# LOG#013. 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 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:

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:

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:

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