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Sunday, October 18, 2015

A beautiful thought (addendum)




You might ask why the speed of light is a universal constant. Specifically, why the speed of light is the same for all observers irrespective of the relative motion of the light source. The most straightforward answer is that this is simply how it measures. That is, when we go out and measure the speed of light, it always measures the same (299 792 458 m/s in a vacuum) whether the light source is beside us, travelling towards us, or travelling away from us. This applies to all particles (not just photons) that travel at the speed of light. That the speed of light plays some fundamental role in the fabric of the universe appears to be consistent with experimental observation. This postulate of Einstein’s special theory of relativity withstands the scrutiny of some of the brightest minds - not to mention the calculations of some of the most complex and accurate instrumentation - humanity can throw at it. 

But there is another way to think about it. And it involves going back to basics (if you will, the flip-side of complex calculations and instrumentation). Let’s consider a lightbulb hanging from the ceiling of a room. We can determine the lightbulb’s location within that room by setting up a coordinate system using three points of reference (say we use the lightbulb’s distance from the ceiling, its distance from one wall, and its distance from another wall). If we know the dimensions of the room then these coordinates will determine the exact position of the lightbulb within that room. But the lightbulb exists in that room regardless of how we choose to measure or relate to its position. We impose a construct of coordinates so we may understand how the lightbulb relates to us (as a sentient being within that local environment). A deeper understanding is that the lightbulb simply exists. To relate to it we choose to define its position within our experience of the three-dimensional space in which we find it. 

The same goes with the speed of light. A photon is what it is. When it travels through a vacuum we can determine its speed by measuring the distance it travels through space and dividing it by the time taken. To do this we may impose a construct (like we did with the example of the lightbulb) through which we can understand how a beam of light (or, more accurately, a beam of electromagnetic radiation) relates to us. The basis of that construct is an agreeably consistent, three-dimensional Euclidean space with time that reliably tempos in one direction - to which we apply some universally agreed measure of space and time (a lifetime of being told where we need to be and at what time tends to reinforce such a construct). All well and good for the most part but keep in mind that this is a perception of reality and not reality itself. And that this construct is learnt, not innate. If the speed of light appears unchanging regardless of the relative velocity of the observer’s frame of reference, then this means that there is a problem with the tools we have at our disposal. Either the instrumentation we use to measure the speed of a photon is faulty or inaccurate (the speed of light is, after all, ridiculously fast) or the construct on which we base our measurement is wrong. A photon exists. It just “is”. Our attempts to define it (and measure it) revolutionised our (deeper) understanding of space and time.

For the hundred years since Einstein’s general theory of relativity found its mathematical construct we have understood the speed of light to be a universal constant. One of a select number of physical constants that determines the laws of physics in the known universe (https://en.wikipedia.org/wiki/Physical_constant). The question we should ask is not why the speed of light is a universal constant (although it is worthwhile continuing to check its validity using clever experiments) but how much things would change should the speed of light, or the value of any other physical constant for that matter, be different from that which we have calculated. 

One of my favourite skits from the wonderful Blackadder series is the one where Edmund Blackadder tries to teach Baldrick how to count.


It’s funny because nothing Baldrick says is actually wrong. Blackadder creates an expectation (if you will, a basic mathematical construct which we share with Blackadder) that Baldrick is unable to meet. 

In truth, some things are not nearly as obvious as they first appear (http://scienceblogs.com/goodmath/2006/06/17/extreme-math-1-1-2/). 


While others are not as they seem.





Saturday, September 12, 2015

A beautiful thought






You are standing in a room. You feel your weight push through your feet and onto the floor below. The floor pushes back with equal force. A light shines through a window and onto the wall behind you. You yawn and stretch out your arms. Nothing special here.

Now imagine you are floating in the vacuum of space. Then imagine you are floating in a room that itself is floating in space. The room then accelerates through space pushing you to the floor. You get up and stand on the floor. The floor pushes back. A light shines through the window and onto the wall behind you.

Imagine that both scenarios are equivalent.

From an observer sitting outside the room the light beam travels, as it always does, in a straight line through the vacuum of space. Then your room appears out of nowhere and accelerates across the light beam. The light passes through the window and hits the wall behind you. The outside observer sees the light travel in straight line - through space, through the window, through the space in the room, then hitting the wall.

From your point of view - as an observer inside the room - the light appears to bend towards the floor as it travels from the window to the wall. For the wall has moved since the light first entered through the window.

