#lorentz factor

today my teacher (he's awesome) taught me about the lorentz factor which has to do with time dilation, length contraction, and mass expansion. they all have to do with einstein's special theory of relativity (which is different from his general theory of relativity (how? not sure yet :P)) and are SUPER COOL. a few takeaways:

we can't travel at the speed of light because that would take infinite energy and uhhh that's kinda impossible

photons do not have mass! completely forgot that- the thing is its energy is affected not by its mass unlike other things that follow the law of E=mc^2

and related to that, the full formula actually is something like E^2=(mc^2)^2+(pc^2)^2 (don't quote me on this) so since a photon doesn't have mass, its energy depends on its momentum (represented by p)! and then that also has to do with wave-particle duality because its energy also depends on the wave's frequency, as I understand it.

Here I focus only on the magnitude of the vectors of force and momentum.

Force is generally defined as the time derivative of momentum.

where F is force, P is momentum, and τ is proper time.

Both sides of the above formula are multiplied by the speed of light in order to convert momentum to energy.

where c is the speed of light and Es is momentum energy.

Therefore, the formula of momentum energy is expressed as follows:

Momentum energy is obtained by multiplying the time integral of force by the speed of light.

When the direction of the force to be applied to a matter is the same as the direction of motion of the matter, the force is expressed by the following formula.

where R is the vacuum resistance, γ is the Lorentz factor, and v is the relative space speed of the matter with mass m.

Therefore, if the acceleration (dv/dt) is constant, as the space speed v increases, the Lorentz factor γ increases as shown in the figure below.

This means increased vacuum resistance. Hence the force required for the same acceleration increases, resulting in an increase in the momentum energy.

The Swimmer in the Stream

This story illustrates a point that will become very important in the foundations of Special Relativity.

Imagine a broad river flowing gently and evenly at speed r and that there are two markers anchored in line with the flow and at a set distance L apart. A swimmer who swims at a constant speed s is going to swim from the upstream marker to the downstream marker and back again. How long will this take?  

Let us denote the duration of the downstream lap by T1 and the duration of the upstream lap by T2. Swimming with the current the swimmer will have a combined speed of (s+r) and so T1 = L/(s+r). Swimming against the current the swimmer will have a net speed of (s-r) and so T2 = L/(s-r).  Total time T1+ T2 = 2Ls / (s2 – r2) =  g2 . 2L/s     where g= √(1 – r2/s2) . 

Now rearrange the two markers so that they lie directly across the stream but at the same distance L apart. If the swimmer does not allow for the current he will be swept downstream of his destination. He has to angle his swim upstream a bit. Using a bit of Pythagoras, the distance he will have to cover on each lap is  L√(1 + r2 T)  where T is the time it takes. His swimming speed is this distance divided by T. We can easily solve these two equations for T to discover that  T = L / (s2 – r2 )  and so his total time is  2Ls/(s2 – r2 ) =  2L/s /√(1 – r2/s2) = g. 2L/s

In other words, the duration of the swim down and up the river flow is longer than the duration of the swim across and back the river flow by a factor of g = √(1 – r2/s2).

Let us have a look at the factor g.  If the river flow is zero then g=1 so the swim takes the same duration of time in either direction, as you would expect. Also note that g is a positive dimensionless number that is always  ≥1.

What does all this aquatic athleticism have to do with Special Relativity? Well it is relevant to a whole lot of experiments that were taking place towards the end of the 19th century in an effort to discover more about the nature of light. It had already been discovered that light travels at a huge but finite speed close to 300 million meters per second. It had also been discovered that light was essentially electro-magnetic in nature. (See Faraday effect, Kerr effect etc).

But scientist were asking themselves, if light is an electro-magnetic wave, what medium is it travelling in?  The thinking was that there must be a type of invisible “lumiferous aether” enveloping everything and this provided the medium for the electromagnetic wave.

