Relativity for Beginners

Before we can deep–dive into Relativity, we need to understand the problem everyone was trying to solve.

The Problem   ↑

How does light travel through space? Eg. from the Sun to us here on Earth?

Easy to state, devilishly difficult to answer.

Above us the waves   ↑

Thanks to Huygens, Young and Fresnel, by the mid 19th century we were certain light was a wave. Not “hypothetically” nor “probably” nor “most likely”, but fact. Young’s elegant twin slit experiment finally settled the issue. Light undergoes diffraction, dispersion and interference, all things particles cannot. Case closed (for now…).

So light is a wave. The question then becomes, what medium does it move through? Here is where things get tricky…

Think of a wave, any wave. Say ripples on the surface of a lake. They are indisputably waves as they move across the surface of the water with observable wave crests (ie. peaks) and troughs (the valleys between the crests). You can combine the waves to make the crests larger or subtract them to cancel out the waves entirely. You can even combine two sets of ripples at different angles to get interference patterns. So far, so unremarkable.

Now remove the water. The ripples also vanish since there is nothing for them to “ripple” through. The water is the medium which conveys the wave, so if it’s gone then so too must the waves. Ditto waves on the ocean or sound through air or in a solid. Take away the medium through which they propagate (ie. ocean, air or solid) then waves must immediately disappear since there is nothing to carry them. Waves and media are inextricably linked.

I think you can see where I’m going here.

On Earth this makes sense, but what about in the vacuum of space? Is there some kind of substance which can carry light waves? During the 19th century the answer was — unequivocally — yes. Logically there had to be, for how else can light exist without a material for it to propagate through? But what?…

The “Luminiferous Æther”   ↑

Cast your mind back 150 years. We had no knowledge of space other than assumptions it was transparent, mainly due to thousands of years of observing stars and planets in the night skies. We also knew air pressure on Earth decreased with altitude. While at sea level it is, on average, 1.013 Bar. At the top of the Jungfrau in Switzerland (4158m) it is 0.65 Barr. It took a while to get to the top of Mount Everest (8848m), but up there it’s 0.381 Bar. Join the dots and air pressure will eventually reach zero if you go high enough (≈ 90km). Zero air pressure = no atmosphere. In other words, a vacuum. Although everyone was convinced this was true, it was only confirmed in the 1940s by NΑZI V2 rockets at the top of their ballistic arcs.

We have known since Torricelli's experiments with mercury filled glass tubes in the 17th century that light can easily shine through a vacuum. So there must be something in space which allows light to transit. But what?…

In the 19th century we had no idea, but assumed it had to exist. Whatever it was, it couldn’t be very substantial for its mass would interfere with the motions of planets, moons, comets etc.

This idea of cosmic “stuff” to act as a wave–medium was supported by no less a God Tier physicist than Maxwell, he of Maxwell Equations and electricity/ magnetism unification fame [ Note I ].

The existence of an all–pervading medium was such a commonly held belief it became axiomatic. It even acquired a distinguished sounding name: The Luminiferous Æther. It was believed it was everywhere and necessary for light to propagate, either on Earth or else between the us and the stars.

Alright, but how do you prove Æther exists?…

Science is an empirical activity. You make observations, inferences, test hypotheses and gather evidence to support your ideas. Only when there is an overwhelming amount of evidence can you confidently say a theory is valid.

In the 1880s everyone believed Luminiferous Æther was real. Even Maxwell. But unlike the humanities, “feels” ≠ “evidence”. The race was on to shift the dial from untested hypothesis to fact.

The road ahead   ↑

You are probably wondering what any of this has to do with Relativity. In a nutshell:

1. Light is a wave
2. Waves require a medium to propagate through
3. Experiment(s) were done to prove this medium exists for light
4. To everyone’s amazement the experiments didn’t yield the expected results. Which suggests (1) maybe there is no medium and (2) the speed of light might also be constant
5. Relativity was developed to explain these findings
6. A myriad of new fields became available to explore

Chasing phantoms   ↑

I will use an analogy to explain the concepts behind the Michelson–Morely (“M&M”) Æther experiment, before detailing the experiment itself.

Imagine being in a small boat on a calm lake. How can we determine we are moving if we can’t see out of the boat and aren’t allowed to touch the water? How about sending a pair of sonar waves at right–angles from the boat to bounce off distant targets back to the boat? When the waves return you combine them to get an interference pattern. Then send the right–angled pair of sonar waves out in a different direction: say the first time they went out at 12 and 3 o'clock, the second time send them out at 5 and 8 o'clock.

If the boat was moving, waves propagating away and back through the water would be affected by the boat’s motion. They would be dragged through the water so their wavelengths would change. If the 2nd and 1st interference patterns differ, then the boat must be moving. Neat!

To prove the existence of Æther in the 1880s, Michelson–Morely did a similar thing. They reasoned the Earth is moving through space at 30km/s as it orbits the Sun, so as it moves through the Æther, light beams shot out at right–angles and bounced back off mirrors should be affected, like the sonar waves in our boat example. Combine the beams and you should get a change in interference patterns.

The experiments   ↑

Originally in 1881 while visiting Potsdam, Michelson set up his apparatus with light beams at right angles, then created an interference pattern by combining the reflected rays. He used monochromatic light to ensure its wavelengths were identical (the light source was a sodium lamp but we would use lasers today). The path too/ from the mirrors was kept relatively short, ≈ 1m, so everything could fit onto a bench–top. When the experiment was run there was no change between the 1st and 2nd interference patterns. Michelson kept repeating the experiment, even a few months later when the Earth was moving through a different part of space. No matter what he did, there was no change.

Sceptics said his methodology was flawed in that the light–path was too short at only ≈ 1m. Also the light he used wasn’t 100% coherent (ie. the wave crests/ troughs were not exactly aligned, which is why we would use lasers today). Despite the nay–saying and nitpicking, Michelson didn’t give up. When he returned to the USA he teamed up with Morley in Cleveland and they completely redesigned the apparatus. The light–path was increased to ≈ 11m by bouncing it back and forth from 16 mirrors. They also reduced the impact of external vibrations by building everything onto a sandstone base floating on a pool of liquid mercury.

In 1887 they ran their new experiment and… nothing. To their dismay there was again no change in the interference patterns. They repeated at different times of the year and (annoyingly) still there was no change. They concluded that either the Æther was being dragged along by the Earth as it went around the Sun, or — more outrageously — that maybe there was no Æther and light moved through space of its own accord (the currently accepted view, see [ Note II ]).

In the 1890s opinions about this varied and people argued wildly [ Note III ]. Yet M&M’s results also gave rise to another, more troubling suggestion… Along with the possibility of there being no Æther, perhaps the speed of light in a vacuum never varies?…

We typically associate relativity with Einstein, but there were other players: Galileo, Lorenz and later on Minkowski.

I won’t explore the minutiae of every twist and turn prior to the publication of Einstein’s seminal Special Theory of Relativity in 1905 as the relevant Wikipedia article is sufficient. Instead my goal is to provide a broad overview of SR for the general reader, not a blow by blow account of who said what to whom and when. As Einstein supposedly said, everything should be as simple as possible, but not simpler.

Section Quick Links

• Two Postulates
• Constant “$ c $”
• Galilean Relativity
• Lorenz Transformations
• Length contraction
• Momentum dilation
• Time dilation
• Thought experiments
• “Miraculous Year” papers
• SR aftermath

Pick a pair of prickly postulates   ↑

A “postulate” is a statement or proposition which is assumed to be a fact. There may be no evidence to support it, it can be completely made up, it can even be wildly illogical. Nevertheless it is meant to act as a starting point for an argument, without which discussion cannot meaningfully proceed.

Einstein employed two postulates to underpin SR. He had no evidence (at the time) to support them. They were merely logical assumptions upon which he could build his theory:

Postulate #1

The laws of physics can be written in the same simple form for all inertial frames of reference (inertial observers) that are moving uniformly with respect to each other.

The first part is so obvious it is remarkable it needs stating — the same laws of physics apply everywhere to everyone regardless what they are doing. It is a restatement of Galilean Relativity and has been known since the early 17th century. For example, I may be sliding across an ice rink and you may be flying in an aeroplane, but the same physical laws apply to us both. Me on ice and you in your aircraft, the conservation of momentum, energy, Baryon number and everything else all applies and remains unchanged. Obvious.

The second bit is the one most people miss: all inertial frames of reference. This is where things are either stationary or moving at a constant velocity. To the non–physicist, constant velocity is when you are not changing direction and neither speeding up nor slowing down.

It is this “inertial frame of reference” which makes the 1905 version of Relativity “special”. Since, realistically, how can anything in the universe ever be stationary or travelling in a perfectly straight line with unchanging speed? Indeed it was in order to address this (crippling) limitation, along with the exclusion of gravity, that Einstein developed General Relativity ten years later.

Postulate #2

The velocity of light in a vacuum “$ c $” is a constant, and the same for all inertial observers, regardless of any relative motion between the source and the observer.

Unlike the first postulate, this one raised eyebrows. Einstein interpreted the M&M results to mean the speed of EMR (light) “$ c $” in a vacuum is fixed, irrespective of what the light emitter and observer are doing or how they are moving about…

Which is completely contrary to a Newtonian world, where velocities add or subtract. For example if you and I are stationary and I throw a tomato at you at (say) 2m/s, then it will hit you at 2m/s, ignoring air resistance. If I am instead travelling in a car at 15m/s towards you when I throw the tomato, it will hit you at (15+2)m/s. Ouch. Conversely if I am stationary and you are in a car moving away at 15m/s, then you will laughingly outrun the tomato and it will never hit you.

