbf{v}}{c^2}}\left[\mathbf{v}+\dfrac{\mathbf{u}'}{\gamma_{\mathbf{v}}}+\dfrac{1}{c^2}\left(\dfrac{\gamma_{\mathbf{v}}}{\gamma_{\mathbf{v}}+1}\right)\left(\mathbf{v}\cdot \mathbf{u}'\right)\mathbf{v}\right]\end{cases}\]

These 2 cases can be seen as particular examples in the addition rule of velocities as a “gyrovector sum”, the nonlinear addition rule given by the formula:

    \[ \mathbf{u}\biguplus_{REL}\mathbf{v}=\dfrac{1}{1+\dfrac{\mathbf{u}\cdot \mathbf{v}}{c^2}}\left[\mathbf{u}+\dfrac{\mathbf{v}}{\gamma_{\mathbf{u}}}+\dfrac{1}{c^2}\left(\dfrac{\gamma_{\mathbf{u}}}{\gamma_{\mathbf{u}}+1}\right)\left(\mathbf{u}\cdot \mathbf{v}\right)\mathbf{u}\right]\]

This formula is usually written in a more intuitive expression with the following arguments. Suppose some object moves with velocity \mathbf{v} in some inertial frame S. S is moving itself with relative velocity \mathbf{V} respect to another frame S’. In the S’-frame, the velocity is given by:

    \[ \boxed{\mathbf{v}'\equiv \mathbf{V}\biguplus_{REL}\mathbf{v}=\dfrac{\mathbf{v}_\parallel+\gamma^{-1}(V)\mathbf{v}_\perp + \mathbf{V}}{1+\dfrac{\mathbf{V}\cdot \mathbf{v}}{c^2}}}\]

and where we have defined the projections of \mathbf{v} in the direction parallel and orthogonal to \mathbf{V}. They are given by:

    \[ \boxed{\mathbf{v}_\parallel =\dfrac{(\mathbf{V}\cdot \mathbf{u})\mathbf{V}}{V^2}\;\;\;\;\;\; V^2=\vert \mathbf{V}\vert^2\;\;\;\;\;\mathbf{v}_\perp =\mathbf{v}-\mathbf{v}_\parallel}\]

We will talk about gyrovectors more in a future post. They have a curious mathematical structure and geometry, and they are not well known  by physicists since they are not in the basic curriculum and background of SR courses. Of course, the non-associative composition rule for velocities is not a standard formula you can find in books about relativity, so I will write it here:

    \[ \boxed{\mathbf{u}\boxplus\mathbf{v}=\dfrac{\mathbf{u}+\mathbf{v}}{1+\dfrac{\mathbf{u}\cdot \mathbf{v}}{c^2}}+\dfrac{\gamma_{\mathbf{u}}}{c^2(\gamma_{\mathbf{u}}+1)}\dfrac{\mathbf{u}\times\left(\mathbf{u}\times\mathbf{v}\right)}{1+\dfrac{\mathbf{u}\cdot \mathbf{v}}{c^2}}}\]

and where we used the previous formula for \mathbf{u}\biguplus_{REL}\mathbf{v} and after some algebra we used the known relationship for the cross product of three vectors, two being the same,

    \[ u\times(u\times v)=(u\cdot v)u-(u\cdot u)v\]

Let’s go back to the beginning. Suppose now we imagine a train (our S’-frame) travelling at velocity v. Suppose that Special Relativity matters now. We are in the tracks as observers “in relative rest” with respect to the train ( we are the S-frame) and suppose that we take into account the SR corrections above to the addition of velocities. Inside the train some object is being thrown with velocity u'_x in the direction of motion. What is the velocity u_x in the S-frame? That is, what is the velocity we observe in the tracks? In this simple example, we use the easier addition rule of velocities ( named addition rule SR(I) above). Firstly, we note some expected features from the mathematical structure of the relativistic addition rule of velocities (valid propterties as well in the general case (II) with a suitable generalization):

1st. For low velocities, i.e., if u'_x/c<<1 and/or \beta=\dfrac{v}{c}<<1, the result approaches the nonrelativistic “ordinary life” experience: u_x=u'_x+v.

2nd. For positive velocities u'_x>0 and a positive relative velocity between frames v>0, the addition of velocities is generally u_x<u'_x+v, i.e., we get a velocity smaller that in the non-relativistic (ordinary or “common” experience) limit.

Now, some easy numerical examples to see what is going on bewteen the train (S’) and the track (S) where we are:

Example 1. Moderate velocity case. We have, e.g., velocities v=u'_x=30 km\cdot s^{-1}=10^{-4}c. This gives, using (I):

u_x=\dfrac{2 \cdot 10^{-4}c}{1+10^{-8}}\approx 60 km\cdot s^{-1} - 0.6 mm\cdot s^{-1}

Then, the deviation with respect to the non-relativistic value ( 60km/s) is negligible for all the practical purposes! This typical velocity, 30km/s, is about the typical velocities in 20th and early 21st century space flight. So, our astronauts can not note/observe relativistic effects. The addition theorem in SR is not practical in current space travel (20th/early 21st century).

Example 2. Case velocities are “close enough” to the speed of light. E.g.: One quarter and one half of the speed of light. In the first case,

    \[ v=u'_x=0.25c\]

and then

    \[ u_x=\dfrac{0.25c+0.25c}{1+0.0625}=\dfrac{0.5c}{1.0625}\approx 0.47c\]

In the seconde case, we write v=u'_x=0.5c. This provides

    \[ u_x=\dfrac{0.5c+0.5c}{1+0.25}=\dfrac{c}{1.25}=0.8c\]

Thus, we observe that a higher velocities, e.g., those in particle accelerators, some processes in the Universe, etc, the relativistic effects of the non-linear addition of velocities can NOT be neglected. The effect is important and becomes increasingly important when the velocity increases itself ( you can note how large the SR effect is if you compare the 0.25c and 0.5c examples above).

Example 3. Speed of light case. Extreme case: we are trying to exceed the velocity of light. Suppose now, that the train could move with relative velocity equal to c. The object is thrown with relative speed u'_x=c and u'_y=u'_z=0. What we do see on the track. Naively, ordinary life would suggest the answer 2c, but we do know that velocities transform non-linearly, so, we plug the values in the formula to get the answer:

u_x=\dfrac{c+c}{1+1}=c and u_y=u_z=0. Therefore, if a train is travelling at the speed of light, and inside the train an object is thrown forward at c, we DO NOT observe/measure a 2c velocity, we observe/measure it has velocity c!!!! if we stand at rest on the track. Amazing!

Suppose we try to do it in a “transverse way”. That is, suppose that the velocities are now v=c, u'_x=0 and the transverse speed is not u'_y=c. This case results in the numbers:

    \[ u_x=c\]

and

    \[ u_y=\dfrac{c}{\gamma}=0\]

since

    \[ \gamma \rightarrow \infty\]

and thus u=c.