In 1886, Michelson and Morley set up a clever experiment to detect this aether. The apparatus was an interferometer designed to detect any effects of the earth’s movement through the aether field. It consisted of a beam splitter (half silvered mirror angled at 45 degrees to incoming light), plus two mirrors mounted on orthogonal arms exactly the same length as each other. The whole apparatus was mounted on a massive stone block floating in mercury in order to reduce vibrations and to allow everything to be rotated.

The experiment is very similar to our story of the swimmer in the stream. The observers have two carefully measured lengths set at right angles to each other, much like two axes in a reference frame, and the observers remain at rest in this frame. The role of our swimmers is taken by photons of light. The role of the stream is taken by the movement of the observers and their apparatus through the lumiferous aether. The speed at which this happens is the speed at which the Earth is moving through space as a result of its own rotation (460 m/sec at the equator) plus its orbit around the Sun (30,000 m/sec). Note that the speed of light is 10,000 times faster than the speed of the Earth’s orbit.

In principle, light travelling with the aether drift and then back again would take longer to complete such journey than light travelling across the aether wind and back again (simply because the light would lose more time a swimming against the tide on the way back than it gained getting on the way forward). Any travel time differences would show up as an interference pattern.

But try as they might, no such interference pattern could be generated, no matter which way the apparatus was oriented. Light travelling in one arm of the experiment always took the same amount of time to travel as light travelling in the other arm. No aether stream effects could be detected. The experiment became perhaps the most famous null experiment in physics. Albert Michelson received the Nobel Prize for physics in 1907.

The results of this experiment and others like it were very perplexing. Hendrik Lorentz, George Fitzgerald, Henri Poincaré and others worked hard to interpret what it all meant. What it meant was that the round trip duration of light over the two arms of the experiment was exactly the same, and since the two arms were designed to be exactly the same length as each other then the round trip speed of light must be the same in both arms.  

Let us go back to our swimmer in the stream parable again. Michelson and Morley and the two orthogonal swimming courses are fixed in place. The aether stream is flowing from East to West. But the photons swimming the East-West course take the same time as the photons swimming the North-South course. How can this be?  

Ultimately the experimenters were forced to make some hard decisions. Either the speed of light is an invariant for all inertial observers, with all sorts of consequences for the meaning of time, length speed and in fact all physical properties of Nature. Or maybe it is just that the length of the East-West course has shrunk by the factor of g.

The experimenters interpreted the results as proof that the aether theory was wrong. This may have been a little hasty. What they actually demonstrated was that our fundamental notions of length, time and speed had to be re-examined. Moving observers get different measurements from measurements made by stationary observers. Measurements are affected by how you take them. Everything becomes relative.

Hendrik Lorentz

Hendrik Antoon Lorentz was born in Arnhem Holland in 1853 and became Professor of Physics at Leiden University at the impressive age of 24. With George Fitzgerald, Oliver Lodge, Oliver Heaviside and Heinrich Hertz, Lorentz refined James Clerk Maxwell’s equations and theories of electromagnetism and established the electromagnetic nature of light. Lorentz and his former student Pieter Zeeman received the Nobel Prize for Physics in 1902. He became a major influencer to Albert Einstein.

Lorentz and friends struggled to understand it the results of the Michelson-Morley experiment. There should have been a factor of g between the travel times of light in the two arms of the Michelson Morley experiment, but there wasn’t.

There are about half a dozen potential reasons for the result. Maybe the east-west photons interfered with the north-south photons before detection. Maybe the north-south photons slowed down so that arrived at the same time as the east-west photons. Maybe the east-west photons sped up somehow.

The simplest answer was that there just isn’t any lumiferous aether. This view was championed by Einstein and Herman Minkowski. Einstein just accepted that the speed of light was the same in any inertial reference frame and it did not matter which way such an inertial frame was moving. He made this a starting postulate for his theory of Special Relativity.

