Relativity Theory with No Equations

Why is the Theory of Relativity interesting?

It is interesting to me because it is the root of my understanding of many things – from complicated things like “How does an electric motor work?”, to simple things like “Why does an apple fall from a tree?”.

But I am surprised at how much I read online about Relativity Theory that is either not very helpful, or sometimes downright misleading.

People write about “time slowing down”, and present images of a bowling ball making a depression in a rubber sheet. I have even read that gravity curves something called “spacetime” because Einstein’s equations predict that it does. (What?!)

When I saw this info-graphic online (Figure 1), it occurred to me that almost every single part of this info-graphic is either misleading or just wrong.

So, I decided to put my money where my mouth is, and document what I think – by means of critically assessing each of the six statements, and re-formulating them.

I am not an expert, and if any of my understanding is wrong, I sincerely that hope someone will correct me. (Contact Page).

So here is an introduction to Relativity Theory, with no equations, and certainly no rubber sheets…

Figure 1

“Light Speed Remains Constant”

This statement could easily be confusing. Whilst “light speed” is a phrase in common usage in sci-fi movies, it is actually not a well defined notion.

It could be taken to be mean the speed of light in any circumstance – in which case the statement is false. Light propagates in the form of a wave, and does so at different speeds in different materials. Also, the speed of light through a material medium will be measured differently by observers who are moving relative to the medium.

I often read incorrect statements about light propagation in a medium – such as:

  • “The light still travels at the same speed that it does in a vacuum, it just bounces around inside the medium, so takes a longer path.”
  • “The light doesn’t slow down, it gets absorbed and then re-emitted by electrons.”

Light passing from a vacuum into and through a medium (such as glass) interacts with the charged particles in the glass – creating an additional electromagnetic wave that combines with the original wave, slowing down its speed of propagation. As it emerges, the light becomes free of that interaction, and so resumes its usual speed through the vacuum. This was first demonstrated by Hippolyte Fizeau in 1851, (although he misinterpreted his results in terms of the prevailing theories of his time).

A good modern example of the variable speed of light in a material medium is that a charged particle (for example, an electron) can travel through water faster than light waves propagate through the same water. Such is the origin of the blue glow called Cherenkov Radiation that is familiar to people working with nuclear reactors.

However, we could alternatively assume that the writer really meant “The speed at which light propagates in a vacuum is a constant.” – in which case the statement is correct, but was a fact well known long before Albert Einstein’s time, and – in and of itself – is not an axiom (an a-priori assumption) of Relativity Theory. Understanding why the speed of light in a vacuum must be a constant is useful historical context:

In 1865 James Clerk Maxwell published a set of equations describing the nature of electric and magnetic fields. He noticed that one consequence of his formulations predicted a situation where an oscillating electric field would induce an oscillating magnetic field, and vice-versa, and so the two fields acting together could act to propagate energy through space in a wave-like manner – as an electromagnetic field. His equations showed that in a vacuum, the velocity of that self-propagating wave depended upon only two things:

  1. The permeability of the vacuum (the extent to which free space resists or allows magnetic fields)
  2. The permittivity of the vacuum (the extent to which free space resists or allows electric fields)

These two things are simply universal physical constants. Nobody knows why they take the values that they do, they are just observable and measurable properties of our universe. It may be that in other universes, the properties of the vacuum are different, and so the maximum speed would be different. We cannot investigate other universes in order to find out.

When he did the sums, Maxwell obtained the answer 299,800 Km per second – which is very fast, and astonishingly close to the measurement of the speed through space of starlight, made in 1728 by James Bradley (301,000 km/s). Thus, Maxwell was the first person to realize that light must be a propagating electromagnetic wave.

Summary

The speed of light varies depending on the medium it is travelling through. It varies according to the refractive index of the medium.

In one very particular situation – when light propagates through a vacuum (refractive index = 1) – it is free to adopt the maximum speed allowed by the permittivity and permeability of empty space. That speed is the universal physical constant known as “c”, or 299,792 Km per second).

