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Spacetime! Relativity and Beyond

Updated: Feb 22


Figure 1. What is spacetime? [1]


Chained inside a cave, a man sits watching shadows flit across the wall. Behind him is the cave’s entrance but he has never left the cave, has never even turned his head; he is left to understand the outside world from only the shadows it casts.


This is Plato’s allegory of the cave, that we are chained to human experience and see only a shadow of some greater reality. The goal of physics is to understand this reality, and in the last century we have finally glimpsed its underlying structure: spacetime.


It has taken thousands of years to get there as physicists travelled on a treacherous path called the theory of relativity, inciting public outrage and even death at the hands of the Roman Inquisition [2]. Let us embark on this daring journey now. It involves paradigm shifts, a discarding of common-sense beliefs, and perspective changes, mental games in which we imagine perspectives other than our own [3]. It is an exhilarating adventure, which will at the end provide a glimpse of a reality beyond the world we see.

 

We begin with how space and time naturally appear.


Space is three-dimensional. Objects are separated by distance, and we can decompose this separation into three directions: left/right, forward/backward, up/down, specified by three numbers  called spatial coordinates.

Time is one-dimensional. We see objects moving in space and compare how fast they move with a standard clock, an object which ticks at a constant rate. Each clock tick is given a single number t called the time coordinate. The amount of space an object traverses per clock tick is called its speed, or velocity when we also know what direction it is moving in.

 

The first two paradigm shifts are that the origin of space and time is relative, and that direction is relative. There is no such thing as the centre of the universe, and there is no single moment in time which is the centre between past and future [4]. The universe on the whole has no notion of ‘upwards’ or ‘leftwards’; the cosmos acts without prejudice in all directions. For millennia people thought the Earth was the centre of the universe, and although now that seems nonsensical, it makes sense from a personal level. It is natural to judge distances based on our location (which is the Earth on cosmological levels), equivalent to treating our location as the centre of space. Likewise, when we claim that an event took place ‘yesterday’ or will take place ‘tomorrow’ we are judging time intervals relative to the present moment, treating the present as the centre of time from which time intervals into the past or future are determined. Neither of these notions have any absolute meaning.


The first perspective change is to imagine any point in space and any moment in time as both the place you are currently at (the centre of space) and as the present moment (the centre of time), a practice often employed by historians. The second perspective change is to convince your brain that any arbitrary direction is ‘up’; right now, you are hanging upside down like a bat relative to people on the opposite side of the Earth.

 

The third paradigm shift is that motion is relative, or equivalently space is relative. We cannot tell if we are moving uniformly or not, and there is no preferred state of rest. It initially seems obvious that the ground is at rest and that planes flying over the ground really are moving. However, when we are on an airplane without turbulence we may as well be sitting still. Galileo realised this four hundred years ago when he observed that butterflies flying in the hull of a fast-sailing ship drifted easily in all directions, and never bunched up at the back of the ship as if tired from keeping up with it [5]. Any object moving with constant speed can be deemed as ‘at rest’. An airplane travelling huge distances relative to the ground is at rest relative to itself and claims it has travelled no distance at all; therefore, motion makes space relative. It is impossible for everyone to identify a point in space as remaining the same point over time.


The corresponding perspective change is to imagine any uniform velocity as at rest. On the next train trip convince yourself that the train is sitting perfectly still, and rather it is the outside world that is rushing past you at a great speed.

 

The fourth paradigm shift is that acceleration and gravity are locally relative. It is impossible to tell if we are uniformly accelerating, or if gravity locally exists around us. This principle seemingly defies common sense, and to see why, imagine being in a car. The car is uniformly moving if neither the accelerator nor brake pedal are pressed and the steering wheel is straight. The car is accelerating if either the accelerator or brake pedal are pressed or if the steering wheel is turned. Even with our eyes closed we can tell the difference between uniform and accelerated motion because we can feel acceleration. When the accelerator pedal is pressed we are pushed back into our seats, and when the steering wheel suddenly turns to left we sway to the right. There is another thing we can detect without seeing, which is gravity. When we sense the weight of a heavy bag pulling on our hands what we are actually feeling is gravity. When the lunar astronauts leapt easily over the moon’s surface they could feel the weakness of gravity. Einstein realised a century ago that uniform acceleration and local gravity are really two sides of the same coin; what we think is the effect of one could well be the effect of the other.


However, on a large enough scale gravity does establish itself as a real entity which places the planets in orbits around the Sun. An object moving freely under gravity is called inertial and always travels along the straightest path [6], but sometimes this path can curve. The Earth is inertial since it moves freely under gravity as it orbits around the Sun in a circle (more accurately, a very circular ellipse), but how can a circular path possibly be straighter than a line? [7]


This perspective change involves imagining uniform gravitational fields appearing or disappearing at will. Next time your car suddenly turns left making you sway, imagine instead that you were always travelling in a straight line and for a moment some devious schemer turned on a gravity field pulling you to the right. Imagine that gravity suddenly stops existing in your vicinity and the entire ground begins rushing upwards, faster and faster.