If both scenarios are equivalent then the force otherwise known as gravity causes light to bend towards it. Or, to Einstein’s way of thinking, mass causes a curvature in the unified (indivisible) entity of space-time.






2015. 100 years of General Relativity. (short version)

A photon travels in a straight line within the same (gravity-free) frame of reference as its light source. When it enters another frame of reference that is travelling at a constant acceleration it appears to curve as seen by an observer within that accelerated frame of reference. As the photon itself has not been acted on by a force to displace its position in space-time it must continue in the straightest possible line within space-time (as Newton’s laws of motion are respected in both these frames). The simplest conclusion (without having to resort to theoretical modifiers to explain why different observers see different outcomes) is to accept that the acceleration of a frame of reference causes a curvature in space-time. If uniform acceleration is equivalent to being at rest in a gravitational field then the gravitational field of a mass causes a curvature of space-time. The larger the mass, the greater the gravitational field (acceleration), and so the greater the curvature of space-time.



2015. 100 years of General Relativity. (longer version)

If the speed of light is immutable - ie the speed of light is unchanging irrespective of where the observer sits - then this has profound consequences on how we view the world. This was one of Einstein’s two postulates in 1905 when he came up with his special theory of relativity (Special Relativity, SR). By accepting that the speed of light is the same for all observers regardless of the relative motion of the light source conveniently explained such conundrums as the null result of Michelson & Morley’s experiment and the Lorenz transformation. As speed is distance divided by time then this postulate must affect our conventional understanding of space (distance) and time. Simply said: if the speed of light is a universal constant then space and time cannot be universally constant.

Einstein’s other postulate in SR was actually an idea Galileo introduced a couple of hundred years earlier. It states that the laws of physics are identical in all non-accelerating frames of reference. In other words, playing table tennis on a smoothly running train travelling in a perfectly straight line at a steady 100km/h (ie not accelerating or decelerating) would be exactly the same as playing table tennis at the station platform. Indeed, if the background were to be obscured, then you would not be able to tell the difference. 

In 1905 Einstein also described light as a particle. And that nothing could travel faster than the speed of light. And that time dilates (slows down) with increasing velocity (and also that length contracts in the direction of travel). And that mass and energy could be thought of interchangeably using the formula E = mc2

But let’s go back to that room jetting through space and talk about time dilation.

Imagine that the room you are in is allowed to stop accelerating and thereby maintain a constant velocity through space. You now float off the floor and flap your arms in the vacuum of the room - your body is weightless within the room as it travels at the same velocity as the room relative to the observer sitting in the vacuum of space outside. Perchance you have a special clock that consists of two mirrors between which a photon of light bounces. Each time the photon bounces from one mirror to the next is a measure of one second. Let’s say you set up the mirrors parallel to the direction the room is travelling so that the photon’s path is at right angles to the direction of travel.



What you see is a photon of light bouncing (at 90 degrees) up and down between the two mirrors. But what an observer outside sees is a light path that takes into account both the photon’s path between the mirrors and the travel of the room.




That means the observer outside in space sees the light travel in an oblique path. That is, the distance travelled by the photon is greater for an observer outside the room than for you, the observer inside the room. Imagine the chap outside the room has an identical clock. He has one eye on your clock on one eye on his. If the speed of light is constant for all observers regardless of the relative motion of the light source then in the time the chap in space sees the photon of his clock tick one second he will also note that the photon of your clock hasn’t quite made it from one mirror to the other. That is, your one second is slower than his one second. Your clock ticks slower. And, the faster the room travels (which causes the photon of light to pass with greater obliquity), then the slower time goes inside the room as observed by the chap floating outside in space. This is time dilation. 

The concept of light behaving as a particle with an invariable speed (in a vacuum) regardless of the frame of reference of the observer is key to understanding SR. It’s pure simplicity did away any need for a messy, blah-blah aether medium, explained a lot of experimental physics that, until that time, didn’t really make much sense (it resolved the fudge factor for physics involving very high speeds i.e. the physics of electromagnetic waves including visible light), and allowed the possibility of meaningful time travel. Time travel into the future that is. For if you leave your moving frame of reference then you effectively enter a frame of reference somewhere further ahead of where you would have been had you stayed fixed within that frame. Time travel into the past requires some imagination, a lot of maths, and may not actually be possible.