However, Lorentz was never comfortable with abandoning the aether idea. He thought that perhaps the speed of the Earth’s movement affected electromagnetism in some way. In 1892 he proposed that moving bodies contracted in the direction of their motion by the factor of g. This at least explained the Michelson Morley result in a simple way. George Fitzgerald had suggested the same idea in a letter to the editor of Science in 1889.  In 1899 and 1904 Lorentz added the concept of time dilation to his thinking as well.

Lorentz developed the equations for changing from one set of (inertial) coordinates to another.  These equations are now known as Lorentz transformations. The factor g is usually a written as the Greek letter Gamma in lower case and is called the Lorentz factor. Lorentz’s work paved the way for Einstein’s approach and Einstein regarded Lorentz as his greatest influence.

Note that the idea by Fitzgerald and Lorentz that moving through the aether might cause lengths to contract by the factor g is not the same as the very similar idea that will turn up in Einstein’s Theory of Special Relativity. Lorentz was looking for a physical explanation for an experimental result and the contraction would apply in the reference frame of the observers moving with the apparatus. The Einstein version of events is that a contraction would exist, but only when measured in the reference frame of observers moving with respect to the whole experiment, and the answer would depend on such relative velocity and had nothing to do with any aether drift.

Special Relativity – (my attempt to explain its derivation)

At the start of the 20th century, Albert Einstein developed a remarkable description of physics and in doing so developed some remarkable insights. His approach is now called the Theory of Special Relativity.  Special Relativity has a set of simple foundations:

Approach the concepts of length, time, speed and simultaneity very carefully. 

Define events in terms of the usual 3 spatial dimensions, but add in time as well.

Make sure that time is well defined and coordinated (synchronized) throughout this reference frame. 

Choose a convenient system of units e.g. meters and seconds.

Make sure the reference frame is inertial i.e. the laws of classical physics hold true in a conveniently simple way.

These five points just create clarity about definitions and measurements.

However, there are three key postulates as well:

Insist that the laws of classical physics and the outcome of experiments do not depend on the inertial reference frame that the observer happens to choose to describe them in.

Recognize “classical relativity”, i.e. if body A is moving at velocity V with respect to another body B, then body B is moving at velocity (–V) with respect to body A. 

Postulate that the speed of light is always the same in any inertial reference frame, irrespective of how that frame is moving (providing it is not accelerating or rotating) or oriented.

Einstein did not speculate about why the speed of light is always the same - he simply took an axiomatic approach and assumed it to be invariant so that he could investigate the consequences.

It is a bit odd that his 1905 papers Einstein did not reference the work of Michelson and Morley, but is implausible that Einstein did not know of their experiments and ones similar.

Einstein did wonder about why some reference frames are inertial and others are not, and he attempted to deal with this in a later extension of Special Relativity that he called the Theory of General Relativity. But that is for another essay.

Special Relativity – time slows down

Consider a very simple clock constructed in the form of an evacuated tube in which a pulse of light travels from one end to the other in exactly one second. Every time this happens the clock ticks. It is a simple clock – no moving parts, no electric fields, no outside disturbances. It is an improbably long clock but that doesn’t matter.  It is c meters long, where c is the distance that light travels in one second.

Now choose an inertial reference frame and arrange the clock with its base at (0,0,0,0) and its length aligned to the z axis and fix this reference clock in place. Now imagine that an identical clock with an identical orientation whizzes by at time t = 0, travelling in the x direction at a very high and constant velocity of v meters/second. We are going to describe this second clock using our reference frame as described, and using the postulates of Special Relativity.

In one second the moving clock is v meters to the right. In our reference frame the photons inside the moving clock are travelling up a sloping line. A ramp if you like, with slope equal to c/v. The sloped path is longer than the vertical path being followed by the photons in our reference (stationary clock). But all photons must travel at the speed c (Postulate #3). So the photons moving up the sloping path cannot yet have reached the top of their moving clock. When all the clocks in our reference frame have just ticked to indicate that 1 second has elapsed, the photons in the moving clock have not yet reached the top of their clock, and so that moving clock has not yet ticked.  