That such a speed is also relativistically invariant (i.e. measured to be the same by all observers regardless of the observer’s constant motion relative to the light source being measured) is an important axiom of the Special Theory of Relativity.

Albert Einstein always maintained that the work of Fizeau and Maxwell was very influential in his own thinking.

“Faster you move through space, slower you move through time”.

This is a poor choice of words. The striking consequences of constant motion being relative are not about your movement rather, they are about the effect upon how you measure other moving things.

No-one and nothing “moves through time – such a thing is impossible. If you doubt this, then consider: How would one measure how fast an object is moving through time? For that to be possible, there would necessarily have to be some sort of meta-time by which to measure the rate of actual time. Am I “moving through time” at a rate of 1 meta-second per second? This inevitably leads to a next question – am I also moving through this meta-time at a rate of 1 meta-meta-second per meta-second? When you arrive at an infinite regression like this, you can be pretty sure that something is wrong in your thinking.

So what does change with constant relative motion?

The Theory of Special Relativity begins with an axiom that the speed of light in a vacuum is invariant – which is to say, measured to be the same by all observers who are either standing still or moving at constant velocity relative to the light source. This is a perfectly reasonable assumption; for if the laws of physics are to be useful at all, then surely they must apply equally to every observer – otherwise how could we ever agree what the laws are, or even call them “laws” at all? Since the speed of light in a vacuum is defined by universal constants, then that speed should be invariant.

Einstein realized that if the assumption about invariance was true, then a clock that is moving relative to an observer must count fewer ticks between the same two events when compared to the observer’s own identical but non-moving clock.

Figure 2 (courtesy of Physics Stack Exchange) represents a clock that counts a “tick” each time light bounces off the topmost mirror of two mirrored surfaces mounted facing one another :

Figure 2

The little wiggly arrows (light pulses) are moving at the same speed, but the moving clock counts just one bounce off the top mirror in the same time that the stationary clock counts two bounces.

So, invariance of the speed of light in a vacuum has the consequence that if you compare information about clocks with others who are moving at a constant velocity relative to you, then you will find that their clock counted fewer ticks (between the same two events) than your clock . The phenomenon is called time dilation.

This is not just appearance – from your point of view, their clock really has counted fewer ticks than yours. And it is not just their clock – everything is affected, even bio-chemical processes. Consequently, a person moving relative to you really ages less than you do.

The disagreement over elapsed times has a significant knock-on effect; if our clocks disagree, then (obviously) we will disagree over which events are simultaneous. Since our measurement of (for example) the length of a stick depends upon us measuring both ends of the stick simultaneously, then our relatively moving observers also disagree about how long a stick is. A moving stick measured by a stationary observer is shorter than expected. The phenomenon is called length contraction.

The two observers do not disagree about their relative velocity. Consequently, when the moving observer arrives at their destination, and finds that their clock has counted fewer ticks than expected between the departure event and the arrival event, then they can only conclude that the distance to their destination was shorter than expected. For the moving observer, distances in the direction of motion are contracted.

Furthermore, because constant motion is relative, there is no privileged frame of reference – one cannot say which clock/observer is really standing still. Consequently, the above applies vice-versa – the information that a moving observer receives about your measurements will show your clock counting fewer ticks, and your sticks being shorter. Both observers are correct from their point of view, and there is no arbitration that can prove either of them wrong. What we measure is reality. The startling realization arising from this is that all the things we thought we could agree upon, such as:

  • How long things take to happen.
  • How far it is from one place to another.
  • Which events are simultaneous.

are in fact not fundamental to the universe, and are measured to be different depending upon the observers’ relative motion.

All this is however, about what we measure – not about what we see. Normally, when we look at an object, light from the left-hand side of the object reaches our eye at the same time as light from the right-hand side. We can (usually) also assume that both those rays of light set out from the object at the same time.

When talking about how a very fast moving object would look, we need to take into account the fact that light from one part of the object may have set out at a different time compared to light from another part of the object, and the time-of-travel of the light on its way to the observer is significant.