Therefore, if the train travels at c, an inside of the train an object is launched at right angles to the direction of motion at the velocity c, the object itself is measured/seen to have velocity c measured from a rest observer placed beside the tracks. Amazing, surprise again!

Example 4. Case: Superluminal relative motion. Suppose, somehow, the relative motion between the two frames provides v=2c (even you can plug v=nc with n>1 if you wish). Suppose the object is measured to have the extremal limit speed u'_x=c (imagine we consider a light beam/flash, for instance). Again, using the addition law we would get:

    \[ u_x=\dfrac{3c}{1+2}=c\]

and

    \[ u_x=\dfrac{n+1}{1+n}c=c\]

Even if the inertial frames move at superluminal velocities relative to each other, a light beam would remain c in the S-frame if SR holds! Surprise, again!

In this way, we can conclude one of the most important conclusions of special relativity ( something that it is ignored by many Sci-fi writers, and that we would like to be able to overcome somehow if we have to master the interstellar travel/interstellar communications as Sci-fi fans, or as an interstellar civilization, you should get some trick to avoid/”live with” it.):

    \[ \boxed{\mbox{The speed of light can never be exceeded by adding velocities in SR.}}\]

If SR holds, the velocity (or speed) of light is the maximum speed attainable in the Universe. You can like it or hate it, but if SR is true, you can not avoid this conclusion.

There is another special case of motion important in practical applications: two dimensional motion. I mean, imagine that in the S’-frame, an object has the velocity \mathbf{u}'=(u'_x,u'_y,0). The velocity subtends an angle \theta ' with the x’-axis. See the figure below:

2dVeladditionSR

What is the angle \theta in the frame that we observe between \mathbf{u} and the x-axis? For the S-frame we find:

    \[\tan \theta= \dfrac{u_y}{u_x}=\dfrac{\dfrac{u'_y}{\gamma \left( 1+\dfrac{u'_x v}{c^2}\right)}}{\dfrac{u'_x+v}{1+\dfrac{u'_x v}{c^2}}}\]

and after some easy algebraic manipulations we get the important result

    \[ \boxed{\tan \theta =\dfrac{1}{\gamma}\dfrac{u'_y}{u'_x+v}}\]

We observe that, according to this last equation, with teh exception of l v=0, \tan \theta is smaller in the S frame than \tan \theta '=\dfrac{u'_y}{u'_x} in the S’-frame. In the non-relativist limit, we recover the result that our ordinary intuition and experience provides (\gamma \rightarrow 1):

    \[ \theta_{nonrel}=\dfrac{u'_y}{u'_x+v}\]

It is logical. In the nonrelativistic limit we do know that u'_y=u_y and u_x=u'_x+v, so the result agrees with our experience in the low velocity realm.

LOG#009. Relativity of simultaneity.

Other striking consequence of Lorentz transformations and then, of the special theory of relativity arises when explore the concept of simultaneity. Accordingly to the postulates of relativity, and the structure of Lorentz transformations we can understand the following statement:

    \[ \boxed{\mbox{Simultaneity is a relative concept. It depends on the inertial reference frame.}}\]

What does it means? Not surprisingly, if two arbitrary events happening in space and time, E_1, E_2, take simultaneously in one inertial reference frame, they do NOT do so in any other inertial reference frame. Indeed, Einstein himself guessed an alternative definition of simultaneity:

“(…) Two events E_1, E_2 taking place at two different locations are said to be simultaneous if two spherical light waves, emitted with the events, meet each other at the center of the tie line connecting the locations of the events(…)”

A proof can be done using Lorentz transformations as follows. In certain frame S’, two events E'_1 and E'_2 are found to be simultaneous, i.e., they verify that t'_1=t'_2. Therefore, using the Lorentz transformations (in the case of parallel motion of S’ with respect to S without loss of generality) we get

    \[ ct_1=\gamma (ct'_1+\beta x'_1)\]

    \[ ct_2=\gamma (ct'_2+\beta x'_2)\]

and thus, since t'_2=t'_1, the substractiong produces

    \[ c(t_2-t_1)=\gamma \beta (x'_2-x'_1)\]

We can recast this result as:

    \[ \boxed{\Delta t=t_2-t_1=\dfrac{\beta}{c}\gamma (x'_2-x'_1)=\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}\dfrac{v}{c^2}(x'_2-x'_1)}\]

We can also derive this equation with a LIGHT CLOCK gedanken experiment. A light clock of proper length L'=l' is at rest in the S’ frame. Its x’-axis moves at speed v parallel to the x-axis in teh S-frame. We attach some mirrors (denoted by M) to the short ends of the clock, and light travels parallel to the direction of motion. See the next figure of this device:

SimultaneityLightClock

At the initial time, we synchronize the clocks in S and S’, meaning that t=t'=0 when x=x'=0 and the light flash is emitted. In the S’-frame, light propagates in both directions at speed of light, and then, it reaches the mirrors at the ends at the same time

    \[ t'=\dfrac{s'}{c}=\dfrac{L'}{2c}\]

In the S-frame, by the other hand, the length of the light clock is contracted due to length contraction L=\sqrt{1-\beta^2}L' but light still propagates at speed c. However, the left-hand mirror is moving toward the light at speed v. In the time interval t_1 required for light to reach the left mirror, light travels a distance

    \[ ct_1=\dfrac{L}{2}-vt_1\]

In the same way, the right-hand mirror is running away from the light flash. In a certain time t_2 required for light to reach it, light should travel a total length

    \[ ct_2=\dfrac{L}{2}+vt_2\]

Substracting both equations, we obtain

    \[ t_2-t_1=\dfrac{L}{2(c-v)} - \dfrac{L}{2(c+v)}=\dfrac{L}{2}\dfrac{2v}{c^2-v^2}=L \dfrac{v}{c^2} \dfrac{1}{1-\dfrac{v^2}{c^2}}\]

Using the contraction lenght result, i.e., using that

    \[ L=\dfrac{L'}{\gamma}=\sqrt{1-\dfrac{v^2}{c^2}}L'\]

and that

    \[ L'=x'_2-x'_1\]

in the S’-frame, we get the previous result

    \[ \Delta t= t_2-t_1=\dfrac{1}{ \sqrt{1-\dfrac{v^2}{c^2}}}\dfrac{v}{c^2}(x'_2-x'_1)=\dfrac{\beta}{c}\gamma (x'_2-x'_1)\]

Q.E.D.

Therefore, the meaning of this boxed formulae is straighforward:

If two events are simultaneous in the S’-frame are simultaneous in the S-frame in the case they do happen at the same position (x'_2=x'_1) and/or the relative velocity v between the two frames is zero (v=0).