There’s more. Now imagine the same situation of the room moving at a constant velocity through the vacuum of space. You decide to float to the very centre of the room and and turn on a light that emits photons evenly throughout the room. Say that the leading wall in the room and the trailing wall both glitter brightly when a light photon strikes it. From your point of view (in the middle of the room) both walls will glitter simultaneously. However, for the observer floating in space outside the room the trailing wall glitters before the leading wall as the photons have to travel a shorter distance to the trailing wall (because, from the point of view of the observer in space, the trailing wall is moving towards the light source and the leading wall is moving away from it). In other words, the time an event occurs is dependent on the reference frame of the observer.

Classical mechanics of the sort Newton played around with treats space and time as universal constants and works perfectly well when talking about cannon balls, gas lamps, water mills and steam engines. It cannot entirely explain a world of guided missiles, nuclear power, microwave ovens, and x-ray machines. What Einstein said by introducing his special theory of relativity in 1905 was that space together with time were relative (at least when dealing with non-accelerating frames of reference - hence the term “special”). The maths is complicated and quite unnecessary when dealing with things at relatively low speeds but becomes increasingly relevant when dealing with things that travel close to the speed of light. Things such as electricity (or rather, electromagnetic waves) and the stuff that happens when atomic bombs explode.

In essence what Einstein concluded in SR is that what we see every day as space (as defined by an object within three dimensional Euclidean space) and time cannot be separated and needs to be thought of together as a unified entity of “space-time”. More accurately, in 1908 Einstein’s observer-dependent relativity of inertial reference frames evolved into a deeper, observer-independent structure of all possible events that can occur within a four dimensional manifold known as Minkowski spacetime (we’ll get back to this).

It gets even better.

Let’s get back to the example of time dilation. There is an apparent paradox here as velocity is but a relative thing. That is to say that what you see with the photon clocks and what the observer in space sees is identical but in reverse. You see time slow down for the chap in space just as he sees your time slow down in the moving room. At a constant velocity neither of you can tell the difference. So if the chap in space walked into your room shouldn’t he then be walking into your future? Einstein resolved this paradox by saying that there is a difference between the two frames of reference: to get to your constant velocity your frame of reference had to accelerate away from the frame of reference of the chap in space (and to get back to his frame of reference you will have to decelerate). So, although both frames of reference appear identical at a constant velocity they are, nonetheless, different. It is time in your frame of reference that slows down when compared to his because it is your frame of reference (the room) that accelerated out of his frame of reference (outer space).


Time-travelling machines. Reference frame:  Oz Comic Con, Brisbane Convention Centre, Australia. 
Saturday, 19 September @ 1247hrs.


And Optimus Prime.


Special Relativity is special in that it is specific to non-accelerating frames of reference (in physics terms you are an inertial observer in an “inertial frame of reference” as Newton’s law of inertia is not violated - see *). General Relativity came into being because Einstein laboured to reconcile his special theory of relativity with Newton’s law of universal gravitation. The brilliance of Einstein - his eureka moment if you will - was his concept of “equivalence” described at the start of this indecently long waffle. It states that what we perceive as gravity is nothing more than a perception of uniform acceleration (at 9.8 m/s2). In essence, Einstein’s general theory of relativity (General Relativity, GR) is relativity applied to a world where things are constantly accelerating/decelerating (in physics terms it is relativity applied to “non-inertial frames of reference”). General Relativity is considered by many to be the greatest contribution to science made by a single person.

If the maths of SR is hard, the maths needed for GR is much harder again (it took Einstein 8 years to provide the mathematical evidence for GR - and it would have taken longer if not for Marcel Grossmann and David Hilbert). For a person who never did more than high school physics and maths it will be impossible for me to fully understand the beauty of the concept. Words are kind but malleable when trying to describe the purity of thought that sees its truest form in mathematics. Nonetheless, it is the centennial of GR and I would like to understand it more. And I have internet connection.

Newton’s law of universal gravitation postulates that gravity is an attractive force between two bodies that is proportional to the product of their masses and inversely proportional to the square of the distance between them. Newton provides an equation that can be used to make this calculation but key to the understanding of this gravitational force is that it is attractive over an infinite range and that its effect is instantaneous. This concept of force propagating instantaneously between two bodies at a distance troubled Newton himself. It troubled Einstein even more. What was the nature of this force? How does one body “know” the presence of another body and through what medium is this force conveyed?