Obvious, right? Except according to Postulate #2 this does not apply to light (and likewise all EMR) in a vacuum. If we are both stationary and I shine a beam of light at you, its speed is obviously “$ c $”. But if I now race towards you at 200 000m/s, you will still observe the beam arriving at “$ c $”. Alternatively if we move away from each other at tremendous speeds, we will still measure beam between us as being unchanged at “$ c $”. What?!

The basis of “c”   ↑

It wasn’t as though Einstein pulled this out of his hat. He didn’t decide one day, while goofing off as a Technical Expert Class III at the Patent Office at Bern, Y’know what? From now on the speed of light in a vacuum is constant. Among a multitude of influences and contradictory ideas for the constancy of “$ c $”, he mostly relied upon the following:

A. Maxwell’s Equations 1862

Maxwell’s Equations are one the great triumphs in the understanding of light and the unification of electricity and magnetism into electromagnetism.

If you rearrange the Maxwell equations and do some math, you can derive an expression for the speed of light “$ c $” thus:

Where:
“$ \varepsilon_{0} $”   Vacuum Electric Permittivity (8.85419 ×10⁻¹² F/m)
“$ \mu_{0} $”   Vacuum Magnetic Permeability (1.25664 ×10⁻⁶ N/A^-2)
“$ c $”   Speed of light in a vacuum (299 792 458 m/s)
 

Since “$ \varepsilon_{0} $” & “$ \mu_{0} $” are Fundamental Physical Constants and are fixed, “$ c $” (in a vacuum) must also therefore be constant [ Note IV ].

Measurements done in 2005 by Stephan Schiller's team in Dusseldorf, obtained consistent values for “$ c $” with an upper limit of 6 parts in 10E16 (UNSW). No one has yet found a scenario in which the speed of light in vacuum ever changes. Hence M&M’s results and the Æther being an illusion.

B. Michelson–Moreley 1887

Admittedly it wasn’t a huge influence, but M&M’s null–result also implied “$ c $” , regardless of whether the light beam was pointing towards or away from the direction of the Earth’s motion around the Sun. The Earth only moves at 30km/s, whereas “$ c $” is 299 792km/s, but M&M’s Potsdam apparatus would have been sensitive enough to detect a difference. That it didn’t strongly indicates there was something very unusual about the nature of “$ c $”.

C. The geometry of spacetime

Einstein adopted a concept of spacetime in his SR theory, but it was his mentor/ university lecturer Minkowski who fleshed it out more fully in 1907.

Space and time are combined into a single 4–dimensional entity, called (unsurprisingly) “spacetime”. What is unusual is that every object is considered to be moving through this 4D spacetime at the speed of light, with different objects having differing amounts of “time” and “space” components. A stationary object, far away from the Earth or Sun, is not moving through space, therefore its “time” component is at a maximum while its “space” component is (almost) zero. EMR waves however, which move at “$ c $”, have zero “time” so therefore only their “space” component can vary WRT an external observer.

Which means (1) you cannot ever move faster than “$ c $” and (2) since everything is already moving through spacetime at “$ c $”, the speed of light must be identical for all observers.

Heavy stuff. For a clear visual explanation, see Fermilab (2017) Why can't you go faster than light? on YouTube.

Galileo First   ↑

I mentioned Galilean Relativity above. Einstein would not have relied upon it directly, but would have been aware of it as part of his physics education.

Galilean Relativity is useful here, as it illustrates an important principle of how different reference frames can validly interpret the same event with differing results.

Yacht frame of reference

Imagine you are on the deck of a yacht travelling at 3m/s (ie. just under 6 knots). A sail–clamp on the mainmast is winched up at a constant speed to the top. You do some measurements: 4m tall mast, takes 1s to reach the top, so the clamp has a velocity of 4m/s up. Unremarkable, right?

Shoreline frame of reference

Now switch to an observer standing onshore. They see the same yacht moving forward at 3m/s, the same metal clamp, same 4m rise, same 1 second to reach the top. This time however the onshore person also notices the yacht is moving forward while the clamp rose. For them the clamp did not follow a straight line up the mast, but rather a diagonal line WRT the sky behind. Consequently, for them, the velocity of the clamp now has two components: one for the vertical rise and another for forward motion. Do a simple calculation and the velocity of the clamp, in the frame of reference of our onshore observer, is now 5m/s.

Reconciling the two frames

How could that be? Same event, but the velocity is 4m/s when observed on–board but 5m/s when observed from the shore.

If you think about it, the two results are easy to reconcile because they use differing criteria. When on–board the observer ignores the forward motion of the yacht since in their frame (of reference) everything on–board is moving forward. For them the clamp rises straight up the mast. However from the frame of the stationary shoreline observer, the rising clamp has both vertical and forward motion, resulting in a greater overall velocity. Which is “correct”? Both are, according to their individual viewpoints.

The Einsteinian twist

But say there is some kind of universal law which requires all moving sail clamps to always have a speed of 4m/s, for all observers irrespective of where they are.

For the person on the yacht, nothing changes. The clamp still rises at 4m/s.

But for the shoreline observer things are different. If the clamp is to have a total velocity of 4m/s, then the time required for it to rise has to be longer (1.25 seconds in our grossly oversimplified example).

Both observers can now agree the clamp moves at 4m/s, although the shore–based observer will observe it takes a little longer and travels in a different direction.

This the essence of Special Relativity. Time and space must distort to ensure the speed of light is constant for all inertial observers, regardless of where they are and how they are moving.

Which if you think about it, is mind–blowing (!)

Is there a mathematical way to describe how this Space/ Time distortion is applied? Of course — they are known as Lorenz Transformations…

Lorenz–FitzGerald transformations   ↑

In physics it is not enough to make a statement of If X then Y will happen. You need to support your theory with a (usually mathematical) model, which others can test. Newton didn’t merely say Masses will attract each other due to gravity but also supplied the (famous) equation…

… and then invented calculus to prove the his expression worked for planets and stars, if you assumed their masses were concentrated into their Centre of Mass. It’s not every day someone will invent an entirely new branch of mathematics just to prove a point [ Note V ].

Einstein knew he was on the right track when he deduced space/ time had to distort to keep “$ c $” constant. But how? Lorenz–FitzGerald to the rescue.

Einstein’s work was made vastly easier by the earlier development of the Lorenz factor “$ \gamma $”, which described the geometry of moving objects as they approach the speed of light:

Where:
“$ \gamma $”   Lorenz factor (dimensionless)
“$ v $”   speed of moving ojbect (m/s)
“$ c $”   Speed of light in a vacuum (299 792 458 m/s)

At normal velocities “$ \gamma $” has negligible effect. It is only at substantial fractions of “$ c $” when it kicks in. It really takes off when you get “up into the nines” (eg. 0.9999$ c $):

Distorting Space & Time

All the components are now in place. Thanks to Michelson–Morely we can retire the concept of Æther. We further have Postulate #2 and the Lorenz transformations, so we can assemble the pieces into a coherent theory.

Einstein submitted Special Relativity (“SR”) for publication in the prestigious Annalen der Physik on 30 June 1905, and it was published three months later on 26 September. He was twenty–six.

To keep the speed of light constant in all frames, SR has to impact three things: (a) Length, (b) Momentum & (c) Time

(a) Length Contraction   ↑

Length Contraction is where a fast-moving object appears squeezed in the direction of motion. Einstein’s formula is identical to the Length Contraction formula developed by Lorenz and FitzGerald in 1892:

Where:
“$ l_{obs} $”   observed contracted length (m)
“$ l_{mf} $”   moving–frame length (m)
“$ v $”     speed of moving ojbect (m/s)
“$ c $”     Speed of light in a vacuum (299 792 458 m/s)

Put simply, the faster something moves the more contracted it appears to be.

It is amazing Einstein could make this equation his own. Lorenz–FitzGerald got the expression right, but their explanatory architecture was totally wrong. They assumed Newtonian absolute space was still valid, while the Æther and time stayed as fixed frames of reference in space, with only the moving objects themselves becoming physically contracted.

At the time Thomson had not yet to discovered the electron, so everyone assumed atoms, if they existed at all (more on this below), were indivisible spheres with no internal structure. So squashing them would have no impact. Of course by 1920s we knew this was spectacularly incorrect.

Einstein maintained, thanks to Michelson–Morely, that “absolute space” and Æther were as defunct in physics as epicycles were in astronomy [ Note VI ]. Length appears contracted in the frame of the observer because spacetime (and not the object) is contracted. If you could somehow ride on the object, nothing would be contracted within your moving frame, although due to Relativity the stationary world outside would appear horizontally compressed.

A couple of examples to illustrate this:

Particle Accelerators

Crude versions were developed in the 1920s, although high speed accelerators had to wait until the 1960s. Because accelerated particles travel at incredible velocities, you need to allow for relativistic length contraction if the accelerators are to work. When your particles (usually electrons or protons) get “up into the nines”, length contraction becomes so significant that, in the frame of the moving particles, the space between the accelerator’s magnets becomes so contracted they cannot accelerate particles any further. For example, the CERN Large Hadron Collider (“LHC”) in Geneva accelerates protons to 0.999999991$ c $. This is a “$ \gamma $” of 7453.56, which means for the particles in their frame to see external magnets separated by (say) 1cm, they need to be 7.345m apart in our stationary frame. This spacing helps explain why the LHC needs an enormous 27km diameter, making it the largest machine in the world.

Force between parallel conductors

I used to do a classroom demonstration with my Y8 students. Set up a couple of parallel strips of aluminium foil, place them close to each other (but not touching). Turn on the current and the strips will attract. The classical explanation is the magnetic fields around each conductor interact in such a way to apply a force to pull the conductors together (“Ampére’s Right Hand Grip Rule”). Years later, when some of those students are now in my HSC Physics class, I run the experiment again, except this time as an illustration of SR. Identical results, but the actual reason the conductors attract is due to the moving electrons in one aluminium strip seeing a different number of electrons per unit length in the other, due to a tiny amount of SR length contraction. This creates a charge imbalance and the conductors pull together due to electrostatic attraction — see Fermilab (2023) How Einstein saved magnet theory on YouTube. Students are always amazed when I explain why.