Moreover, we have this interesting additional result:

The larger the spatial separation is between two simultaneous events in the S’-frame, and/or the higher the relative velocity is between the S and the S’ frame, the greater is the temporal separation of the events in the S-frame.

LOG#008. Length contraction.

Once we introduce the postulates of special relativity and we have deduced the generalization of galilean transformations for electromagnetism and mechanics, the Lorentz transformation. We can deduce some interesting results.

Suppose we have two events E_1 and E_2, whose coordinates of space and time are given generally by E_1(x_1,y_1,z_1,t_1) and E_2(x_2,y_2,z_2,t_2). We also suppose, for simplicity, that the relative motion is along the x-axis. Imagine a rod, whose ”rest” length is

    \[ L_0= x_2- x_1\]

It is evident from the structure of Lorentz transformations that time depends on the observer frame (S or S’) and we have to fix the notion of “simultaneity” to measure the rod length in a meaningful way. Therefore, we can set t'_1=t'_2 to “syncronize” our stick measurements in “motion”, i.e., we have to measure the position of the rod in the same time in order to determine its length at motion!

Using the inverse Lorentz transformations:

    \[ x_1=\gamma (x'_1+vt'_1)\]

and

    \[ x_2=\gamma (x'_2+vt'_2)\]

Therefore, substracting both equations, we get, using the temporal condition (simultaneity) as well:

    \[ x_2-x_1=L_0=\gamma (x'_2-x'_1)\]

or equivalenty

    \[ \dfrac{x_2-x_1}{\gamma} = (x'_2-x'_1)\]

i.e.

    \[ \boxed{L'=\sqrt{1-\beta^2}L_0}\]

This result is known in Special Relativity (SR) as length contraction. Bodies in motion have “dimensions” that are shorter than those “in rest”. Of course, according to the postulates of relativity, it is relative. From the S’ frame, objects that were “in rest” in S appear to be shorter as well. In that case, from the viewpoint of S’ L'_0=x'_2-x'_1 and L_0=x_2-x_1 if we set t_1=t_2 for our “stick”. In this case we get:

    \[ x'_1=\gamma (x_1-vt_1)\]

and

    \[ x'_2=\gamma (x_2-vt_2)\]

and then

    \[ L'_0=x'_2-x'_1=\gamma ( x_2-x_1)\]

so, according to S’, the length of the rod in the S frame is

    \[ L_0=L'_0\sqrt{1-\beta^2}\]

i.e., again, the body in motion is “shorter” than the same body “in rest”. The only careful point is to realize if the proper “length” is known or the contracted length, and then use the suitable expression to obtain the contracted length or the proper length, respectively.

There is an alternative proof of this result using what Einstein himself called LIGHT CLOCK. The light clock is a nice Gendankenexperiment using frames and light signals between the S and S’ frames. S is at rest relative to S’. S’ is in motion relative to S. At t’=t=0 a light signal ( a “flash”) is emitted in S’, where there is an object with lenght equal to l’ (since we are in the S’ frame).

lightclockSprime

By the other hand, in the S frame we have the following events:

SframeLightclock

Now we will see the physical picture behind the two frames.

A) S’ frame. The light flash travel till the end ob the object, where is put a mirror and it comes back. The path traveled by the light is equal to 2l’ during a time t’, and thus, the length observed by the S’ frame is equal to:

    \[ l'=\dfrac{1}{2}ct'\]

B) S frame. The light flash from t=0, but the object is in motion too, so it travels an additional quantity t=t_1 until the ray is reflected, and another extra quantity vt_2 till it arrives to our origin when we measure the light time of arrival. There are two differents contributions to the total length l:

    \[ l_1=l+vt_1\]

    \[ l_2=l-vt_2\]

But, of course, l_1=ct_1 and l_2=ct_2, and then

    \[ (c-v)t_1=l\]

and

    \[ (c+v)t_2=l\]

The total running time of the light path in the S-frame will be:

    \[ t=t_1+t_2=\dfrac{l}{c-v}+\dfrac{l}{c+v}=\dfrac{l(c+v)+l(c-v)}{c^2-v^2}=\dfrac{2lc}{c^2-v^2}=\dfrac{2l}{c}\dfrac{1}{1-\frac{v^2}{c^2}}\]

or

    \[ t=\gamma^2\dfrac{2l}{c}\]

Therefore, the light clock in the S-frame measures

    \[ l=\dfrac{1}{\gamma^2}\dfrac{1}{2}ct\]

Dividing the results from A) and B), we get

    \[ \dfrac{l'}{\gamma l}=\dfrac{\gamma t'}{t}\]

But as we know that the time contraction implies

    \[ \gamma t'=t\]

so we get at last

    \[ l=\dfrac{l'}{\gamma}\]

The main conclusion is the following:

    \[ \boxed{\mbox{Length is relative. Lengths measured in the direction of motion are shorter.}}\]

This phenomenon is known as LENGTH CONTRACTION in special relativity.

Of the whole set of inertial observers moving along a certain direction, an observer at rest relative to an object extended in that direction measures the greatest length for that object. This length is commonly called PROPER LENGTH of the object. Lengths in TRANSVERSE( or orthogonal) directions of motion are not subject to length contractions.

LOG#007. Time dilation.

Suppose two events happening in the S’-frame at the same point at different times. E'_1(x',y',z',t'_1) and E'_2(x',y',z',t'_2). What is the temporal separation in the S-frame? According to Lorentz transformations, it is:

c(t_2-t_1)=\gamma \left[ c(t'_2-t'_1)+\beta(x'-x')\right]

or equivalently

t_2-t_1=\Delta t=\gamma (t'_2-t'_1)=\gamma \Delta t'

i.e.

\boxed{\Delta t = \gamma \Delta t'}

Of course, in the parallel case, if two events are happening in the S-frame at the same point but different times, i.e., if E_1(x,y,z,t_1) and E_2(x,y,z,t_2), their temporal separation in the S’-frame will be, using the inverse Lorentz transformations will be:

c(t'_2-t'_1)=\gamma \left[ c(t_2-t_1)-\beta(x-x)\right]

Thus,

t'_2-t'_1=\Delta t' = \gamma (t_2-t_1) =\Delta t

i.e.

\boxed{\Delta t' = \gamma \Delta t}

The boxed equations are called the TIME DILATION.

We can use an alternative procedure to derive this deep result in relativity (a result with important phenomenology!). We can use a LIGHT CLOCK device. In the S’-frame, we send light rays to a mirror as a “clock”. In the S-frame, in motion relative to S’ with velocity parallel to it we have the following scheme for the “light-clock”:

clock2

Thus, we have the above Pytaghorean relationship. The tic-tac in the S-frame is \Delta t=t/2. The tic-tac in S’ is equal to \Delta t'= t. Moreover, accordingly to the S’-frame, a light flash will travel a distance 2d (forward and back, reflected on a mirror). Therefore:

2d=ct'

Therefore, by the Pythagorean theorem in the triangle with the sides given in the above picture, we have

\dfrac{v^2t^2}{4}+c^2\dfrac{t'^2}{4}=\dfrac{c^2t^2}{4}

and then

c^2\dfrac{t'^2}{4}=\dfrac{c^2t^2-v^2t^2}{4}

so

t'^2=\dfrac{t^2}{\gamma^2 }

i.e., since we can choose initial tic-tacs in S and S’ equal to zero, we obtain t'=\Delta t' and \Delta t= t, and we also get

\Delta t = \gamma \Delta t' as before!