For a start SR dictates that nothing can travel faster than the speed of light. According to SR time stops altogether for a particle travelling at the speed of light. If you look at the thought experiment with the photon clocks, you can see that if the room went at the speed of light then, from the perspective of the observer in outer space, time in the room stops as the photon in the room appears stationary between the mirrors. In other words, from the point of view of the photon, time does not exist (it takes about 8 minutes for a photon of light from the sun to reach your eyeball, from the photon’s point of view the moment it was created is the moment it hits your retina - a timeless instant). A photon does not have a rest mass (it has relativistic mass as it has energy), does not carry a charge, and is stable. If you will, the photon is an easily handled elementary particle whose characteristics make it useful for furthering our understanding of the universe (it is not the only particle that travels at the cosmic speed limit in a vacuum). In SR the speed of light is a fundamental constant of the universe. But from the photon’s point of view its speed is infinite - it can travel any distance in no time at all as time does not exist at the speed of light. So the speed of light is only finite (and invariable at 299,792,458 m/s when travelling in a vacuum) to an outside observer. 

Theoretical physics currently understands four fundamental forces. The strong interaction (which holds atoms together), the weak interaction (responsible for radioactive decay and neutrino interactions), electromagnetism (which operates between particles with an electric charge), and gravity (a weak but long-ranged force caused by matter/ mass). These forces can be described either as the action of a force field generated by one particle on another or in terms of force-carrier particles that exchange between them. For example, electromagnetism can be described as a continuous electromagnetic field or as the exchange of force-carrier particles (which happen to be photons). Physicists will use either the field or quantum descriptions depending on which happens to be more workable in their calculations. The graviton is the virtual force-carrier particle hypothesized for gravity. Two decades earlier, in November 1915, Einstein described the field equations for gravity that encompassed and expanded on his special theory of relativity. These are the equations for the general theory of relativity.

The idea came to Einstein in 1907.

“In 1907 I was sitting in a chair in the patent office in Berne when all of a sudden there occurred to me the happiest thought in my life. If a person falls freely [to Earth] he will not feel his own weight! I was startled. This simple thought made a deep impression on me. It impelled me towards a theory of gravitation.” 


This evolved into the thought experiment where the man is placed into a room and both are placed into free fall caused by the pull of Earth’s gravity. Because both of them fall at the same rate (as Galileo showed with his - almost certainly apocryphal - experiment when he dropped two balls of different weights off the Leaning Tower of Pisa) then there is no way that the man within the room can tell whether he is floating in space or plummeting to his doom. Specifically, there is no experiment that he can do within the confines of that room that can tell him whether he is in free fall, at a constant velocity, or floating stationary in space (assuming that this is all taking place in a vacuum). 

I know what you are thinking because I didn’t get it either. I’m thinking that there are plenty things that tell me when I am in free fall: (1) I can see my motion, (2) I can feel the rush of air against my skin, (3) the accelerometers in my inner ear tell me I am in a state of acceleration, and (4) my inside bits push up against my backbone just as my back pushes into my seat as I accelerate off at the traffic lights. Putting me in a vacuum and into a closed room (that is also in free fall) might rid me of the first two sensory inputs but what about the other two? This mistake of logic arises from thinking that the Earth we walk upon is static and thereby the reference frame for everything else. Think of the inner ear and the abdominal cavity around your guts as boxes that hold stuff inside. The reason why we can feel acceleration without having to see a rush of movement or feel the passage of air is because the stuff moving around in those boxes provide us with sensory information that tell us we are accelerating. This is a form of sensory input known as proprioception. But the information is referenced from a starting point that we perceive as static and (mostly) dependable. That reference frame being Earth (or, more specifically, the space around Earth). It’s the same as you floating in space in a box that itself in floating in outer space. You know when the box accelerates in outer space because you are pushed up against the trailing wall of that box. In the same way the fluid in the inner ear and the organs in your abdominal cavity reference the space you walk around in. When your body accelerates that stuff sloshes in the opposite direction giving you an indication of the direction and speed of that acceleration.


“With great power..”


“comes great responsi..”

“What the fuck?”


By using the thought experiment of a free falling man and then enclosing him in a room that is also in free fall allows us to ignore the reference frame of Earth. If we ignore a preference bias for any specific reference frame then we will see that the observations made by Newton and Galileo are preserved whether one is at a constant velocity or in free fall (free fall implying that all objects within the room are subjected to the same constant acceleration). If I throw a ball into the air when standing on the edge of a cliff I will see it travel in a parabolic curve. If I jump off the cliff just as I throw the ball in the air I will see it continue in a straight line away from me (until air resistance slows its velocity). It’s worth saying it again: there is no experiment that we can do within the confines of a closed frame of reference that can tell us whether we are in free fall, travelling at a constant velocity, or floating stationary in space. If you throw a ball in any direction within that frame of reference it will behave in exactly the same way (in observance of Newton’s laws of motion) in any one of those scenarios. In other words, a free falling frame of reference is also an inertial frame of reference.