(b) Momentum Dilation   ↑

Momentum is where you take an object’s mass and multiply it by its velocity:

Where:
“$ \rho $”   momentum (N.s)
“$ m $”   mass (kg)
“$ v $”  velocity (m/s)

Since relativistic speeds affect length and time, it will affect momentum as well:

Where:
“$ \rho_{obs} $”   observed momentum (N.s)
“$ m_{mf} $”   moving–frame mass (s)
“$ v $”    speed of moving ojbect (m/s)
“$ c $”    Speed of light in a vacuum (299 792 458 m/s)

To walk it through:

Thus to keep changing an object’s momentum (Δρ/Δt) — ie. to make it speed up/ down or change direction — you have to apply a force (F).

This is usually fine, but at relativistic speeds things go awry since momentum blows up the closer you get to “$ c $”. Consequently the force you need to apply has to be overwhelmingly enormous as well. Which leads to a few conclusions:

Effect of mass

If something has mass then it cannot move at “$ c $”, since it requires infinite force (energy) to get there. There is no problem if the object is massless, eg. the rest–mass of photons.

Acceleration

If mass≠0, then you cannot even accelerate your object to “$ c $”. To merely get “up into the nines” requires amounts of energy we are incapable of creating for macro sized objects. The LHC can do it for protons or atomic nuclei, but spaceships?… nyet.

Spaceships

Currently we have chemical rockets and ion thrusters, but neither can deliver sufficient impulse (FΔt) to get to 0.001$ c $, let alone “12 nines”. As of 2026 the fastest human–made object is the Parker Solar Probe, with a velocity of 191 km/s or 0.0006371$ c $. It took 7 years to reach this speed via gravity assists from repeatedly looping around Venus, not from an engine blowing stuff out the back.

The upshot of relativistic momentum is that nothing (with mass) can travel at “$ c $” or even be accelerated to it. So whenever you see sci–fi movie where our heroes jump to light speed, cue an informed eye–roll.

(c) Time Dilation   ↑

Have saved the best for last. The most stunning thing about SR is how it messes with time, a theme Einstein returned to in greater detail in his General Theory of Relativity in 1916.

Through a series of elegant “thought experiments” (see below), Einstein realised any concept of simultaneity cannot exist. If something appears simultaneous in one frame (say two bolts of lightning striking the ground), then it could not be so when viewed by an observer moving past at extremely high speed.

So if events cannot be simultaneous everywhere, then Newton’s concept of Absolute Time is wrong and therefore time must be relative to the individual observer.

According to SR, the faster things move the more its observed time appears to slow down. This Time Dilation can be calculated via the following expression:

Where:
“$ t_{obs} $”   observed dilated time (s)
“$ t_{mf} $”   moving–frame time (s)
“$ v $”    speed of moving ojbect (m/s)
“$ c $”    Speed of light in a vacuum (299 792 458 m/s)

To put it colloquially: moving clocks appear to run slow, while at “$ c $” they appear to stop.

Again similar to L/C, clocks work just fine on the moving objects. They only appear to run slow when measured by an external observer — see the concise visual examples by ScienceClic (2021) Visualising Time Dilation on YouTube.

Say you have a spaceship moving at 0.999$ c $, which in Earth time ($ t_{obs} $) travels for one year. According to SR the spaceship will only experience a flight–time ($ t_{mf} $) of 0.04471 years, which works out to slightly more than a fortnight [ Note VII ].

This is the stuff of Science Fiction. The on–board astronauts have only aged a couple of weeks yet their counterparts on Earth have experienced a full year. Sounds great, but unfortunately by Momentum Dilation, the staggering amount of energy required to reach these speeds is well beyond our current (and even future) abilities 😕

The evidence for T/D

T/D first confirmed experiment

Time dilation provided the earliest direct evidence of SR via the Ives–Stilwell experiment in 1938–41, three decades after the publication of SR. Technology of the time did not allow for moving physical objects, let alone clocks, at relativistic speeds, so instead they measured the Doppler shift of hydrogen emission spectra. The observed values only made sense when SR was taken into account.

Muon Extended Life

Further evidence for T/D was obtained by the discovery of muons. These sub–atomic particles — similar to an electron but with 200× greater mass — are created when cosmic rays smash into nitrogen and oxygen molecules in the upper atmosphere. Due to momentum, some muons continue streaming toward the Earth’s surface, where they can be detected. The problem is muons only have a lifespan of 2.2µs in the laboratory, much too brief to cover the distance through the atmosphere. However due to T/D, since muons are typically moving at 0.9997$ c $, they have plenty of time in the Earth’s frame to reach the surface (≈ 90µs).

Neutrinos Have Mass

Time dilation also proved neutrinos must have (tiny) mass. Neutrinos were originally proposed in 1930 by Pauli as massless particles to explain conservation of energy during β decay. This caused a lot of amusement in the Physics community (A fermion with zero mass… yeah, right) until neutrinos were confirmed by experiment in 1956. In subsequent decades it was observed neutrinos change type or “flavour” as they move through space from the Sun to Earth. These “neutrino oscillations” are still being studied by Fermilab, IceCube and CERN, but the upshot is, if a neutrino is capable of changing while transiting through space, then it must also experience time within its moving frame. If so, then it must be moving slower than “$ c $” and subsequently have some mass. Current estimates put neutrino mass at 0.45 eV, vastly smaller than that of a proton (typically 238 000 000 eV), but not zero.

Photons everywhere all at once

Unlike neutrinos, photons are bosons with zero rest–mass. Which means they can move at “$ c $” without any SR limitations. But it also means by L/C in their frame, the entire universe shrinks to zero length, or by T/D they experience zero time when “moving” from any point to anywhere else. Zero length, zero time… This logically means a photon in a vacuum is simultaneously everywhere in the universe… until it interacts with matter somewhere. This concept led Feynman to propose photons instantaneously explore infinite pathways when undergoing refraction, reflection or interference (!)

Twin Paradox?

We have twins. One of them flies off in space at relativistic speeds, the other stays on Earth. When they are reunited the astronaut, due to T/D, is now months younger than their earthbound counterpart. The paradox is that, from the point of view of the traveller, shouldn’t the earthbound twin have also experienced T/D and thus be exactly the same age?…

No. T/D only applies to an object which physically moves through spacetime and not one which merely appears to do so when viewed from a different frame of reference. The explanation is not due to the traveller being in a non–inertial frame of reference (ie. experiencing acceleration/ deceleration away/ towards Earth) but rather due to the mathematical logic built into the Lorenz Transformations, upon which SR is founded. See Fermilab (2023) Does acceleration solve the twin paradox? on YouTube.

SR Thought Experiments   ↑

The main reason people didn’t fall about swooning when SR was published was there was no way to test it. Without concrete evidence and data, it was yet another elegant mathematical abstraction in a world full of them.

Einstein wasn’t a shrinking violet, but he knew he had to prepare for critics shouting, Wherefore art thou goddamn evidence?!!!!

Unfortunately there wasn’t any. Other than Michelson–Morely’s suggestion the speed of light was (maybe) constant, there was no data.

To make things worse, in 1905 there was no way of obtaining any:

• No atomic clocks with nanosecond accuracy (mechanical devices in 1905 could only measure to 1/10 sec)
• No fast aeroplanes (Wright Flyer, 48 km/h, December 1903)
• Land-speed record was 169 km/h (Darracq V8, 1905)
• No particle accelerators
• The only known subatomic particle was the electron (1897)

Yet Einstein had to deliver something to support SR, so he presented a few “Thought Experiments” (Gedankenexperiment) — imaginary exercises to show SR was logically consistent. This satisfied no one, but it was better than nothing.

I won’t review them as they are now of historical interest only — see the Wikipedia link above. From the 1940s onwards various effects of SR have been measured and confirmed countless times (eg. E=mc², muons, Hafele–Keating etc.), but in 1905? Nothing.

The thought experiments also show Einstein had an extraordinary lucid visual imagination. Similar to great writers, he had an ability to visualise in intricate detail the scenarios he described, whether it was observing lightning bolts at relativistic speeds or mirrors on a train, or riding alongside a beam of light. Apparently fellow physicist Dirac shared the same skill.

In the classroom I sometimes did a short exercise with my HSC students to illustrate Einstein’s ability to visualise things.

I would ask them to imagine five things, then sketch what they imagined. Most of them drew circles, some shaded spheres, once a girl drew a banjo fretboard with five strings (brilliant!). I then told them what Einstein would have hypothetically imagined: five die (God does not play dice with the universe, but I do…), each a different colour (green, brown, red, yellow & black) and each with the upper face showing, in sequence, “1”, “2” etc. up to “5” dots. Of course, for fun, he would arrange them touching in a cross shape with “5” in the middle and “1”, “2” etc. going around the outside, anticlockwise naturally.

Now comes the hardcore stuff, for he would then describe — accurately — what numbers were visible on the vertical faces of each die as he walked around them in his mind, then the texture of the lace tablecloth on which they lay, the grain and colour of the oval wooden table they were on, the room, the wallpaper, the titles of the books on the shelves, his wife pestering him from another room to not forget to pay the housekeeper and so on.

Bricks without straw…

What is particularly fascinating is Einstein managed to develop Relativity without any observations or data. He figuratively pulled a phantom rabbit out of an imaginary hat, basing his theory upon little more than intuition, imagination, deductive reasoning and mathematical logic.