CONCLUSION:

\mbox{Time is relative. Time measurements of clocks (tic-tacs) in motion are longer.}

Indeed, of the whole set of inertial observers, an observer at rest relative to a process measures the shortest possible time for that process. It is called the PROPER TIME, and it is generally denoted by the greek letter \tau.

CAUTION: It seems that the above two formula for the time dilation are contradictory, but it is wrong. They are OK. We must keep in mind that the time variables on the right-hand side of those equations mean “proper time” of a concrete process, i.e., it stands for the time that an observer measures being at rest relative to that process. The left-hand side of both equations denotes the time an observer measures being “in motion” relative to that process. When S and S’ are in motion, there can only be ONE frame where the process is taken as “being in rest”. We have to select one of the equations, we can NOT choose both!

LOG#006. Lorentz Transformations(II).

    \[ \boxed{ \begin{cases} x'_0=ct'=\gamma (ct - \mathbf{\beta} \cdot \mathbf{r}) \\ \mathbf{r'}=\mathbf{r}+(\gamma -1) \dfrac{(\mathbf{\beta}\cdot \mathbf{r})\mathbf{\beta}}{\beta^2} -\gamma \beta ct \\ \gamma = \dfrac{1}{\sqrt{1-\beta^2}}= \dfrac{1}{\sqrt{1-\beta_x^2-\beta_y^2-\beta_z^2}} \end{cases}}\]

    \[ \boxed{\left( \begin{array}{c} ct' \\ x' \\ y' \\ z' \end{array} \right) = \begin{pmatrix} \gamma & -\gamma \beta_x & -\gamma \beta_y & -\gamma \beta_z \\ -\gamma \beta_x & 1+(\gamma -1)\dfrac{\beta_{x}^{2}}{\beta^2} & (\gamma -1)\dfrac{\beta_x \beta_y}{\beta^2} & (\gamma -1)\dfrac{\beta_x \beta_z}{\beta^2} \\ -\gamma \beta_y & (\gamma -1)\dfrac{\beta_y \beta_x}{\beta^2} & 1+(\gamma -1)\dfrac{\beta_{y}^{2}}{\beta^2} & (\gamma -1)\dfrac{\beta_y \beta_z}{\beta^2} \\ -\gamma \beta_z & (\gamma -1)\dfrac{\beta_z \beta_x}{\beta^2} & (\gamma -1)\dfrac{\beta_z \beta_y}{\beta^2} & 1+(\gamma -1)\dfrac{\beta_{z}^{2}}{\beta^2} \end{pmatrix} \left( \begin{array}{c} ct\\ x\\ y\\ z\end{array}\right)}\]

These equations define the most general (direct) Lorentz transformations  and we see they are not those in the previous post! I mean, they are not the one with the relative velocity in the direction of one particular axis, as we derived in the previous log. We will derive these equations. How can we derive them?

The most general Lorentz transformation involves the following scenario (a full D=3+1 motion):

1st) The space-time coordinates of an event E are described by one observer (and frame) A at rest at the origin of his own frame S. The observer B is at rest at the origin in a second frame S’. S and S’ have parallel axes.

2nd) The origin of the S and S’ frames coincide at t=t’=0.

3rd) B moves relative to A with a velocity  3d vector (space-like) given by

    \[\mathbf{v}=(v_x,v_y,v_z)\]

.

4th) The position vector of the event in the S-frame is \mathbf{r}=(x,y,z). It is decomposed into  “horizontal/vertical” or parallel/orthogonal pieces as follows

    \[ \mathbf{r}=\mathbf{r}_\parallel + \mathbf{r}_ \perp \]

The following transformation is suitable for the S’-frame, defining \beta=\mathbf{v}/c=(v_x/c,v_y/c,v_z/c):

    \[ ct'=x_0=\gamma(ct-\beta r_\parallel)=\gamma (ct - \mathbf{\beta} \cdot \mathbf{r})\]

    \[ \mathbf{r'}_\parallel = \gamma (\mathbf{r}_\parallel - \mathbf{\beta}ct )\]

    \[ \mathbf{r'}_\perp=\mathbf{r}_\perp \]

where the dot represents scalar product. Using the elementary knowledge and application of the scalar product with projections of vectors, we calculate the projection of the position vector onto the velocity vector in any frame with the scalar product of the position vector with a normalized velocity vector, \mathbf{\hat{v}}=\mathbf{v}/v . In the S’-frame we will get the projection \mathbf{\hat{v}}\cdot \mathbf{r}. Therefore,

    \[ \mathbf{r}_\parallel = (\hat{\mathbf{v}}\cdot \mathbf{r})\hat{\mathbf{v}}\]

and thus, the component of the position vector with respect to the parallel direction to the velocity will be:

    \[ \mathbf{r}_\parallel = (\hat{\mathbf{v}}\cdot \mathbf{r})\hat{\mathbf{v}}= \dfrac{(\mathbf{v}\cdot \mathbf{r})\mathbf{v}}{v^2}\]

or

    \[ \mathbf{r}_\parallel = \dfrac{(\mathbf{\beta}\cdot \mathbf{r})\mathbf{\beta}}{\beta^2}\]

Then, since \mathbf{r}_\perp = \mathbf{r}-\mathbf{r}_\parallel, we have

    \[ \mathbf{r}_\perp = \mathbf{r}- \dfrac{(\mathbf{\beta}\cdot \mathbf{r})\mathbf{\beta}}{\beta^2}\]

Finally, we put together the vertical/horizontal (orthogonal/parallel) pieces of the general Lorentz transformations:

    \[ \mathbf{r'} =\mathbf{r'}_\parallel + \mathbf{r'}_ \perp = \gamma \left( \dfrac{(\mathbf{\beta}\cdot \mathbf{r})\mathbf{\beta}}{\beta^2} -\beta ct \right)+\mathbf{r}- \dfrac{(\mathbf{\beta}\cdot \mathbf{r})\mathbf{\beta}}{\beta^2}\]

Then, the general 4D=3d+1 Lorentz transformation (GLT) from S to S’ are defined through the equations:

    \[ \boxed{GLT(S\rightarrow S') \begin{cases} x'_0=ct'=\gamma (ct - \mathbf{\beta} \cdot \mathbf{r}) \\ \mathbf{r'}=\mathbf{r}+(\gamma -1) \dfrac{(\mathbf{\beta}\cdot \mathbf{r})\mathbf{\beta}}{\beta^2} -\gamma \beta ct \\ \gamma = \dfrac{1}{\sqrt{1-\beta^2}}= \dfrac{1}{\sqrt{1-\beta_x^2-\beta_y^2-\beta_z^2}} \end{cases}}\]

Q.E.D.