Well, actually, physicists will point out that this is not entirely true as the acceleration in a gravitational field is directed towards a single point representing the centre of gravity of that mass. That means two bodies free falling in a gravitational field caused by a large mass will converge (in violation of Newton’s law of inertia). This effect, known as a tidal force, is best explained here: http://www.einstein-online.info/spotlights/geometry_force. But given a small enough reference frame and a short enough time frame then tidal effects become negligible.

What Einstein saw in his thought experiment was something that no-one else before him had ever considered. It was the equivalence principle described at the start of this post. That is to say: standing in a room on Earth is equivalent to standing in a room accelerating at 9.8 m/s2 through outer space. Or, more generally, being at rest in a gravitational field is physically equivalent to that of uniform acceleration (in gravity-free space). 

To Einstein free falling objects can be considered to be at rest (after all they are in an inertial frame of reference) and everything on the surface of Earth - human observers included - are accelerating towards them (ie Earth pushing up against us puts us in a non-inertial reference frame). When we throw a ball in the air it goes up in a straight line obeying Newton’s law of inertia. The movement appears to be parabolic only because the Earth we stand on accelerates us (the observers) up towards the ball. When we sit in a chair we don’t push down on it, rather the chair pushes up against us to keep us in place. Newton understood that the act of acceleration leads to fictitious forces (see *). Einstein tells us that gravity is merely one example of a fictitious force seen in non-inertial frames


In the mathematical language of physics what Einstein was saying in his equivalence principle was that Newton’s second law of motion (which states that the “acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object” ie a = F/m) often stated as F = ma is equivalent to Newton’s law of universal gravitation. That is, inertial mass and gravitational mass are one and the same.


Not since 1543, when Nicolaus Copernicus published his heliocentric theory, had planet Earth taken such a downgrade in its preeminence: for this lump of rock and water not only failed to be the centre of the universe, it also fails to be the correct frame of reference for creatures that crawl upon its surface. In 1905, Einstein showed that space and time were relative depending on the inertial reference frame of the observer. In 1907, Einstein came to a realisation that Earth was a non-inertial reference frame and gravity was its side effect. Einstein understood that fictitious forces are the characteristic feature of non-inertial frames as described by classical mechanics. If space and time are relative in inertial reference frames then, maybe, space and time are somehow distorted in non-inertial frames thereby accounting for the occurrence of fictitious forces. Fictitious forces such as gravity.

In essence what Einstein was proposing was this: what if the fundamental nature of the fictitious force (gravity) seen in the non-inertial reference frame (Earth) is actually a curvature of space-time? 

At this point it is not too hard to follow a line of reasoning that gravitational fields cause a curvature of space-time based on the thought experiment outlined at the start of this post. A photon travels in a straight line within the same (gravity-free) frame of reference as its light source. When it enters another frame of reference that is travelling at a constant acceleration it appears to curve as seen by an observer within that accelerated frame of reference. As the photon itself has not been acted on by a force to displace its position in space-time it must continue in the straightest possible line within space-time (as Newton’s laws of motion are respected in both these frames). The simplest conclusion (without having to resort to theoretical modifiers to explain why different observers see different outcomes) is to accept that the acceleration of a frame of reference causes a curvature in space-time. If a uniform acceleration is equivalent to being at rest in a gravitational field then the gravitational field of a mass causes a curvature of space-time. The larger the mass, the greater the gravitational field (acceleration), and so the greater the curvature of space-time. 

It is, of course, a lot more complex than that. Physics requires the precision of mathematical equations to be testable in the real world. And until such equations are provided any idea, no matter how neat, novel or logically argued, will remain vague and unworkable. It took Einstein eight years and a number of false starts to fully explain how relativity of space and time could be applied to non-inertial frames. To do this he had to first understand then further develop some very complex applied mathematics (Einstein, for all his genius, was primarily a physicist and secondarily a mathematician). But even before Einstein got his final formulations together there were some testable consequences from combining SR with a novel take on gravity. One such consequence was gravitational frequency shifts as described here by Wikipedia https://en.wikipedia.org/wiki/Introduction_to_general_relativity (which is equivalent to gravitational time dilation described here - starting at 9:49 - on youtube https://www.youtube.com/watch?v=4rBkXVuT95U). Both links provide an excellent introduction to GR and the curvature of space-time (not to mention executed with the clarity and brevity of experts). If you have ever relied on GPS (global positioning systems) - on your phone, in your car, or in a plane taking you to Honolulu - then gravitational time dilation is relevant to you.