Incredible stuff, but also a negation of the Scientific Method which formed one of the pillars of the Enlightenment. Over centuries there had been a concerted effort to rid Science of the hocus pocus of dogma, superstition and blind obedience to authority (Aristotle says…). Unlike “Natural Philosophy”, the entire point was to base theories upon objective experiment and direct evidence. Now here was this new theory, without any means to test it, where the only proof was Einstein says…. The irony was not lost and many scientists unsurprisingly baulked.

1905 — paper miracles   ↑

Another remarkable thing about SR was it was merely one of four papers published in 1905, all of which presented radically new ideas, and all from the pen of a lowly 26–year–old patent clerk (!)

They were obviously not all written from scratch within a single year, but rather the outcome of 2–3 years work in Einstein’s spare time. They were developed either at the Bern Patent Office or in his tiny 2nd floor apartment on Kramgasse 49, which he shared with his first wife Mileva Marić, infant son Hans Albert and a daytime aforementioned housekeeper.

The Annus Mirabilis papers are summarised in chronological order below:

June → Photoelectric Effect

In order to explain the shape of the Blackbody Radiation curve, Planck spent six years trying different ideas before publishing his famous expression in 1900:

Where:
“B_ν(ν,T)”   Radiance as a function of Frequency “ν” &
      Temperature “T”
“$ h $”    Planck Constant (6.626 ×10^-34 J s)
“$ \nu $”   EMR frequency (Hz)
“$ k_B $”   Boltzmann Constant
“$ {\color{blue}{h\nu}} $”   energy of 1 quanta (J)

A lot of physicists were suspicious of Planck’s theory because it assumed energy was exchanged in discrete chunks or “quanta”, rather than continuously as per classical thermodynamics. Planck himself was dubious and considered it a mathematical hack, which later researchers would undoubtedly correct. Not Einstein. Less than five years after Planck’s published, Einstein seized upon quanta theory and used it to explain why some metallic elements generate an electrical current when exposed to coloured light, a phenomena known as the “Photoelectric Effect”.

Einstein proposed electrons and electromagnetic radiation exchanged energy via discrete energy quanta (later called “photons”). If these packets had sufficient energy then electrons in the metal would flow according to a “Work Function” relationship, which can be presented in a simple form thus:

Where:
“$ K_{\text{max}} $”   electron maximum kinetic energy (J)
“$ h $”     Planck Constant (6.626 ×10^-34 J s)
“$ \nu $”     EMR frequency (Hz)
“$ \Phi $”    Work Function of the metal (J)

If the energy of incoming EMR (“hν”) was insufficient to overcome to work function of the metal (“Φ”), then no photoelectrons could be released since “K_max” is either negative or zero and thus no energy was available to trigger a current. Einstein’s insight was that an electron’s energy depended solely upon the incoming light’s frequency (“ν”) and not its intensity (which only determined the number of electrons freed).

BTW do not think of light as a stream of photons. When moving from place to place, light always was and always will be a wave. It only acquires a particulate nature (ie. photons) when interacting with matter.

Einstein’s explanation was straightforward enough to be tested, which it eventually was by Mikikan in 1916, who confirmed both theory and equation. The Nobel Prize Committee, after spending a decade rejecting Physics Prize nominations for Einstein, finally (grudgingly?) awarded him one in 1921, […] for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.

Planck also had to patiently wait for his prize. He was finally awarded the Nobel Prize in Physics in 1918, almost twenty years after he upended classical physics and ushered in the age of Quantum Mechanics.

July → Brownian Motion

It is astonishing that as late as 1905 there was still argument over whether atoms were an abstraction or real. Einstein was firmly in the “real” camp and the bickering annoyed him no end, for he wrote this paper, along with his 1906 PhD dissertation, to prove once and for all the existence of molecules and atoms.

Brownian Motion is the mysterious jiggling of pollen grains in water when viewed through a microscope. In hindsight the reason is obvious — the pollen is jostled by random thermal collisions with invisible water molecules — but at the time the exact cause was uncertain. Einstein presented a simple statistical interpretation of the molecular motion by borrowing from the Kinetic Theory of Gases. He provided expressions which experimentalists could use with measurements of pollen displacement to unambiguously prove the motion was caused by atom–based entities.

Einstein’s approach was controversial for it relied upon statistical mechanics, an unpopular theory at the time and the criticism of which ultimately drove its discoverer Boltzman to suicide in 1906.

September → Special Relativity

Not much to add other than observe the paper was not known as “Special Relativity”, but rather On a Heuristic Viewpoint Concerning the Production and Transformation of Light. It was only following the publication of General Relativity in 1916 that the 1905 version became known as “Special”.

November → Mass–energy Equivalence

Einstein realised if SR was valid, then it would also impact the conservation and transfer of energy. This led him to deduce mass and energy were equivalent via his famous equation:

Where:
“$ E $”   Energy (J)
“$ m $”   Mass (kg)
“$ c $”   Speed of light in a vacuum (299 792 458 m/s)

An unintended consequence was the unification of “mass” and “energy”, whereby sometimes things are expressed as energy, other times as mass. It is also the core concept for the production of energy in stars and why we can validly measure subatomic particle masses in electron volt energy (eV) instead of grammes.

Although in 1905 the relationship between the inertia of a body (ie. its mass) and its energy seemed an intriguing if arcane idea, by the 1930s it became commonplace with nuclear physics, and from the 1940s onwards, the production of nuclear bombs and million–degree fireballs over Hiroshima and Nagasaki.

Einstein’s 1905 papers have been gathered into a recent book with a forward by Roger Penrose. It features new English translations, along with Einstein’s PhD dissertation on how to calculate Avogadro’s Number. See… Stachel, J. (2005) Einstein's Miraculous Year: Five Papers That Changed the Face of Physics (Princeton University Press).

Facsimiles of the original German papers, along with English introductions, can be downloaded for free from the Universität Zürich (UZH) website (2023, PDF 9MB).

Reception   ↑

The physics world was not upended in 1905. As mentioned earlier, SR was too theoretical and — let’s be honest — too unorthodox for general physicists to accept. The material on the Photoelectric Effect fared better because, despite being based on Planck’s untested quantum theory, was straightforward enough and dealt with familiar (if unpopular) concepts. It was also far easier to confirm by experiment — which was indeed done in 1916.

What did attract attention was the publication of five significant and original physics papers in merely one year (the four “Miracle” papers plus Einstein’s PhD dissertation) — a staggering achievement. Planck immediately took notice, as did Lorenz. Even the sleepy Bern Patent Office offered Einstein a promotion to Science Specialist Class II, with a corresponding increase in pay.

In 1905 Einstein’s marriage to Mileva was still loving and strong, despite later admitting he treated her little better than an unpaid servant. Mileva was a skilled mathematician in her own right and thus a brilliant second pair of eyes to check his ideas and work. How extensively she contributed to his early work is a matter of contention, but her complete omission from Einstein’s 1905 publications, despite thanking his friends Besso and Grossmann, borders upon churlish.

SR took years to catch on (eg. 1928 for Dirac’s equation), although people started taking it far more seriously after the publication of GR in 1916.

Knowing a good thing when they saw it, the Universität Zürich (UZH) created an associate professorship for Einstein in 1909, which he immediately accepted. Thereafter he moved to Zürich, then Prague, then back to Zürich and then to Berlin for a full professorship in 1914, where he remained during WW1 while completing his theory of General Relativity…

Einstein was uneasy about the forced limitations of SR, so after 1905 he got to work incorporating acceleration and gravity into an expanded and more comprehensive version of Relativity, eventually known as General Relativity (GR). It required almost 10 years to develop, not only due to its extraordinary complexity, but also because of major interruptions in Einstein’s life, eg. changing jobs every few years and moving countries three times.

The birth of his second son and his failing marriage didn’t help, although he found solace in various infidelities plus starting a relationship with his first cousin Elsa Einstein, whom he eventually married in June 1919, three months after divorcing Mileva.

Thus around this time Einstein’s personal life was a mess, in stark contrast to his idyllic time in Bern. This turbulence is reflected in the intricate complexity of GR, as opposed to the simple elegance of his 1905 work.

The Theory of General Relativity was originally published in November 1915, again in the Annalen der Physik, although a more comprehensive review was later published by Einstein in 1916.

Section Quick Links

• Field Equations
• Cosmological Constant
• Spacetime Curvature
• Black Holes
• GR Time Dilation
• Angels dancing on the head of a pin
• Gravity ≠ Force
• Fame is the Prize

Field Equations   ↑

While still working at the Patent Office in Bern in 1907, Einstein had his “happiest thought” when he realised there was no difference between acceleration and motion in a gravitational field. Einstein seized upon this idea and developed it, with support from his more mathematically adept friend Grossmann, from 1907 until the consolidated publication of GR in 1916.

Unlike E=mc², there is no simple equation which encapsulates the entirety of GR. In his 1916 paper Einstein provided a set of non–linear partially differentiated Field Equations which people had to solve. With his constrained mathematics, Einstein considered them unsolvable and he could only supply approximations [ Note VIII ]. Luckily in 1916 Schwarzschild suggested a complete solution, which came to be known as the Schwarzchild Metric. Einstein eagerly (and gratefully!) incorporated this into GR.

As noted above, GR is a set of Field Equations where only those parts are used which are relevant for what people wish to calculate. The whole system is tied together by the following expression:

Where:
“$ R_{\mu\nu} $”   Rici tensor
“$ R $”    Rici scalar
“$ g_{\mu\nu} $”   Spacetime metric
“$ \Lambda $”    Cosmological Constant
“$ G $”    Gravitational Constant (6.6743 ×10^-11 m³/(kg s²))
“$ c $”    Speed of light in a vacuum (299 792 458 m/s)
“$ T_{\mu\nu} $”   Energy momentum tensor
 

Conceptually it is straightforward: The left side of the equal–sign (the Einstein Tensor) deals with mathematical geometry and the curvature of spacetime; The right side (the Stress–Energy Tensor) concerns the physics of mass and energy. Their interrelationship was neatly summarised by Wheeler:

Spacetime tells matter how to move;
Matter tells spacetime how to curve.