The inverse GLT (IGLT) will be:

    \[ \boxed{IGLT(S'\rightarrow S) \begin{cases} x_0=ct=\gamma (ct' + \mathbf{\beta} \cdot \mathbf{r}') \\ \mathbf{r}=\mathbf{r}'+(\gamma -1) \dfrac{(\mathbf{\beta}\cdot \mathbf{r}')\mathbf{\beta}}{\beta^2} +\gamma \beta ct' \\ \gamma = \dfrac{1}{\sqrt{1-\beta^2}}= \dfrac{1}{\sqrt{1-\beta_x^2-\beta_y^2-\beta_z^2}} \end{cases}}\]

Indeed, these transformations allow a trivial generalization to D=d+1, i.e., these transformations are generalized to d-spatial dimensions simply allowing a d-space velocity and beta parameter, while time remains 1d. Indeed, the Lorentz transformations form a group. A group is a mathematical gadget with certain “nice features” that physicists and mathematicians love. You can imagine the Lorentz group in D=d+1 dimensions as a generalization of the rotation group  called Lorentz group. The Lorentz group involves rotations around the spatial axes plus the so-called “boosts”, transformations involving mixing of space and time coordinates. Indeed, the Lorentz transformations involving relative motion along one particular axis IS a (Lorentz) boost! That is, the simplest Lorentz transformations like the one in the previous posts are “boosts”.

With the above transformations, the GLT can be easily written in components:

    \[ \boxed{\left( \begin{array}{c} ct' \\ x' \\ y' \\ z' \end{array} \right) = \begin{pmatrix} \gamma & -\gamma \beta_x & -\gamma \beta_y & -\gamma \beta_z \\ -\gamma \beta_x & 1+(\gamma -1)\dfrac{\beta_{x}^{2}}{\beta^2} & (\gamma -1)\dfrac{\beta_x \beta_y}{\beta^2} & (\gamma -1)\dfrac{\beta_x \beta_z}{\beta^2} \\ -\gamma \beta_y & (\gamma -1)\dfrac{\beta_y \beta_x}{\beta^2} & 1+(\gamma -1)\dfrac{\beta_{y}^{2}}{\beta^2} & (\gamma -1)\dfrac{\beta_y \beta_z}{\beta^2} \\ -\gamma \beta_z & (\gamma -1)\dfrac{\beta_z \beta_x}{\beta^2} & (\gamma -1)\dfrac{\beta_z \beta_y}{\beta^2} & 1+(\gamma -1)\dfrac{\beta_{z}^{2}}{\beta^2} \end{pmatrix} \left( \begin{array}{c} ct\\ x\\ y\\ z\end{array}\right)}\]

Q.E.D.

These transformations can be written in a symbolic way using matrix notation as \mathbb{X}'=\mathbb{L}\mathbb{X} or using tensor calculus:

    \[ x^{\mu'}=\Lambda^{\mu'}_{\;\nu} x^\nu\]

The inverse GLT (IGLT) will be in component way:

    \[ \boxed{\left( \begin{array}{c} ct \\ x \\ y \\ z \end{array} \right) = \begin{pmatrix} \gamma & \gamma \beta_x & \gamma \beta_y & \gamma \beta_z \\ \gamma \beta_x & 1+(\gamma -1)\dfrac{\beta_{x}^{2}}{\beta^2} & (\gamma -1)\dfrac{\beta_x \beta_y}{\beta^2} & (\gamma -1)\dfrac{\beta_x \beta_z}{\beta^2} \\ \gamma \beta_y & (\gamma -1)\dfrac{\beta_y \beta_x}{\beta^2} & 1+(\gamma -1)\dfrac{\beta_{y}^{2}}{\beta^2} & (\gamma -1)\dfrac{\beta_y \beta_z}{\beta^2} \\ \gamma \beta_z & (\gamma -1)\dfrac{\beta_z \beta_x}{\beta^2} & (\gamma -1)\dfrac{\beta_z \beta_y}{\beta^2} & 1+(\gamma -1)\dfrac{\beta_{z}^{2}}{\beta^2} \end{pmatrix} \left( \begin{array}{c} ct'\\ x'\\ y'\\ z'\end{array}\right)}\]

and they can be written as

    \[ \mathbb{X}=\mathbb{L}^{-1}\mathbb{X'}$, or using tensor notation \[ x^\rho=(\Lambda^{-1})^\rho_{\;\mu'} x^{\mu'}\]

in such a way that

    \[ x^{\mu'} = \Lambda^{\mu}_{\; \nu} x^\nu \rightarrow (\Lambda^{-1})^{\rho}_{\; \mu'}x^{\mu'} = (\Lambda^{-1})^{\rho}_{\;\mu'}(\Lambda)^{\mu'}_{\;\nu} x^{\nu} = x^{\rho} = \delta ^{\rho}_{\; \nu}x^\nu\]

Thus,

    \[ (\Lambda^{-1})^{\rho}_{\;\mu'}(\Lambda)^{\mu'}_{\;\nu} = \delta ^{\rho}_{\; \nu}\]

or equivalently

    \[\mathbb{L}^{-1}\mathbb{L}=\mathbb{L}\mathbb{L}^{-1}=\mathbb{I}\]

.

\delta ^{\rho}_{\; \nu} is the “unity” tensor, also called Kronecker delta, meaning that its components are 1 if \rho = \nu and 0 otherwise (if \rho \neq \nu). The Kronecker delta is therefore the “unit” tensor with two indexes.

NOTATIONAL CAUTION: Be aware, some books and people use to assume you know when you need the matrix \mathbb{L} or its inverse \mathbb{L}^{-1}. Thus, you will often read and see this

 

    \[ x^{\mu'}=\Lambda^{\mu'}_{\;\nu} x^\nu \rightarrow x^\nu= \Lambda^\nu_{\;\mu'} x^{\mu'}\]

where certain abuse of language since it implies that

    \[ \Lambda^\nu_{\;\mu'} = (\Lambda^{-1})^{\mu'}_{\;\nu}\]

and because we have  to be mathematically consistent, the following  relationship is required to hold

    \[ \Lambda^\nu_{\;\mu'}\Lambda^{\mu'}_{\;\nu}=1\]

or more precisely, taking care with the so-called free indexes

    \[ \Lambda^\rho_{\;\mu'} \Lambda^{\mu'}_{\;\sigma}=\delta^\rho_\sigma\]

as before.