Coming to terms with GR means first coming to terms with the geometry of curved surfaces then coming to terms with non-Euclidean geometry. Think of a coordinate system in which to track what happens to a particle when it travels from an inertial frame of reference to a non-inertial frame. Lifting a statement straight out of the Wikipedia link we learn that Einstein (and others) saw analogies for this when folding folding a Cartesian coordinate system (in which the coordinate lines are straight lines) to a curved coordinate system (where coordinate lines need not be straight). The simplest, archetypal examples of curvature are a circle in two dimensions and a sphere in three dimensions. In particular, the surface of a sphere brings up some interesting and easily understood properties. The straightest line (shortest distance) between two points becomes a geodesic and parallel lines can meet.



Two particles travel in parallel on a planar surface.


If they appear to meet without any apparent external force then classical physics suggests that they are under the influence of a fictitious force (ie they are travelling in a non-inertial frame).


However, if the surface itself is curved into a sphere, then particles that are parallel at the equator meet at the poles even though both particles have travelled in straight lines. If space is curved in non-inertial frames then this could be postulated as the real reason for the appearance of a fictitious force.


Similarly a geodesic is the shortest distance between two points on a curved surface. In the case of a spherical surface a geodesic is a segment of great circle whose centre corresponds with the centre of the sphere.


Einstein understood that this property of curved surfaces is analogous to the appearance of tidal forces seen in gravitational fields. It was this understanding that stirred him to develop his theory further by using the mathematics of geometry. Einstein already knew that three-dimensional space alone cannot define the position of a particle: time adds a fourth dimension that needs to be factored into the equation. The condition of SR where space and time are relative when comparing the local environment of observers in different reference frames adds further complexity. For this Einstein had to progress to the non-Euclidean geometry of four-dimensional Minkowski spacetime.

Before I proceed further I need to backtrack on the concept of space-time glossed over earlier in this post. Although the term space-time and spacetime are used interchangeably I’ve used the hyphenated “space-time” to represent a relativistic, observer-dependent view of space and time. I now want to talk about “spacetime” as an observer-independent structure that shows all possible events that can occur within a four dimensional manifold of space and time.

Special Relativity and the universal constancy of the speed of light was not a proposition solely forwarded by Albert Einstein. In 1905 Einstein showed the world that space and time were relative depending on the reference frame of the observer. That same year Henri Poincaré published two papers on the “dynamics of the electron” which tackled the same problematic areas in physics albeit from a completely different angle. Poincaré’s interpretation of relativity came from solid physics and mathematical background: an approach much less accessible to those who are not conversant in theoretical physics and its evolution. His particular genius (in the area of SR as Poincaré made many important contributions to mathematics and theoretical physics) was his ability to provide a mathematical solution to problems seen in the study of electromagnetism first put into field equations by James Clerk Maxwell then pondered upon by Hendrik Lorentz. The solution used by Poincaré abandoned the notion of absolute space and time whilst accommodating the tenet of an aether medium. It was Poincaré’s solution that first outlined space and time in four-dimensions. Einstein’s insight into the same problems (published a few months after the first of the two papers by Poincaré) did away with the aether medium altogether and allowed the layperson a glimpse of the wonders of relativity - an enlightenment that complex equations steeped in tradition could not possibly hope to achieve. 

By 1905 the concept of SR came into being albeit in the form of some funky equations that could explain away some perplexing aspects of physics involving waveforms (ie fields) that travel at the speed of light but not in a form convenient nor cohesive enough to take it beyond the setting of inertial systems. It was Hermann Minkowski who brought the work of Einstein and Poincaré/Lorentz together and provided the mathematical tools for a four-dimensional manifold where space and time are inseparable (now known as Minkowski spacetime). By 1908 four key players - Hendrik Lorentz, Henri Poincaré, Albert Einstein, and Hermann Minkowski (James Maxwell should also be included as the father of the applied mathematics used for all of this) - had reinterpreted everything we thought we knew about space and time. The stage was set for Einstein’s “happiest thought” to evolve into the mathematical equations that became the general theory of relativity.