For this reason, the above expression is often simplified to:

Where:
“$ G_{\mu\nu} $”   Spacetime curvature
“$ \Lambda $”    Cosmological Constant
“$ g_{\mu\nu} $”   Spacetime metric
“$ k $”    Einstein gravitational constant (2.07665 ×10⁻⁴³ N^-1)
“$ T_{\mu\nu} $”   Energy momentum tensor
 

As many have pointed out, the GR expression is merely the tip of a mathematical iceberg and… the maths is murder. For example, the spacetime metric “$ g_{\mu\nu} $” is defined by a set of partial differential equations — which you solve courtesy of the Schwarzschild metric…

Whereby you can (eventually) calculate the geometric and causal structure of spacetime, which of course is non–Euclidean but for simplicity is (usually) assumed to be symmetrical. Then you need to calculate the Ricci Tensor “$ R_{\mu\nu} $” via the overarching Riemannian tensor. Easy enough (cough). Don’t forget the Cosmological Constant “$ \Lambda $”, which we downplayed for decades but since the 1990s has acquired new significance. Then we move to the right hand side and start playing with the “$ T_{\mu\nu} $” energy momentum tensor…

Science is rarely tidy.

See:

• Sperhake, U. (2018) Introduction to Numerical Relativity (PDF, 41MB)
• Hirvonen, V. (2025) General Relativity For Dummies: An Intuitive Introduction (Profound Physics website)
• Eigenchris. (2021) Relativity 108a: Schwarzschild Metric - Derivation on YouTube

Small wonder people recoiled in horror when GR was first published.

Asked in 1919 whether it was true that only three people in the world understood the theory of General Relativity, Eddington allegedly replied: Who’s the third?…

Cosmological Constant “$ \Lambda $”   ↑

Upon publication of GR in 1916, mathematicians realised if spacetime was malleable, then it could also expand and contract.

Einstein noticed this himself. So in 1917 he promptly injected a term into the GR expression he called the Cosmological Constant (“$ \Lambda $”), to circumvent any expansion/ contraction mischievous and superfluous stunts. He did this because he believed in a Steady State model of the universe, whereby globally nothing much changed. Of course local stars and galaxies form and disappear, but overall the structure and size of the cosmos remains forever the same. What you have now is more or less what you had 10 billion years ago and likewise in 10 billion years hence.

Yet mathematicians like de Sitter, Friedmann, Lemaître and Robertson realised Einstein might be wrong. If you altered “$ \Lambda $” then the universe could expand. Einstein was unimpressed:

When Father Lemaître [in 1927] presented his concepts of the “primeval atom” and an expanding universe, Einstein told him, Your mathematics is perfect, but your grasp of physics is abominable.

By the late 1920s however, the “expansionist” case acquired significant impetus by the consistent measurement of cosmological red–shift by Hubble, followed by the development of the Big Bang Theory. Due to the growing amount of evidence and scientific consensus, Einstein eventually backed down, sheepishly admitted his mistake and apologised. Thereafter “$ \Lambda $” became something of a curiosity, to be tinkered with for decades according to prevailing cosmological theory.

Which is where the story would have ended, except in the 1990s it was confirmed by Perlmutter, Schmidt & Riess (“PSR”) that the expansion of the universe was actually speeding up. Previously it had been assumed the rate was constant, or maybe even slowing down due to gravity — the main reason for doing their measurements. Instead we all got a big surprise, expansion was accelerating. Cosmology was turned on its head and PSR were subsequently awarded the Nobel Prize in Physics in 2011.

To accommodate this unexpected cosmological acceleration, a new concept of Dark Energy was invoked and, after decades in limbo, “$ \Lambda $” reacquired significance in GR calculations (hence its inclusion in the expressions above).

So it transpired Einstein was right after all, although for completely the wrong reasons 🙃

Spacetime Curvature (STC)   ↑

As noted earlier, the left hand side of the GR expression concerns the mathematical geometry of spacetime. Contrary to popular understanding, Einstein did not invent spacetime, but rather based his work upon earlier theories from Poincaré and Minkowski (the latter his university mathematics lecturer).

What Einstein did was incorporate Gravity into Minkowski’s notion of 4D spacetime, such that its curvature arose dynamically from the effect of mass, whereby the greater the mass, the greater the curvature. Hence the Left/ Right structure of the GR expressions above.

Testing GR

Einstein remembered the collective shrugs which greeted his work in 1905. So for GR he wanted to show his more comprehensive theory could be reasonably tested using available technology.

He proposed three different ways:

Mercury’s precessing orbit

The planet Mercury does not merely orbit the Sun in an elliptical orbit, but also precesses such that its long axis slowly shifts for each orbit. This is known as Precession of the perihelion of Mercury and was measured to be 43” (arc–seconds) every 100 years. It had puzzled astronomers for centuries and could not be explained by Newtonian orbital mechanics. On the other hand GR calculations explained it exactly when you took into account the rotational curvature of spacetime. You would imagine this feat would impress people but… shrugs.

Gravitational deflection of light

Another prediction was light would be deflected by spacetime curvature as it passed near a massive object. Einstein did a few GR calculations and showed it should be double what is predicted by Newtonian methods (which assumed light was comprised of “corpuscles”). The only way to test this prediction would be by observing stars near the Sun, which during daytime would obviously be impossible. Luckily however there was a full solar eclipse due in May 1919 in Brazil and Príncipe, a small island off the west coast of Africa. British astronomers Eddington and Dyson set out on expedition. Photographs were taken and measurements from them seemed to confirm the apparent positions of stars were shifted to where GR predicted they should be. It was a tremendous success and newspapers, including the stodgy New York Times, heralded the findings as the start of a new era: Lights All Askew in the Heavens. Yet a handful of astronomers remained unconvinced since Eddington’s measurements were not without flaws. After decades of low–level controversy and polite bickering, Eddington’s 1919 results were reassessed in 1980 and claims of selection bias were made. The Royal Society had to interject with their own analysis to confirm the validity of Eddington’s data and conclusions.

Gravitational red–shift

Here light from an intensely bright source, say a super–dense White Dwarf star, should have its spectra red–shifted due to the intense gravitational field it escapes. Although proposed in 1916 as a feasible test, it was only confirmed unambiguously fifty years later by Greenstein, Oke & Shipman in 1971 via analysis of the spectra of Sirius B.

Æther 2.0?   ↑

Having thoroughly demolished the concept of “Luminiferous Æther” in 1905, Einstein became troubled about the nature of spacetime following the publication of General Relativity in 1916.

What bothered him was the incongruity of an empty vacuum which was able to bend and flex. For how can you contort “nothing”?… It bothered him so much he began to believe there must be some kind of Æther after all. Not to act as a medium for the transmission of light, but rather as a physical entity which could be acted upon by matter/ energy.

The brilliant seer who convinced (almost) everyone Æther was a fairytale, now swapped sides and became its passionate advocate. Eyebrows rose. Phasers were set to “stun”.

We still don’t have an answer, although virtual particles which comprise Wheeler’s Quantum Foam is a likely candidate.

Gravitational Lensing   ↑

If spacetime curvature in the proximity of the Sun can bend light, then something really massive, like a galaxy, should be able to bend it vastly more. The effect is so pronounced it has come to be known as Gravitational Lensing. It was only hypothesised in 1917, but since the 1970s we have produced numerous photographic examples from space based telescopes like HST and JWST.

A famous example is The Einstein Cross, where light from a distant quasar (8 billion ly) is lensed by a much closer galaxy (400 million ly) such that four images are produced of the single astronomical source.

Gravitational Waves   ↑

If GR shows spacetime can curve, then conceptually it should also be able to modulate or “ripple”. Ergo Gravitational Waves.

Einstein realised this early and even calculated these waves must have a speed of “$ c $”. He also admitted any detection of them would be phenomenally difficult as their amplitudes are so small to be beyond the capabilities of foreseeable technology. Furthermore the gravitational disturbances which generate them would need to be vast, say the result of neutron stars or black holes merging. This would ensure they are rare events, making them even harder to detect.

Accordingly for decades Gravitational Waves remained an interesting idea supported by GR theory, but without any supporting evidence. Then in the 1970s the first attempts were made to directly measure them. Weber built aluminium cylinders approximately the size of a pickup truck and claimed that by having them resonate at certain frequencies, they would detect gravitational waves when they passed. He even claimed to measure a few.

The physics world got (very) excited and tried to replicate his results. They couldn’t. They tried again and… failed. It quickly became apparent Weber had claimed to measure things he never did. You would imagine he would be hounded from polite physics company, but not really. Eventually he lost public funding, but despite the opprobrium many researchers still credited him with “kick starting” gravitational wave research. Indeed there is a Weber Bar is on display honouring him at the LIGO facility in Hanford.

We had to wait until 2015, in the centenary year of GR’s initial publication, to directly measure Gravitational Waves. It was achieved by LIGO, using laser based interferometry detectors considerably larger than a truck. To quote from their website:

LIGO exemplifies extreme engineering and technology. It consists of:

• Two L-shaped detectors with 4 km long vacuum chambers
• Situated 3002 kilometres apart operating in unison
• Over the 4km length of each arm, the Earth curves away by nearly a meter
• All to measure a motion 10,000 times smaller than an atomic nucleus
• Caused by the most violent and cataclysmic events in the Universe, occurring tens-of-millions or billions of light years away!

Upon confirmation of LIGO’s results, the usually overcautious Nobel Committee immediately awarded them the Nobel Prize in Physics in 2017.

The waves LIGO detected were generated by the merger of a pair of super–massive black holes 10 billion years ago. After the first confirmed detection in 2015, hundreds have since been found by LIGO and other research teams. Gravitational Waves are no longer hypothetical.