LOG#005. Lorentz transformations(I).

LorTransformations

For physicists working with objects approaching the light speed, e.g., handling with electromagnetic waves, the use of special relativity is essential.

The special theory of relativity is based on two single postulates:

1st) Covariance or invariance of all the physical laws (Mechanics, Electromagnetism,…) for all the inertial observers ( i.e. those moving with constant velocity relative to each other).

It means that there is no preferent frame or observer, only “relative” motion is meaningful when speaking of motion with respect to certain observer or frame. Indeed, unfortunately, it generated a misnomer and a misconception in “popular” Physics when talking about relativity (“Everything is relative”). What is relative then? The relative motion between inertial observers and its description using certain “coordinates” or “reference frames”. However, the true “relativity” theory introduces just about the opposite view. Physical laws and principles are “invariant” and “universal” (not relative!).  Einstein himself was aware of this, in spite he contributed to the initial spreading of the name “special relativity”, understood as a more general galilean invariance that contains itself the electromagnetic phenomena derived from Maxwell’s equations.

2nd) The speed of light is independent of the source motion or the observers. Equivalently, the speed of light is constant everywhere in the Universe.

No matter how much you can run, speed of light is universal and invariant. Massive particles can never move at speed of light. Two beams of light approaching to each other does not exceed the speed of light either. Then, the usual rule for the addition of velocities is not exact. Special relativity provides the new rule for adding velocities.

In this post, the first of a whole thread devoted to special relativity, I will review one of the easiest ways to derive the Lorentz transformations. There are many ways to “guess” them, but I think it is very important to keep the mathematics as simple as possible. And here simple means basic (undergraduate) Algebra and some basic Physics concepts from electromagnetism, galilean physics and the use of reference frames. Also, we will limite here to 1D motion in the x-direction.

Let me begin! We assume we have two different observers and frames, denoted by S and S’. The observer in S is at rest while the observer in S’ is moving at speed v with respect to S. Classical Physics laws are invariant under the galilean group of transformations:

    \[ x'=x-vt\]

We know that Maxwell equations for electromagnetic waves are not invariant under Galileo transformations, so we have to search for some deformation and generalization of the above galilean invariance. This general and “special” transformation will reduce to galilean transformations whenever the velocity is tiny compared with the speed of light (electromagnetic waves). Mathematically speaking, we are searching for transformations:

    \[ x'=\gamma (x-vt) \]

and

    \[ x=\gamma (x'+vt')\]

for the inverse transformation. There \gamma=\gamma(c,v) is a function of the speed of light (denoted as c, and constant in every frame!) and the relative velocity v of the moving object in S’ with respect to S. The small velocity limit of special relativity to galilean relativity imposes the condition:

    \[ \displaystyle{\lim_{v \to 0} \gamma (c,v) =1}\]

By the other hand, according to special relativity second postulate, light speed is constant in every reference frame. Therefore, the distance a light beam ( or wave packet) travels in every frame is:

    \[ x=ct\]

in S, or equivalently

    \[ x^2=c^2t^2\]

and

    \[ x'=ct'\]

in S’, or equivalently

    \[ x'^2=c^2t'^2\]

Then, the squared spacial  separation between the moving light-like object at S’ with respect to S will be

    \[ x^2-x'^2=c^2(t^2-t'^2)\]

Squaring the modified galilean transformations, we obtain:

    \[ x'^2=\gamma ^2(x-vt)^2 \rightarrow x'^2=\gamma ^2 (x^2+v^2t^2-2xvt) \rightarrow x'^2-\gamma ^2x^2+2\gamma ^2xvt=\gamma ^2 v^2t^2\]

    \[ x^2=\gamma ^2 (x'+vt')^2 \rightarrow x^2-\gamma ^2x'^2-2\gamma ^2x'vt'=\gamma ^2v^2t'^2\]

The only “weird” term in the above last two equations are the mixed term with “xvt” (or the x’vt’ term). So, we have to make some tricky algebraic thing to change it. Fortunately for us, we do know that x'=\gamma(x-vt), so

    \[ x'=\gamma x -\gamma vt \rightarrow \gamma x'=\gamma ^2 x-\gamma ^2 vt \rightarrow \gamma xx'=\gamma ^2 x^2-\gamma ^2 xvt\]

and thus

    \[ 2\gamma xx'=2 \gamma ^2 x^2-2\gamma ^2 xvt \rightarrow 2\gamma ^2 xvt =2\gamma ^2x^2-2\gamma xx'\]

In the same way,  we proceed with the inverse transformations:

    \[ x=\gamma x'+\gamma vt' \rightarrow \gamma x=\gamma ^2x'+\gamma ^2vt' \rightarrow \gamma xx'=\gamma ^2x'^2-\gamma ^2x'vt'\]

and thus

    \[ 2\gamma xx'=2\gamma^2x'^2+2\gamma^2x'vt' \rightarrow 2\gamma^2x'vt'=2\gamma^2x'^2-2\gamma^2xx'\]

We got it! We can know susbtitute the mixed x-v-t and x’-v-t’ triple terms in terms of the last expressions. In this way, we get the following equations:

    \[ x'^2=\gamma ^2(x-vt)^2 \rightarrow x'^2=\gamma ^2(x^2+v^2t^2-2xvt) \rightarrow x'^2-\gamma ^2x^2+2\gamma ^2x^2-2\gamma ^2xx'=\gamma ^2v^2t^2 \]

so

    \[ x'^2+\gamma ^2x^2-2\gamma ^2xx'=\gamma ^2v^2t^2\]

    \[ x'^2=\gamma ^2(x'+vt')^2 \rightarrow x^2=\gamma ^2(x'^2+v^2t^2+2x'vt') \rightarrow x^2-\gamma ^2x'^2+2\gamma ^2x'^2-2\gamma ^2xx'=\gamma ^2v^2t'^2 \]

and then

    \[ x^2+\gamma ^2x'^2-2\gamma ^2xx'=\gamma ^2v^2t'^2\]

And now, the final stage! We substract the first equation to the second one in the above last equations:

    \[ x^2-x'^2+\gamma ^2(x'^2-x^2)=\gamma ^2v^2(t'^2-t^2) \rightarrow (x'^2-x^2)(\gamma ^2-1)= \gamma ^2v^2(t'^2-t^2)\]

But we know that

    \[ x^2-x'^2=c^2(t^2-t'^2)\]

, and so

    \[ (x'^2-x^2)(\gamma ^2-1)= \gamma ^2v^2(t'^2-t^2) \rightarrow c^2(x'^2-x^2)(\gamma ^2-1)= \gamma ^2v^2(x'^2-x^2)\]

then

    \[ c^2(\gamma ^2-1)= \gamma ^2v^2 \rightarrow -c^2= -\gamma ^2c^2+\gamma ^2v^2 \rightarrow \gamma ^2=\dfrac{c^2}{c^2-v^2}\]

or, more commonly we write:

    \[\gamma ^2=\dfrac{1}{1-\dfrac{v^2}{c^2}}\]

and therefore

    \[ \gamma =\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}\]