If we describe the moving clock in our reference frame, we would have to conclude that it is running slow. It completes a fraction of one of its seconds in the time that our clocks complete one second.  

Express this as t’ g = t  where t’ is a time duration time in the moving clock and g is a number bigger than 1. A simple application of Pythagoras Rule shows that      g = √(1 – v2/c2). This formula is called the Lorentz factor.  

Time intervals are well defined in our reference frame. And we know exactly where the moving clock is in our reference frame at every tick of our reference clocks. So in 10 seconds (say), the moving clock will be 10v meters to the right. In fact we could position a camera at that point on our x axis and take a picture of both the moving clock and the local reference clock at the instance the moving clock whizzes by. Our clocks will show 10 seconds, but the picture will reveal the moving clock to show a lesser time,10/g. 

This is not an illusion. It is a real result. For it not to occur the light in the moving clock would have had to speed up in terms of our reference system so that it could hit the top of its moving tube at the same time as our clocks do. But then we would have to say that some light in our reference frame is moving faster than some other light, which is contrary to experimental results and Postulate #3.

Alternatively the length of the moving clock could have shrunk down to meet its photon in one of our seconds. But over time this would result in the moving clock shrinking to nothing, which is not what is observed either.

It is hard to comprehend. It seems contrary to everyday experience. That is because it is contrary to everyday experience.  

Our brains struggle to fit it into the patterns of what we already know. We look for an explanation in terms of things we already know about. Perhaps the movement of the clock affects its functioning somehow. A bit like running underwater is slower than running on land.

I’m afraid not. We can attach a second reference frame to the second clock so that both the clock and the new reference frame are moving together. In other words the second clock is now stationary and the clock that was previously considered to be stationary now has to be considered to be moving. If we describe the second clock in terms of this new reference frame it would be perfectly normal. One tick every second.  

And if we attempt to describe our first set of clocks in terms of this new set of coordinates, then it is our first set of clocks that will be observed to be running slow. The thought experiment is symmetrical. See Postulates 1 and 2.  

The outcome has been confirmed experimentally many times over. For example, we know that mesons decay at a certain rate in the laboratory and we can count the decay products. However, if a group of mesons originate in the sun, and come whizzing past at a great rate of knots, we find far fewer decay products. The mesons are no different and they are decaying just as fast per second of time in a clock moving with them. But their second is smaller than a second in our earthbound reference frame and so we detect fewer decay products in our detection meters.

The atomic clocks in satellites used for global positioning systems have to be very accurate in order to work. But since the satellites are moving with respect to the earth’s surface, their clocks need to be adjusted so that they keep pace with identical clocks here on earth. In fact they need to be adjusted for several relativistic effects. Time dilation due to relative velocity is just one of the adjustments required. Not even the largest one, which is gravitational time dilation (see General Relativity later on).

We have to let go of our concepts of time developed since we were infants. Time is nothing more than a sequence of events, as measured in a suitable reference frame. If you want a reference time you need some reference events.

We have multiple blows to our normal concepts of time  (1) everything we see is in the past and not in the “now” because of the finite time duration it has taken for the light to reach us, and (2) we will observe time running slow on every fast moving system that we try to observe.

What is the Time?

The simple question “What is the time please?” shows how deeply the concept of an absolute time is ingrained into out consciousness. There are in fact two versions of this question. The first is a request to be told what is the agreed reference standard time for the local location i.e. what would a “correct” clock say if one were available. The second question could be a request about a time duration – as for example if a parent asks a time keeper at the school athletics carnival for the result of their child’s running race.

In science, and particularly in astronomy, we have to remember that time is an artificial man made construct. There is no absolute time. There is no absolute standard of time duration. All we can observe is the passage of events. If those events are regular, such as the vibrations of a quartz crystal in a watch, then they serve a useful purpose in measuring durations of other things as well – but only in the local inertial reference frame.

Special Relativity – lengths contract

Let us switch to a reference frame attached to the moving clock and describe some of the things that are happening from this perspective.  