Einstein was right. Again.

Black Holes   ↑

The escape velocity from the surface of a planet (or star) can be calculated via the following formula (ahem, assuming minimal rotation and insignificant relativistic effects):

Where:
“$ V_{\text{esc}} $”   escape velocity  (m/s)
“$ G $”    Gravitational Constant  (6.6743 ×10^-11 m³/(kg s²))
“$ M $”    mass of planet/ star  (kg)
“$ r $”    radius of planet/ star  (m)
 

For Earth it works out to be 11.1 km/s. For the Sun, 617.7 km/s. For a neutron star it is typically 0.5c and for a Black Hole it is “$ c $”.

And that’s it. They are simply objects with an escape velocity ≥ the speed of light. Since nothing can go faster than "c", nothing escapes, not even light. All you need is a large enough mass “M” packed into a small enough radius “r”.

The existence of Black Holes were hinted at in GR, but they acquired a firmer footing in 1968 with the work of Wheeler. In the 1960s a Black Hole candidate was located via intense X–rays emanating from Cygnus X–1, 7000 ly distant from Earth. In 1974 Thorne & Hawking made a bet over whether it really was a Black Hole. In the early 1990s Hawking paid up.

We have since managed to create (very blurry) images of Black Holes, most famously in 2018 for the super massive black hole M87*, followed in 2022 by the Black Hole at the centre of our Milky Way — Sagittarius A*. Creating these images was non trivial, see Veritassium (2022) How did they actually take this picture? on YouTube.

Other than being simply weird, Black Holes also excite physicists because the intensity of Gravity within them is so strong it matches the three other fundamental forces (Strong, Weak & Electromagnetic). The hope is by studying them we will one day find a means of unifying all these forces into a single overarching “superforce”. Hold not thy breath.

Black Holes need not be vast celestial things. Provided you have sufficient (extreme!) density and meet the requirements of a Schwarzschild radius, you can make them small as (say) a peppercorn. Reddit Scholars love to argue about this stuff.

Tiny Black Holes aren’t purely hypothetical though, for in 2008 Wagner & Sancho filed a lawsuit in Hawaii to prevent CERN’s LHC in Geneva from being switched on. The concern was its high–energy collisions might inadvertently create a microscopic Black Hole which would gobble up the Earth and our solar system. In 2010 the learned judge dismissed the case, a year after the LHC began doing high energy collisions…

But why does matter curve spacetime?

Short answer — despite bombastic pronouncements by Reddit Scholars, nobody knows.

As Newton remarked, Hypotheses non fingo. Einstein agreed. GR describes what happens and how spacetime curves, but not why 😖

Messing with time   ↑

Mixing Bowls Analogy

(Big picture stuff. Broad strokes.) Time for a crude, mathematically incorrect analogy to illustrate GR time dilation…

Imagine two mixing bowls. Place them on a bench beneath a strong light. Both bowls should have the same diameter, say 30cm. One of them should be shallow, say 10cm deep. The other should be much deeper, say 25cm.

Now find a knitting needle or chopstick, long enough to rest across either of these bowls it its widest point. This “stick” will represent an object’s linear timeline (ie. “spacetime interval”) as it moves from the past into the future.

Rest the stick on the rim of the shallow bowl, with one end at (say) at 12 o’clock and the other at (say) 4 o’clock. Use some tape or a felt pen to mark both points on the stick where it makes contact with bowl’s rim. Let’s call these two points “A” and “B”.

Now look inside the bowl and notice the curved shadow of the stick as it runs along the bowl’s concave interior. The shadow starts at the 12 o’clock rim at A, then runs down the interior of the bowl before curving back up to the other end at B.

We can use this analogy to (roughly!) model the effect of spacetime curvature on time. In our crude model the linear distance between A and B is the flat–space time. Now look at the stick’s shadow on the curved interior of the bowl. Notice how its length is longer than that of the straight stick, even though they both travel from A to B?

Now place the stick on the rim of the deeper bowl such that the markers A & B are resting on its rim at 12 and 4 o’clock again. Once more we have a certain linear stick length, identical to the shallow bowl example. But look inside at the curved shadow running along the inside of the deeper bowl. Notice how it is more pronounced and its arc length is much longer between A and B. Consequently where there is more curvature, the length of the arc is much longer.

Effect of STC on Time

As for bowls, so too for time in a curved spacetime. A shallow curvature will barely have any effect on a linear timeline, but a deep curvature (due to a planet or star or neutron star) will dilate time significantly. From our oversimplified bowl analogy, what appears to be 1 minute for a linear timeline, will measure significantly longer when transiting through curved geometry.

BTW this timeline is known as a “World Line” or “Geodesic” (the latter for freely falling objects). It is straight for stationary objects far away from the influence of gravitational fields, since all the object is doing is moving forward through time. For a stationary object here on Earth, it traces a large curving spiral because the object is not only moving forward through time, but is also moving through space on a spinning planet which is orbiting the Sun. For a moving object on Earth, say a car driving from Christchurch to Invercargill or an aeroplane flying from Buenos Aires to Cape Town, it’s… a bit more complicated.

SR causes observed time to dilate when a moving object approaches “$ c $”. GR distorts spacetime through which time moves, also resulting dilated time. The maths is completely different, but conceptually the results are similar. Observed time increases as velocity approaches “$ c $” and/ or the greater the spacetime curvature.

The closer you are to a planet’s (or star’s) centre of gravity the greater the STC. On Earth this means clocks appear to tick slower at sea level than at an altitude of (say) 20 000km. The difference is not vast, but it is measurable.

NOTE don’t forget this from the frame of an external observer. The clocks do not physically run slow nor tick faster. Irrespective of where it is located in curved spacetime, in a clock’s frame 1 second remains 1 second. Time differences will only become apparent when measured from the frame of a distant observer.

Quantifying T/D

This section is summarised from Dialect (2025) What Causes Gravitational Time Dilation? A Physical Explanation on YouTube, along with Controlgroup (2025) What's the gravitational time dilation at the surface of a neutron star? at Astronomy Stack Exchange.

If you look at the Schwarzschild line element…

… you will notice the following expression appears twice:

Take the square root and you have the GR T/D Factor:

Where:
“$ \gamma $”   Time Dilation Factor
“$ G $”   Gravitational Constant (6.6743 ×10^-11 m³/(kg s²))
“$ M $”   mass of space–curving object (kg)
“$ r $”    distance from space–curving object (m)
“$ c $”    Speed of light in a vacuum (299 792 458 m/s)
 

Aside from a Black Hole, “γ” can only ever be less than 1. The smaller its value, the greater the T/D. For example if “γ” is 0.99, then for every 100 ticks of an observer’s clock, only 99 will be observed on the dilated clock. Whereas if “γ” is 0.01, then for every 100 ticks of the observer’s clock only 1 will be observed on the dilated timepiece.

The formula also shows the greater the mass (“M”) and/ or smaller distance (“r”) from a planet/ star, the lower the “γ”. T/D becomes extreme as “γ” approaches 0.

For typical sized objects like (say) planets in our solar system, “γ” will be ≈ 1. You need to go to extremes to achieve substantially lower “γ” values. Eg. for a neutron star of 2 solar masses (2× 1.989 ×10³⁰ kg) and a radius of 12 km (1.2 ×10³ m), along with assuming no rotation to keep the maths simple… “γ” works out to 0.712. Thus a clock on the surface of the star would be observed, from far out in space, to be deliver only 71.2 seconds for every 100 seconds of an observer’s identical clock. This would be trivially easy to measure. Unfortunately the star’s surface gravity would be 18.8 ×10¹² times that on Earth, so any clocks thereupon would get somewhat squished.

T/D Examples

Mr Clock takes a holiday (1971)

STC was confirmed by Eddington in 1919, but we had to wait until 1971 for GR time dilation to be directly measured by Hafele & Keating (“H–K”). Earlier experiments had used subatomic particles to measure the effect, but H–K physically transported caesium–beam atomic clocks using around–the–world commercial flights, by booking seats for themselves and “Mr Clock”. The only reason their experiment worked was their nanosecond accurate atomic-clocks had a precision well within the time changes expected. Not only did their results confirm T/D due to SR motion (time slowed), but also due to GR altitude (time got faster). The westward flight even returned results within 10% of the predicted Relativity values. Project GREAT whimsically repeated the experiment in 2005, this time by a family of mountain climbers using atomic clocks bought off eBay (really? how?!) — they similarly found the higher altitude clocks gained time (22 ns).

GPS (1980s)

Satellites in the Global Positioning System are the goto example for the impact of Relativity on time. GPS satellites orbit at a velocity of 4 km/s at an average altitude of 20 200 km. The latter means satellite clocks tick 45 μs faster due to GR, but due to their velocity they also tick 7 μs slower due to SR. Overall it yields a time difference — WRT a stationary sea–level frame — of +38 μs. Which doesn't sound like a lot, but it’s enough to create multi–kilometre navigational errors every day. BTW don’t forget to further account for the eccentricity of the orbit (perfectly circular orbits only exist in HSC exams…). Hence GPS transceivers need to be regularly recalibrated to operate at the microsecond precision required.

Interstellar (2014)

In the movie we are told GR time effects on the “water planet” (ie. “Miller’s Planet”), is 1 hour = 7 years due to extreme T/D in the vicinity of the super–massive black hole. When our astronauts return to the main spaceship we discover 23 years have elapsed, which means they must have spent slightly more than 3 hours on the planet. Which is feasible, but it would require an insanely strong gravitational field and/ or “high 9s” orbital velocity. How our intrepid heroes could land and leave under such circumstances is “interesting”. Then there is the lethal flux of X–rays they would experience in the proximity of “Gargantua”, or being shredded by high–velocity plasma because they were well within its accretion disc… Never mind, we got to see robot CASE cartwheel through the water to save weepy astronaut Brand — which impressed me far more than the dodgy astrophysics.