Moreover, we usually define the beta (or boost) parameter to be

    \[ \beta = \dfrac{v}{c}\]

To obtain the time transformation we only remember that

    \[ x'=ct'\]

and

    \[ x=ct\]

for light signals, so then, for time we  obtain:

    \[ x'=\gamma (x-vt) \rightarrow t' =x' /c= \gamma (x/c-vt/c)=\gamma ( t- vx/c^2)\]

Finally, we put everything together to define the Lorentz transformations and their inverse for 1D motion along the x-axis:

    \[ x'=\gamma (x-vt)\]

    \[ y'=y\]

    \[ z'=z\]

    \[ t'=\gamma \left( t-\dfrac{vx}{c^2}\right)\]

and for the inverse transformations

    \[ x=\gamma (x'+vt)\]

    \[ y=y'\]

    \[ z=z'\]

    \[ t=\gamma \left( t'+\dfrac{vx'}{c^2}\right)\]

ADDENDUM: THE EASIEST, FASTEST AND SIMPLEST DEDUCTION  of \gamma (that I do know).

If you don’t like those long calculations, there is a trick to simplify the “derivation” above.  The principle of Galilean relativity enlarged for electromagnetic phenomena implies the structure:

    \[ x'=\gamma (x-vt)\]

and

    \[ x=\gamma (x'+vt')\]

for the inverse.

Now, the second postulate of special relativity says that light signals travel in such a way light speed in vacuum is constant, so t=x/c and t'=x'/c. Inserting these times in the last two equations:

    \[ x'=\gamma (1-v/c)x\]

and

    \[ x=\gamma (1+v/c)x'\]

Multiplying these two equations, we get:

    \[ x'x =\gamma ^2(1+v/c)(1-v/c)xx'\]

If we consider any event beyond the initial tic-tac, i.e., if we suppose t\neq 0 and t'\neq 0, the product xx' will be different from zero, and we can cancel the factors on both sides to get what we know and expect:

    \[ \gamma^2(1-v^2/c^2)=1\]

i.e.

    \[ \gamma = \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}\]

LOG#004. Feynmanity.

feynman_cern2

The dream of every theoretical physicist, perhaps the most ancient dream of every scientist, is to reduce the Universe ( or the Polyverse if you believe we live in a Polyverse, also called Multiverse by some quantum theorists) to a single set of principles and/or equations. Principles should be intuitive and meaningful, while equations should be as simple as possible but no simpler to describe every possible phenomenon in the Universe/Polyverse.

What is the most fundamental equation?What is the equation of everything? Does it exist? Indeed, this question was already formulated by Feynman himself  in his wonderful Lectures on Physics! Long ago, Feynman gave us other example of his physical and intuitive mind facing the First Question in Physics (and no, the First Question is NOT “(…)Dr.Who?(…)” despite many Doctors have faced it in different moments of the Human History).

Today, we will travel through this old issue and the modest but yet intriguing and fascinating answer (perhaps too simple and general) that R.P. Feynman found.

Well, how is it?What is the equation of the Universe? Feynman idea is indeed very simple. A nullity condition! I call this action a Feynman nullity, or feynmanity ( a portmanteau), for brief. The Feynman equation for the Universe is a feynmanity:

    \[ \boxed{U=0}\]

Impressed?Indeed, it is very simple. What is the problem then?As Feynman himself said, the problem is really a question of “order” and a “relational” one. A question of what theoretical physicists call “unification”. No matter you can put equations together, when they are related, they “mix” somehow through the suitable mathematical structures.  Gluing “different” pieces and objects is not easy.  I mean, if you pick every equation together and recast them as feynmanities, you will realize that there is no relation a priori between them. However, it can not be so in a truly unified theory. Think about electromagnetism. In 3 dimensions, we have 4 laws written in vectorial form, plus the gauge condition and electric charge conservation through a current. However, in 4D you realize that they are indeed more simple. The 4D viewpoint helps to understand electric and magnetic fields as the the two sides of a same “coin” (the coin is a tensor). And thus, you can see the origin of the electric and magnetic fields through the Faraday-Maxwell tensor F_{\mu \nu }. Therefore, a higher dimensional picture simplifies equations (something that it has been remarked by physicists like Michio Kaku or Edward Witten) and helps you to understand the electric and magnetic field origin from a second rank tensor on equal footing.

You can take every equation describing the Universe set it equal to zero. But of course, it does not explain the origin of the Universe (if any), the quantum gravity (yet to be discovered) or whatever. However, the remarkable fact is that every important equation can be recasted as a Feynmanity! Let me put some simple examples:

Example 1. The Euler equation in Mathematics. The most famous formula in complex analysis is a Feynmanity e^{i\pi}+1=0 or e^{2\pi i}=1+0 if you prefer the constant \tau=2\pi.

Example 2. The Riemann’s hypothesis. The most important unsolved problem in Mathematics(and number theory, Physics?) is the solution to the equation \zeta (s)=0, where \zeta(s) is the celebrated Riemann zeta function in complex variable s=\kappa + i \lambda, \kappa, \lambda \in \mathbb{R}. Trivial zeroes are placed in the real axis s=-2n \forall n=1,2,3,...,\infty. Riemann hypothesis is the statement that every non-trivial zero of the Riemann zeta function is placed parallel to the imaginary axis and they have all real part equal to 1/2. That is, Riemann hypothesis says that the feynmanity \zeta(s)=0 has non-trivial solutions iff s=1/2\pm i\lambda _n, \forall n=1,2,3,...,\infty, so that

    \[ \displaystyle{\lambda_{1}=14.134725, \lambda_{2}= 21.022040, \lambda_{3}=25.010858, \lambda _{4}=30.424876, \lambda_{5}=32.935062, ...}\]

I generally prefer to write the Riemann hypothesis in a more symmetrical and “projective” form. Non-trivial zeroes have the form s_n=\dfrac{1\pm i \gamma _n}{2} so that for me, non-trivial true zeroes are derived from projective-like operators \hat{P}_n=\dfrac{1\pm i\hat{\gamma} _n}{2}, \forall n=1,2,3,...,\infty. Thus

    \[ \gamma_1 =28.269450, \gamma_2= 42.044080, \gamma_3=50.021216, \gamma _4=60.849752, \gamma_5=65.870124,...\]

Example 3. Maxwell equations in special relativity. Maxwell equations have been formulated in many different ways along the history of Physics. Here a picture of that. Using tensor calculus, they can be written as 2 equations:

    \[ \partial _\mu F^{\mu \nu}-j^\nu=0\]

and

    \[ \epsilon ^{\sigma \tau \mu \nu} \partial _\tau F_{\mu\nu}=\partial _\tau F_{\mu \nu}+ \partial _\nu F_{\tau \mu}+\partial_\mu F_{\nu \tau}=0\]

Using differential forms:

    \[ dF=0\]

and

    \[ d\star F-J=0\]

Using Clifford algebra (Clifford calculus/geometric algebra, although some people prefer to talk about the “Kähler form” of Maxwell equations) Maxwell equations are a single equation: \nabla F-J=0 where the geometric product is defined as \nabla F=\nabla \cdot F+ \nabla \wedge F.