So what is it? What is this spacetime, or rather, how can we think about four-dimensions when we can only relate to three? In essence four-dimensional Minkowki spacetime is the collection of all possible events that can occur in three-dimensional space and time. Not a three dimensional Euclidean space that evolves with time (ie three dimensions with a past, present and future) but four-dimensional spacetime (three coordinates of space and one coordinate of time) whose points correspond to all possible events at any time, anywhere, without ascribing a chronological order to these events. There is no motion as such in Minkowski spacetime but it is our experience of it that gives us a perception of past, present and future. As sentient being that grinds through his daily existence in three dimensions it might be possible to imagine this four-dimensional spacetime manifold by removing one dimension of space and replacing it with a time dimension like so:


Mr Square and wife in two-dimensional Flatland.


Flatland in three-dimensional spacetime.


Spacetime according to Mr Square.


Flatland lifecycle in spacetime.


Another way to illustrate this comes via this excellent youtube video https://www.youtube.com/watch?v=sryrZwYguRQ. As the narrator says: “We are evolving through a spacetime which we perceive only in temporal crosscuts - [progressing] from moment to moment. At any moment we only perceive a three-dimensional crosscut of the four-dimensional reality.” Nice. The sad truth is that we can’t draw four-dimensional Minkowski spacetime. Indeed, many argue that we can’t really conceive it outside of its mathematical context. But we can glean some of its essence through words and pictures.

No one can explain why our experience of spacetime means we each appear to carve a path through through this four-dimensional system (and what you see as my path may not actually be what I see). But this isn’t meant to be an existential discussion. Nor some whacked-out idea that the sensation of “deja vu” may be a blip of reality rather than a trick of the mind (although it would be awesome if it was). Nor is this a convoluted introduction to multidimensional universes and such (which may yet be possible). It was meant to be about me (my posts are always about me) coming to terms with the concept of GR without requiring the maths needed for a deeper understanding.


Journey of an innumerate attempting to explain a mathematical concept.
(The time axis is “ct” ie speed of light x time.
According to SR when a particle is absolutely stationary in space it travels through the time dimension at the speed of light, ie space has no meaning. Likewise if a particle travels through space at the speed of light then time does not exist, ie time has no meaning. 
Existence as we know it lies somewhere between these two boundaries.)


What links the events in spacetime is not the distance (space) nor the time interval between events - as SR tells us that these are not universal constants - but causality. And causality is bounded by the cosmological speed limit - which is a universal constant - the speed of light. Spacetime and causality is introduced in yet another clever youtube video linked here: https://www.youtube.com/watch?v=YycAzdtUIko
Indeed, the entire series on relativity and spacetime is so good that I will put a link to it here:
An extension of this series is linked here:
https://www.youtube.com/watch?v=AwwIFcdUFrE&list=PLsPUh22kYmNDRYfImV3BzNZ6yTwhIpe0k 
If I had stumbled upon these videos earlier I wouldn’t have bothered about writing this post in the first place. And if you have read this far I suspect you’ll probably be thinking the same thing (when I say my posts are about me getting my head around things it is a statement of fact not self-deprecating commentary).

But what are we talking about here? What is the nature of causality? And if space and time are dimensions that cannot be universally agreed upon by all observers then what can?  The easiest way to understand this is to first watch the youtube video linked above (https://www.youtube.com/watch?v=YycAzdtUIko) then watch this excellent youtube video linked here: https://www.youtube.com/watch?v=5ODGnbkobNg

Nice. To summarise:

Causality can be seen as possible events bounded by the past and future light cones for any particular event within spacetime. 







The spacetime interval is invariant. Which is to say that while different observers may not agree about the space (distance) and time between events they will all agree on the spacetime interval as represented by this equation:


Something to hang our hats on.


The parabolic function of the spacetime interval means that, for any universally agreed value of it, the closer a described event approaches the speed of light through space then the greater the distortion of time (and space) as seen by an observer stationary within that space.





Wrapping up the basics is another link here: 

And that is just a taste of flat spacetime. Flat spacetime is the mathematical structure for SR. What Einstein went on to do was formulate the field equations that determine the curvature of spacetime under the influence of mass-energy (mass-energy equivalence E = mc2) and linear momentum. This is the curved spacetime of GR. For this Einstein had to learn the intricacies of Riemannian geometry (http://comet.lehman.cuny.edu/sormani/research/riemgeom.html) from an old university buddy, Marcel Grossman. It must be remembered that Einstein already knew what he wanted to achieve in 1907 when he had his “happiest thought” and started in ernest to resolve SR with Newton’s law of universal gravitation. But until Einstein mastered Minkowski spacetime and Riemannian geometry he lacked the mathematical structure for doing so. Spacetime is a necessary tool for giving predictive power to Einstein’s relativity. But it should be recognised that Einstein came to this geometric solution rather late, and, by most reports, rather rushed with David Hilbert almost beating him to the line. The point is that the concept of mass-energy leading to a curvature of space and time preceded the mathematical equations used to describe it. What I’m trying to say (some 6,000 words since I started) is this: to get a flavour for GR does not necessarily mean we need to understand the complex mathematics that has become its language. A deus ex machina of sorts? Well, probably.