Angels dancing on the head of a pin   ↑

Medieval Scholasticism was a form of bogus intellectualism which helped make the Middle Ages “dark”. It was a dumpster fire of febrile theologians arguing over stuff which was entirely made up. The stakes were high, figuratively, for if you crossed the wrong line then you were guaranteed a hot date with a large pile of firewood. Just ask Giordano Bruno.

Mathematical Physics (“M–P”) is our modern day equivalent. Not to be confused with Theoretical Physics, which attempts to explain empirical observations and data within mathematical frameworks. No, M–P ignores that pedestrian nonsense to instead philosophise about the philosophical nature of philosophy.

Over the last couple of decades our abstractionist friends have confected a multiverse of arcane prognostications about what “time” could be. Youtube is littered with them. A particular favourite is the “Block Universe”, wherein every possible moment in the past, present and future all coexist simultaneously. Time is not a river but geometry. Their very small hat is hung upon consoling remarks made by Einstein, when he wrote to the family of a lifelong friend (Besso) who had died recently:

Now he has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion.

We’ll give Einstein a pass because he was a genius and his earlier mathematical models, themselves initially without any data or evidence, turned out to be spookily accurate. Yet this does not excuse those who wish to follow, who invent stuff and then smother it in acres of postdoc mathematics to pass off as fact. We seem to have circled around an entire millennia, with Deus Vult! exuberantly supplanted by The Mathematics Shows!

Gravity ≠ Force   ↑

Einstein consistently maintained Gravity was not a force. The Newtonian approach is to say Gravity exerts a non–contact attraction force on an object to make it fall (eg. the apple) or orbit the Earth (say the Moon). Put colloquially, Gravity Sux.

Einstein repudiated this. He argued Gravity doesn’t “pull” or “attract” anything. By his Equivalence Principle, there is no difference between accelerating an object due to Gravity or an external action. Instead things “fall” in a gravitational field solely because spacetime is curved.

You have probably seen trampoline analogies, where a large heavy object causes a trampoline’s surface to distort. A tiny ball is then rolled nearby to show how the path of the smaller ball is deflected by the heavy ball’s influence. Likewise coin funnels at fast food restaurants — coins roll down in a spiral towards the centre due to conical geometry.

Whilst this simplistic approach is easy to understand, it is conceptually wrong [ Note IX ]. In reality the heavy object (say a planet) causes spacetime to curve such that time is distorted. The closer you get to the planet’s surface — ie. the further you travel down the 4D gravity well — the slower time appears to pass. Likewise the farther away from the surface the faster the observed time. NB. (1) I covered this above and (2) clocks themselves don’t actually slow down nor speed up, they only appear to do so from an external observer’s frame.

Now for the magic — it is this infinitesimal difference in time in a gravitational field which causes objects to “fall”. They appear to accelerate as they fall, not because they are acted upon by an external force, but because they experience increasing time differentials while moving through an increasingly curved spacetime.

(Let that sink in for a moment…)

This time–difference (“ΔT”) need not be vast, it just has to exist. Indeed for an object released (say) 10m above the ground on Earth, the ΔT due to spacetime curvature, between the object and ground, is in the order of mere picoseconds. Nevertheless it is non–zero, so the object falls. Alternatively out in deep space, where there is effectively no spacetime curvature, ΔT ≈ zero and therefore the object maintains a straight path.

Spacetime gets curved by a (say) planet such that a smaller object’s World Line is deflected toward it. ΔT causes the curved trajectory, which to us appears as the object falling toward the ground. Without the planet, the object would continue travelling in a straight line into the future, but since spacetime is curved, so too will be the object’s path.

This can be difficult to visualise, so see the first part of Idea List (2021) How Time Dilation Causes Gravity, and How Inertia Works on YouTube.

Explaining Gravity by getting rid of Newton’s attractive force was certainly provocative and radical for 1916! Yet even today most people refuse to countenance it. But consider the case of a force–based accelerometer: place it onto a firm surface and it will report an acceleration of 9.8m/s². Now drop it from a height and while falling it will report zero acceleration since it experiences no force (!)

Fame is the Prize   ↑

Three Photographs

The Solvay Conferences were gatherings of (mostly) physicists to share research and discuss cutting–edge ideas. They were attended by pre–eminent scientists and, of course, Einstein was invited.

Two group photographs were taken at these conferences, one in 1911 and another in 1927. The before/ after difference in Einstein’s demeanour is remarkable.

At the first conference in 1911, pride of place is occupied by Solvay and Lorenz. Einstein is on the far right, looking distractedly across the room and almost ignoring the camera. His “Miraculous Year” papers had been published six years earlier, yet he is obviously not the most prestigious person present, although at 32 he was the second youngest.

Everything changed at the fifth conference in 1927. In 1919 Eddington had confirmed GR deflection of starlight and Einstein had been awarded the Nobel Prize in 1921. Instant global fame. Consequently the 1927 seating plan has Einstein front and centre, confidently staring into the lens. He is immediately flanked by Lorenz and Langevin, with Dirac hovering over his right shoulder. Other physics luminaries are also present, mostly in the background or periphery: Shrödinger, Pauli, Heisenberg, Bohr and Planck — the who’s who of 20th century Physics and eventually 17 Nobel Laureates — all playing second fiddle to our man Albert.

The third photograph is a formal portrait taken by Karsh in 1948. It was done at Princeton after the war and shows him being uncharacteristically melancholy. The joy and confidence of fame had long faded. By this time he was haunted by ghosts for which he blamed himself. As noted on the Karsh website:

One did not have to understand his science to feel the power of his mind or the force of his personality. He spoke sadly, yet serenely, as one who had looked into the universe, far past mankind’s small affairs. When I asked him what the world would be like were another atomic bomb to be dropped, he replied wearily, ‘Alas, we will no longer be able to hear the music of Mozart.

The Home Front

By most accounts Einstein enjoyed his sudden fame. Initially. In the 1920s he took pleasure from the attention and used it to further his interests in humanitarian and political causes. He got to tour the world to deliver physics lectures. Upon escaping to the USA with his wife and family in 1933, he also set about helping displaced persons and other refugees. As he got older he even became a role model for the “absent minded professor”, despite his contemplative personality being manifestly different. However, as the years became decades and his fame kept growing, the constant attention and scrutiny became tiresome and he had to rely on secretaries to screen him from increasingly unhinged disciples, who forever wanted to hijack his time.

On the marriage front, after the shouty Mileva he was (mostly) happy with Elsa. At least he didn’t have to write any further “Marital Demand” lists. The happiness was unfortunately short–lived, for Elsa died in Dec 1936 from an unspecified illness. He never married again, although apparently he did “horizontally collaborate” with his secretary Helen Dukas plus a Soviet lady spy.

It took a while for Hollywood to start using SR and GR as plot devices. They almost inevitably get it wrong, but kudos for trying.

A note on abbreviations: “SR” and “GR” you already know; “L/C” is length contraction; “M/D” is momentum dilation; “T/D” is time dilation and “STC” is spacetime curvature.

Planet of the Apes (1968)

Three astronauts crash–land their spaceship “Liberty 1” on Earth in the year 3978 after being launched in January 1972, following a near light speed attempt to fly another star. I believe this is the first mainstream Hollywood movie which made use of T/D as a plot device, sixty odd years after SR was published

Close Encounters of the Third Kind (1977)

US Navy airmen, presumably from “Flight 19”, emerge from the alien spaceship near the end of the film. They are still wearing their WW2 uniforms and have not aged a month, despite being abducted in 1945

Einstein & Eddington (2008)

Attempts to dramatise the confirmation of GR by Arthur Eddington via observation of a solar eclipse in Príncipe in May 1919, along with their lives during WW1 and its aftermath. Similar ground was covered by Season 1 of “Genius” (2017), but for both productions the science and basic historical accuracy are ignored to instead focus upon lerv ‘n’ relationships. I only recommend these films if you like playing drinking games: down a shot every time you spot an error or falsehood — I guarantee you will be under the coffee table by the 40–minute mark

Interstellar (2014)

Everyone’s nerdy go–to movie for GR, but my enthusiasm is more constrained. T/D due to severe STC in the vicinity of a super–massive black hole ✓ Employing a Nobel Laureate to create 3D modelling to spectacularly visualise the Black Hole ✓✓ Robots TARS and CASE ✓✓✓ Depicting STC on the “water planet” so severe that 1 hour = 7 years without physiological effects ✗ Weepy astronauts debilitated by not being able to hug their kids or papa ✗✗ Lerv is the one thing that we’re capable of perceiving, that transcends dimensions of time and space ✗✗✗✗✗✗✗✗✗✗✗✗…

Childhood’s End (2015)

Based on Arthur C. Clarke’s 1953 novel, which eventually led to his collaboration with Kubrick to create “2001: A Space Odyssey”. In the third episode of the TV miniseries, Rodericks stows away on Karellen’s spaceship to fly back to their world before returning to Earth. The round journey takes a few weeks, yet 80 years elapse on Earth. In the novel Clarke explains the aliens (and stowaway) can survive the incredible accelerative forces by using a technology which overcomes inertia (!)