Indeed, in the Lorentz gauge  \partial_\mu A^\mu=0, the Maxwell equations reduce to the spin one field equations:

    \[ \square ^2 A^\nu=0\]

where we defined

    \[ \square ^2=\square \cdot \square = \partial_\mu \partial ^\mu =\dfrac{\partial^2}{\partial x^i \partial x_i}-\dfrac{\partial ^2}{c^2\partial t^2}\]

Example 4. Yang-Mills equations. The non-abelian generalization of electromagnetism can be also described by 2 feynmanities:

The current equation for YM fields is (D^{\mu}F_{\mu \nu})^a-J_\nu^a=0.

The Bianchi identities are (D _\tau F_{\mu \nu})^a+( D _\nu F_{\tau \mu})^a+(D_\mu F_{\nu \tau})^a=0.

Example 5. Noether’s theorems for rigid and local symmetries. Emmy Noether proved that when a r-paramatric Lie group leaves the lagrangian quasiinvariant and the action invariant, a global conservation law (or first integral of motion) follows. It can be summarized as:

    \[ D_iJ^i=0\]

for suitable (generally differential) operators D^i,J^i depending on the particular lagrangian (or lagrangian density) and \forall i=1,...,r.

Moreover, she proved another theorem. The second Noether’s theorem applies to infinite-dimensional Lie groups. When the lagrangian is invariant (quasiinvariant is more precise) and the action is invariant under the infinite-dimensional Lie group parametrized by some set of arbitrary (gauge) functions ( gauge transformations), then some identities between the equations of motion follow. They are called Noether identities and take the form:

    \[ \dfrac{\delta S}{\delta \phi ^i}N^i_\alpha=0\]

where the gauge transformations are defined locally as

    \[\delta \phi ^i= N^i_\alpha \epsilon ^\alpha\]

with N^i_\alpha certain differential operators depending on the fields and their derivatives up to certain order. Noether theorem’s are so general that can be easily generalized for groups more general than those of Lie type. For instance, Noether’s theorem for superymmetric theories (involving lie “supergroups”) and many other more general transformations can be easily built. That is one of the reasons theoretical physicists love Noether’s theorems. They are fully general.

Example 6. Euler-Lagrange equations for a variational principle in Dynamics take the form \hat{E}(L)=0, where L is the lagrangian (for a particle or system of particles and \hat{E}(L) is the so-called Euler operator for the considered physical system, i.e., if we have finite degrees of freedom, L is a lagrangian) and a lagrangian “density” in the more general “field” theory framework( where we have infinite degrees of freedom and then L is a lagrangian density \mathcal{L}. Even the classical (and quantum) theory of (super)string theory follows from a lagrangian (or more precisely, a lagrangian density). Classical actions for extended objects do exist, so it does their “lagrangians”. Quantum theory for p-branes p=2,3,... is not yet built but it surely exists, like M-theory, whatever it is.

Example 7.  The variational approach to Dynamics or Physics implies  a minimum ( or more generally a “stationary”) condition for the action. Then the feynmanity for the variational approach to Dynamics is simply \delta S=0. Every known fundamental force can be described through a variational principle.

Example 8. The Schrödinger’s equation in Quantum Mechanics H\Psi-E\Psi=0, for certain hamiltonian operator H. Note that the feynmanity is itself H=0 when we studied special relativity from the hamiltonian formalism. Even more, in Loop Quantum Gravity, one important unsolved problem is the solution to the general hamiltonian constraint for the gauge “Wilson-like” loop variables, \hat{H}=0.

Example 9. The Dirac’s equation (i\gamma ^\mu \partial_\mu - m) \Psi =0 describing free spin 1/2 fields. It can be also easily generalized to interacting fields and even curved space-time backgrounds. Dirac equation admits a natural extension when the spinor is a neutral particle and it is its own antiparticle through the Majorana equation

    \[ i\gamma^\mu\partial_\mu \Psi -m\Psi_c=0\]

Example 10. Klein-Gordon’s equation for spin 0 particles: (\square ^2 +m^2 )\phi=0.

Example 11. Rarita-Schwinger spin 3/2 field equation: \gamma ^{\mu \nu \sigma}\partial_{\nu}\Psi_\sigma+m\gamma^{\mu\nu}\Psi_\nu=0. If m=0 and the general conventions for gamma matrices, it can be also alternatively writen as

\gamma ^\mu (\partial _\mu \Psi_\nu -\partial_\nu\Psi_\mu)=0

Note that antisymmetric gamma matrices verify:

\gamma ^{\mu \nu}\partial_{\mu}\Psi_\nu=0

More generally, every local (and non-local) field theory equation for spin s can be written as a feynmanity or even a theory which contains interacting fields of different spin( s=0,1/2,1,3/2,2,…).  Thus, field equations have a general structure of feynmanity(even with interactions and a potential energy U) and they are given by \Lambda (\Psi)=0, where I don’t write the indices explicitly). I will not discuss here about the quantum and classical consistency of higher spin field theories (like those existing in Vasiliev’s theory) but field equations for arbitrary spin fields can be built!

Example 12. SUSY charges. Supersymmetry charges can be considered as operators that satisfy the condicion \hat{Q}^2=0 and \hat{Q}^{\dagger 2}=0. Note that Grassman numbers, also called grassmanian variables (or anticommuting c-numbers) are “numbers” satisfying \theta ^2=0 and \bar{\theta}^2=0.

The Feynman’s conjecture that everything in a fundamental theory can be recasted as a feynmanity seems very general, perhaps even silly, but  it is quite accurate for the current state of Physics, and in spite of the fact that the list of equations can be seen unordered of unrelated, the simplicity of the general feynmanity (other of the relatively unknown neverending Feynman contributions to Physics)

    \[ something =0\]

is so great that it likely will remain forever in the future of Physics. Mathematics is so elegant and general that the Feynmanity will survive further advances unless  a  Feynman “inequality” (that we could call perhaps, unfeynmanity?) shows to be more important and fundamental than an identity. Of course, there are many important results in Physics, like the uncertainty principle or the second law of thermodynamics that are not feynmanities (since they are inequalities).

Do you know more examples of important feynmanities?

Do you know any other fundamental physical laws or principles that can not be expressed as feynmanities, and then, they are important unfeynmanities?