“When a blind beetle crawls over the surface of a curved branch, it doesn’t notice that the track it has covered is indeed curved. I was lucky enough to notice what the beetle didn’t notice.”

Albert Einstein (from a letter to his son Eduard, 1922)

What Einstein showed with GR in 1915 was that gravity is not so much a force in the classic Newtonian way of thinking but a distortion of spacetime caused by mass and energy. Or as John Wheeler so famously summarised: in GR “Matter tells space how to curve. Space tells matter how to move.” This is not the entire story. Most physicists believe that gravity is one of the four fundamental forces of the known universe (along with the two nuclear forces and electromagnetism). The other three have been unified with quantum theory leaving gravity as the only force to defy unification (http://arstechnica.com/science/2011/04/is-gravity-a-result-of-thermodynamics/). This is not the place to discuss it. Nevertheless the predictive power of GR has been tested by super-accurate atomic clocks put into jet planes and on top of mountains; has described the effect of gravitational lensing and the precession of the perihelion of Mercury; and continues to pervade our daily lives through the use of GPS systems.

In practical terms the unwieldy calculations mean that relativity as proposed by Einstein is applied only when we look at the atomically small (in the case of SR) or the cosmologically large (in the case of GR). For most of us, centennial celebrations or whenever, this is not something we see or apply in any meaningful way as we interact with the rest of the world. Here, on planet Earth, most of us make do with relativity as described by Galileo Gallilei in 1632 and with Mr Newton’s three laws of motion (published in 1687).

I have a conversation inside my head that goes something like this:

Paul: Time dilates in a moving frame of reference. A situation that becomes meaningful as the speed approaches the speed of light. That means we can travel into the future.

John: Cool.

Paul: Don’t you want to know why?

John: No. Not really.

Sally: I would like to know.

Paul: Well it all starts with an experiment into luminiferous aether by Michelson and Morley... (yada, yada, yada)

With so many distractions and priorities we each have a limit of what we allow ourselves the luxury to pursue. The question I have is this: at what point does Sally feel satisfied with the explanation? 










*Newton came up with three laws of motion. One of them, his law of inertia, states that the velocity of an object does not change unless a force acts upon it (although Galileo was actually the first to understand this). An inertial reference frame is one that respects Newton’s law of inertia. Indeed, an inertial reference frame respects all three of Newton’s laws of motion.

If you want to get a handle on inertial frames vs non-inertial frames then this 1960s youtube video is excellent: https://www.youtube.com/watch?v=aRDOqiqBUQY.

In summary the video states:
- The frame of reference in which Newton’s law of inertia holds true is an inertial frame
- The law of inertia holds in the Earth’s frame of reference so it is an inertial frame (it is actually an approximate one - see **)
- A cart moving in a constant velocity relative to the Earth is an inertial frame
- A cart that is accelerated relative to the Earth is not an inertial frame
- Fictitious forces arising in linearly accelerated frames of reference explain the movement of objects within it
- Fictitious forces also arise in rotational frames of reference (which have centripetal acceleration)
**Because the Earth is round and rotates on its own axis as well as revolving around the sun it really isn’t a true inertial frame but the effects of the fictitious forces are very small

This is video was created to introduce the physics of inertial and non-inertial frames to children and underscores two hundred years of accepted wisdom. An inertial reference frame (the Earth, the stars, or whatever) can be used to determine if another frame is inertial or non-inertial by identifying whether fictitious forces are seen within that frame. Newton understood that fictitious forces (also known as inertial forces, d'Alembert forces, or pseudo-forces) are forces that result when a frame of reference accelerates with respect to an inertial frame such as Earth. To Newton Earth was an obvious inertial reference frame. But this did not explain why things fall to the ground (it did not explain gravity). Einstein refused to accept that gravity simply existed because of the presence of a (large) mass. His solution is known as the general theory of relativity.


http://www.scientificamerican.com/article/what-is-a-fictitious-force/ 

http://www.mathpages.com/home/kmath641/kmath641.htm