Passengers (2016)

A movie which attracted a disproportionate amount hostility upon release, although not for its science. Here the spaceship “Avalon” is travelling at 0.5c while transporting colonists to an exoplanet “Homestead II”. 32–years into the journey they perform a slingshot manoeuvrer around the red giant Arcturus, which admittedly looks spectacular, but the producers/ science advisors forgot to allow for L/C and Doppler Shifts. In the movie the flypast looks normal, but in reality the star and its colours would have been greatly distorted due to relativistic effects

Lightyear (2022)

Enjoyable film, but similarly garnered a lot of abuse upon release. Our hero Buzz, after being stranded on exoplanet “T'Kani Prime”, volunteers to test a new spaceship engine. During numerous trials only a few hours elapse in his frame, but a total of 66 years for those on the planet. That he managed to do this at speeds ≤ 0.8$ c $ is mysterious. Cosmic fur–balls from Sox the robotic cat?…

Barbie (2023)

Cinephiles were baffled by the confluence of “Barbenheimer”, but the answer lies deep within the heart of Relativity. For when Barbie and Ken journey between worlds, they are depicted navigating 4D spacetime using cars, boats, spaceships and bicycles. These deceptively clever 2D representations brilliantly encapsulate the GR World Lines our intrepid heroes must traverse in order to obey the Principle of Least Action…

Project Hail Mary (2026)

Based on Andy Weir’s successful 2021 novel. As one of few authors who takes care to get the science right, the film, despite its spectacular visuals and attention to T/D, again forgets L/C distortions when depicting spaceships tearing through space. Mr Weir’s spreadsheet shows the Hail Mary achieved 0.995$ c $ at the acceleration flip–over towards “Tau Ceti”, which means L/C would have been considerable [ Note X ]. Most viewers missed this, instead obsessing over the accuracy of a scene with an unbalanced centrifuge (sigh)

Exodus (2027)

Not a movie but a AAA game. T/D is tightly woven into the plot, where years/ decades elapse whenever you move between worlds, with the effects of your actions/ choices delivering good or terrible outcomes for those you leave behind. Confront time itself to shape the fate of generations. Indeed

It is difficult to convey the nuclear dread which underlay our childhoods in the 1970s and ‘80s. I grew up in Kogarah and Hurstville, both southern suburbs of Sydney. Both easily within the blast damage radius of a 2MT warhead from a Soviet R–36MUTTKh, air–burst over Fleet Base East, a major Australian Navy facility near the Sydney Opera House.

Of course things were far worse for those who lived in London, Berlin, Washington or Moscow — all bullseyes for the Cold War’s estimated 64.1K nuclear weapons. You could see the effect most starkly in Moscow, expressed in the engineering and architecture of Metro railway station entrances. As I note elsewhere on 4020.net:

The reason Metro entrances looked like massive cold-war nuclear bunkers was because… they were entrances to massive cold-war nuclear bunkers.

The closer you got to the Kremlin, the thicker the blast doors became. Building domes went from brick to reinforced concrete to solid steel. Escalators to underground platforms doubled or tripled in length…

Then there was Able Archer ‘83. At the time my mother was visiting relatives at Szombathely on the Hungarian western border, just inside the Iron Curtain. She became so frightened by the constant display of maps with arrows on TV, along with warnings of impending nuclear attack, that it was more than a decade before she dared to visit again.

We (mostly) have Einstein to thank for all this.

Section Quick Links

• Einstein’s Letter to FDR
• Einstein’s Ghosts
• Squaring the Circle

The Letter   ↑

We had known for decades prior to WW2 that radioactivity involved the release of tremendous amounts of energy, albeit spread over long periods of time. E=mc² explained where the energy came from (hint: “nuclear binding energy”), but that’s where things remained. Vast amounts of energy, gradually released by randomly decaying atoms over a period of time. Good for nuclear medicine, not so much for gigawatt power production nor (say) explosives.

As Einstein put it in 1932:

There is not the slightest indication that nuclear energy will ever be obtainable. It would mean that the atom would have to be shattered at will.

He wasn’t alone. In 1933 Rutherford added:

Any one who says that, with the means at present at our disposal and with our present knowledge, we can utilise atomic energy is talking moonshine.

Also in 1933, Szilard speculated about the possibility of initiating faster nuclear chain reactions by the (then) newly discovered neutron. In 1938 came the watershed, when Hahn and Meitner demonstrated the nuclear fission of uranium atoms. Not only was it caused by low energy neutrons, but the reaction released more neutrons for more reactions… We weren’t talking moonshine anymore.

Before we go any further, an analogy is appropriate. Imagine a 55 gallon drum filled with match heads. Not entire matchsticks, just the bulbous ends which ignite when struck. Now take one of these match heads, remove it from the drum, move well away and light it. It will ignite and give off some heat. Repeat every five seconds for ten minutes. The temperature in the room will increase by a degree or two, and it will become a bit smoky, but that’s all. Now ignite one of the match heads atop the pile inside the drum. It will take a moment for the ignition to spread, but it will then be followed by an explosion so ferocious you won’t even have time to run for the door.

The reaction rate changes everything. The first example mimics the behaviour of radioactive decay, with a random atoms splitting sequentially at a leisurely pace. Energy is released, but spread over a long(ish) period. The second illustrates what happens when a chain reaction takes place. All the atoms now split and release energy almost instantaneously. Kaboom.

When Szilard learned the Germans had shattered a uranium atom at will, he joined the dots and realised there was now a pathway to create bombs of unimaginable power. Following the Munich Agreement, it was also obvious the Nazis were about to embark on a conquest of Europe, equipped with brilliant scientists with intellects vast and cool and unsympathetic, who could build such bombs at scale. Szilard, being a recent refugee from Nazi terror, understood the German fission of uranium was not merely an academic exercise, but rather an existential crisis for all humankind [ Note XI ].

He contacted Einstein (by then both had fled to the USA) and in July 1939 they met at Einstein’s holiday house on Long Island. When appraised of the situation, Einstein agreed Szilard had a terrifying point. If the Nazis ever managed to make atomic bombs then the free world would be in desperate trouble. Although Einstein was famously anti–war and (mostly) apolitical, they both decided this needed attention by US authorities. Having dealt with plodding academic administrators all their lives, they knew it was pointless going through usual bureaucratic channels. This issue required immediate action, so only a letter to FDR would cut through. Especially if was co–written and signed by “the world’s greatest scientist” Einstein.

Two letters were composed in Aug 1939, one a shorter “executive summary”, the other longer and more detailed. The latter, with Einstein’s name all over it, was (eventually) hand–delivered to FDR in the Oval Office in October. The president was shocked. The Manhattan Project began. Trinity illuminated the pre–dawn desert. Hiroshima and Nagasaki were incinerated. ≈ 200K people died, although imperial elites remained carefully untouched. WW2 KOHEЦ. The Cold War began.

After years of diligent rubbing, the nuclear genie was well and truly out of the bottle.

The Ghosts   ↑

Claims the scientists at Los Alamos didn’t understand the enormity of what they were doing are disingenuous. They had the Dec 1917 Halifax Harbour explosion — equivalent to 2.9KT — as a model for what a catastrophic blast could do to a city. Following the Trinity test, they learned their “gadgets” would unleash five times what happened in Halifax, maybe more due the air–burst technique they intended to use. Oppenheimer and Groves would have also known about the devastating RAF Fauld underground munitions explosion (3.5 KT) in Nov 1944. None of the scientists were under any illusions and there were no surprises. From the outset they were determined to stick it to the Nazis before the Nazis could stick it to them. When Germany capitulated, their focus shifted to Japan without skipping a beat. Total War. Kill ‘em all.

Figuratively speaking, Einstein’s hands were covered with as much blood and ashes as anyone else. Even more so, for it was him (+ Szilard) who got the nuclear ball rolling to its terrifying conclusion. He was fully aware of this and as a lifelong pacifist it horrified him. As Newsweek in 1947 quoted him saying:

Had I known that the Germans would not succeed in developing an atomic bomb, I would have done nothing for the bomb.

Even in 1954, a year before he died, he was still filled with regret:

I made one great mistake in my life when I signed the letter to President Roosevelt recommending that atom bombs be made, but there was some justification — the danger that the Germans would make them.

(Notice he said signed instead of co–authored…)

It didn’t help that his wife and confidant Elsa had died in 1936, three years before the letter. His first wife Mileva was still alive — she died in Aug 1948 — although they had been estranged for 17 years and she was far away in Zürich. Einstein was surrounded by people yet he was on his own [ Note XII ]. He had to decide quickly, with no time for leisurely back ‘n’ forth correspondence. His choice was stark: he could stay silent or alert the president. With Szilard’s encouragement he chose the latter, which six years later would yield hundreds of thousands of corpses. After WW2, despite making speeches about World Peace and featuring prominently in anit–nuclear movements, Einstein’s powerful visual imagination ensured he spent the remainder of his life haunted by visions of actual and potential nuclear victims.

Squaring the Circle   ↑

Einstein achieved tremendous success with Relativity. But its structure and assumptions were vastly different to that of the other great idea of the Twentieth Century: Quantum Mechanics (“QM”). Indeed the two appeared irreconcilable. GR explained how things behaved on very large scales, while QM did it for the very small. Relativity was built upon a traditional framework of cause & effect, whereas QM relied upon probabilistic statistics, indeterminate states and even outright discontinuity. This bothered Einstein a lot. He became increasingly antipathetic toward QM, even spending the 1927 Solvay conference (the last one he attended) arguing with debating Bohr about it.

Quantum theory yields much, but it hardly brings us close to the Old One’s secrets. I, in any case, am convinced He does not play dice with the universe.

Einstein became determined to overcome QM’s “irregularity” by showing it was merely part of a greater Relativity framework, thereby unifying the two. He dedicated an enormous amount of the last twenty years of his life at Princeton doing so. He failed.

In 2007 a Hawaiian surfer–dude claimed he accomplished it. That we haven’t heard anything since is telling.

For better (or worse) we live in an Einsteinian universe. The concept of spacetime is commonplace. E=mc² a t–shirt slogan. We now understand how Gravity works and have drawn a little closer to understanding Time. Yet the reconciliation of General Relativity with Quantum Mechanics has remained elusive. Maybe one day, in some far–off brilliant future, we will find a way to merge the two. Maybe…