Note that an event in one reference frame is also the same event in the other reference frame, and vice versa. Same event, just from a different perspective and hence measured differently.

According to pPostulate #2, one of the things that we can observe from our moving reference frame is that the original system of clocks and rulers is travelling past at the same speed v but in the opposite direction. We measure that they move v meters (measured in our frame) in one second (measured in our frame) and they measure that we move v meters (measured in their frame) in one of their seconds (measured in their frame), but in the opposite direction.

But v = distance divided by time in both frames. We both agree v is the answer for our relative velocity, and also that c is the speed of light as measured in each of our reference frames.  So  v = d/t = d’/t’  i.e. measurements in both references systems give the same numerical answer for the relative speed v.  (d and t are local distance and local time duration measurements respectively, and the dash indicates measurements in the ‘other’ reference system).

So                d = t/t’ . d’   and since   t = g t’   then       d = gd’  

In plain English, distance measurements between two events, measured in the other reference system, have to be scaled by g to give the same numerical value as the distance for the very same two events, but measured in our reference system.  

In even plainer English, things which we know to be a certain length when they are at rest in our system turn out to have a different (shorter) length when they are moving in that dimension and are measured properly in our reference frame. 

So, for example, even a meter ruler, which we know and expect to be a meter long when at rest in our reference frame, might turn out to measure 0.66 meters long when it is moving at half the speed of light (say) and has to be scaled up by 150% to give the same result as an identical reference meter ruler at rest in our laboratory. But an observer travelling with the moving meter ruler does not observe any variation in his ruler at all.  

Lorentz Transformations

We can choose different reference frames for our perspective. They can be moving (and usually are anyway), but it is simpler if they are not accelerating or rotating. Depending on the choice of reference frame, simple system descriptions can become complicated, and complicated system descriptions can become simple. For example would you rather analyse a trapeze act from the circus ringside, or from a passing train? And if there is a dynamical system on the train, would you prefer to analyse and describe it from a position on the train itself, or from a random viewpoint on the ground outside the train somewhere?

However, if there is a huge difference in speed between one reference frame and another, then we have to use Lorentz transformations for all the data and all the equations.

Other Relativistic Effects

Einstein went on to consider all sorts of physical behaviours using his four dimensional approach, and axioms about relativity and the speed of light. He developed a complete reworking of classical physics based on his new approach. He preserved classical physics but specified it more exactly and precisely, thus explaining some new experimental results involving light and fast moving systems.  

This created some new insights. Considerations of collisions as viewed from different reference frames led to the conclusion that the dynamic response of matter to force decreased when that matter was moving at very high speeds. In other words, it takes increasing amounts of force to generate the same amount of acceleration when the object is already moving at relativistic speeds.

The most famous insight was that energy and mass are intertwined as expressed by the equation  E2 = p2c2 + m2c4  where p is the momentum of the moving object and m is its mass when it is at rest in the reference frame.  

If the object is not moving the equation shortens to E = mc2.  

If the object has no rest mass, the equation shortens to E = pv. Since a photon has no inertial rest mass, this equation applies. It is interpreted as saying that the energy in a photon is equivalent to a certain amount of momentum.

It is a common mistake to shorten Einstein’s equation to E = mc2  and then to apply this to photons. Photons are definitely never at rest. It is the first term on the right hand side that is relevant to photons, not the second, though the numerical result is the same.

Minkowski invented four dimensional diagrams to describe simple systems.  He created a definition of separation called a metric   s = √(x2 + y2 + z2 – c2t2 ).  A Minkowski diagram is usually drawn as a three dimensional diagram with one physical dimension suppressed and replaced by the time dimension.  

(Annoyingly, the time dimension in a spacetime 4 vector is sometimes put first and sometimes put last.  There is also a confusing practice used by most authors in physics to switch away from using meters and seconds and decide to put c=1, with all time and length measurements rescaled accordingly. It is much tidier and shorter, but takes some getting used to.)