 

The last paradigm shift is that the speed of light is absolute, or equivalently time is relative. Even though motion is relative for physical objects, Maxwell showed that light (an electromagnetic wave) is never at rest and always travels at a constant speed [8]; all inertial observers must agree that light travels at exactly 299,792,458 metres per second [9].

Imagine you turn on a flashlight and see that its light has travelled around 300,000 kilometres in a second. Your friend Alice is aboard a rocket moving extremely fast and sees that it has only travelled 3,000 kilometres (remember from Galileo that space is relative). How can both of you possibly agree on the light’s speed? Only if Alice says the light has travelled for 1/100th of a second. Your clock runs 100 times faster than Alice’s clock even though they are built from the same parts and operate under the same laws of physics. Furthermore, you age 100 times faster than Alice. Say you are both twenty years old right now; at your seventieth birthday Alice (aboard the rocket) will still be twenty, having only aged six months [10]. Motion makes time relative, as Einstein realised a century ago. Every person carries their own separate experience of change; it is impossible for everyone to claim that two events happened at the same time.


The final perspective change involves imagining time speeding up or slowing down. Picture an astronaut aboard a spaceship currently whizzing through the Solar System who acts in slow motion; by the time you have turned eighty they have just made their breakfast. Imagine an alien race living in a galaxy far, far away moving incredibly fast away from the Milky Way, and just then as you blinked your eyes the aliens lived through a thousand years of history.



We are approaching the grand finale of our relativistic journey. Galileo has shown that motion makes space relative and Einstein has shown that motion makes time relative, yet the constant speed of light is an absolute tying them together. Space and time can pull and push on each other as if connected by invisible but firmly fixed tendons, and this tells us to join space and time into a cohesive four-dimensional whole called spacetime. Spacetime exhibits hyperbolic geometry (figure 2), an exotic structure which skews the way we add velocities and places the speed of light as the universe’s local speed limit. Spacetime is also endowed with a group of ten gems of symmetry, each being born from a paradigm shift. A symmetry is a perspective change we are allowed to do which leaves the structure of spacetime looking the same. The lack of an absolute origin gives four translational symmetries, the lack of absolute direction gives three rotational symmetries, and the lack of absolute motion gives three velocity symmetries.


Remember the puzzle of the Earth’s straightest path around the Sun being a circle? Einstein solved this when he used the gravity/acceleration paradigm shift to show that spacetime is a warping, rippling fabric which can distort the notions of length and time to the extreme. Spacetime squeezes the region around the Sun, trapping huge amounts of space and time within it. The Earth always follow a straight path relative to the space around it, but it is space itself that has curved around the Sun, guiding the Earth in a circle. [11]



We have finally shaken off the shackles of normal experience through paradigm shifts. We have seen an absolute spacetime structure emerge from relativity, and have held its gems of symmetry in our hands by wielding perspective changes. We have seen this spacetime fabric take on a powerful life of its own, its contractions and expansions governing the motion of the heavens. The spatial world we live in and the flow of time we experience are but cross-sections of some greater four-dimensional structure, silhouettes of an outside reality dancing on the wall of our cave.


 

The year is 1908. The crowd hushes as a man steps up on stage, a mathematical explorer who has found something beautiful underlying the universe, and Hermann Minkowski shares with the world:

“Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” [12]



[1] Figure 1. Coffeekai via Getty Images

[2] Giordano Bruno was burned at the stake in 1600 for various heretical views, which likely included his belief that the Earth is not the centre of the universe.

[3] ‘Perspective changes’ are mathematically known as coordinate transformations.

[4] The centre (origin) of time t=0 is not the same thing as the beginning of time since time also extends backwards into negative numbers. Whether or not there is a beginning of time (and hence a preferred temporal origin t=0) is a different matter not considered here. Treating the present moment as the origin t=0 is also a more subtle issue since it involves a ‘temporally moving’ origin rather than a temporally static origin.

[5] Paraphrased from a quotation taken from Drake’s 1953 translation of Galileo’s Dialogue Concerning the Two Chief World Systems (1632).

[6] ‘Inertial’ is being defined in the Einsteinian sense, not the Galilean sense. An ice puck drifting over a rink with constant velocity is non-inertial (since the Earth’s gravity can be reinterpreted as acceleration), whereas a flying projectile is inertial (since free fall can be reinterpreted as weightlessness, the absence of gravity/acceleration).

[7] More correctly, inertial observers follow the straightest path in spacetime. The Earth’s spacetime path is a helix, its spatial projection being a circular orbit.

[8] See Bedding, From Maxwell To Einstein (2022) for a full derivation from Maxwell’s equations to light’s constant speed. https://www.youtube.com/watch?v=PV2LURcHDUc

[9] Light travels at exactly this speed through vacuum, which is the vast majority of the universe. Inside mediums light can slow down.

[10] This is only true from your perspective. From Alice's perspective you are the one moving, not her (from Galileo's relativity), and hence she sees you age slower than she does. These two perspectives remain consistent since what is simultaneous for you is not simultaneous for her.

[11] Figure 3. ESA/Hubble and NASA, ‘A Horseshoe Einstein Ring from Hubble’,

[12] Hermann Minkowksi, Address to the 80th Assembly of German Natural Scientists and Physicians, 1908.


By Peter Lavilles

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