A real, three dimensional object (say, a large cube for example) passing across our field of vision at a speed close to “c” would actually appear to be not contracted, but twisted and rotated. (Figure 3)

This is due to the fact that rays of light leave the moving object and travel towards the observer with some component of speed that is less than the object’s actual speed. Hence, light from the rear of the cube-shaped object can reach us simultaneously with light from the nearside face. We can think of this as due to the nearside face having “gotten out of the way of the light” due to its very high speed.

Figure 3 – Cubes moving at 0.9 c.
Movie courtesy of : https://www.spacetimetravel.org/

Bizarrely, this twisting would exactly compensate for the real length contraction, so the apparent total length of the cube would appear the same. If the moving object were close to us, additional distortions would be apparent.

If we were vast creatures born and living in interstellar space, communicating by light pulses, and routinely travelling at hundreds of thousands of Kilometres per second between stars, then the relativity of distance, elapsed time and simultaneity, and the complex apparent shapes of fast moving objects would have been obvious to us from birth. But we are not such – we are tiny, tiny, little ultra-slow moving creatures living on a tiny ball of rock; and so it was not until Albert Einstein that we realized the truth.

Summary

If we make the a-priori assumption that the speed of light in a vacuum is invariant for observers moving at constant velocity, then it must follow that if there is a relative speed difference between two observers, and if they are able to compare information about events which they both observed, there will be disagreement about how much time elapsed between the events, and about how much space separated the events. The disagreement on elapsed times necessarily also implies that those observers cannot even agree on which events are simultaneous.

If you measure the motion of any object moving at a constant velocity relative to you, then you will measure its time as dilated and its length in the direction of motion as contracted. The greater the relative speed, the greater the effect will be. Pointing a camera at a very fast moving object produces images that additionally entail peculiar distortions.

“Time is the 4th Dimension”

This is misleading, and amounts to an unjustifiable reification.

The universe we perceive has exactly 3 dimensions of space, and in addition, we experience time as the measure of change in the arrangement of things in the universe.

So why do we read about 4-dimensional “spacetime”?

Analysts often create “2-dimensional” charts in which time is one of the coordinates, and the property being studied is the other coordinate. Figure 4 presents an imaginary 2-dimensional space. It has fewer dimensions than the real world. In this chart, position in the horizontal dimension represents time, and position in the vertical dimension represents the price of gold. The information in this imagined space can be thought of as made up of a lot of tiny little dots connected into a line, each dot signifying the price of gold at a moment in time.

The red line therefore represents a “history” of the price of gold changing over time.

Figure 4

The gold chart is useful tool, providing a very visual and intuitive insight into the underlying events. The imagined space has a geometry, via which one can observe matters of interest such as the slope of the line (rate of change of the price), and other things like degrees of convergence, historic trends and possible future trends, etc. Using such a chart is not essential for market analysis, but the geometry of the imagined space helps us to understand how the price of gold in the real world moves.

Mathematicians also use imaginary spaces that have more dimensions than the real world. The “spacetime” of Relativity Theory is just one example of such a thing. By using 3 coordinates representing space, and 1 coordinate representing time, we can imagine a 4-dimensional “chart” in which each point represents a single event happening at a place and at a time. In this way, a line of connected points can represent the history or “world-line” of an object moving.

Using such a construct is not essential for Relativity Theory (indeed, Einstein himself in his correspondence warned against reading too much into it) – but the geometry of the imagined spacetime does help us to understand how physical objects in the real world move.

However, nobody in their right mind would claim that the 2-dimensional space suggested by the gold chart is somehow real – that the chart on a piece of paper somehow really is the price of gold that is used in a market trade.

Similarly, nobody should claim that the 4-dimensional space suggested by spacetime is somehow more real than the actual universe we experience.

When understanding gravity (see below) the imagined 4-dimensional spacetime comes in very useful, because gravity affects everything equally, and so a lot of very complicated relationships can be reduced to a single equivalent geometry. Such mathematical tools are not available in the same way for other fundamental forces which do not affect everything equally. The charge-to-mass ratio (for example) is different for electrons, protons and neutrons. Each type of particle would have to be assigned a different equivalent geometry.

Summary

When considering relative motion, theorists need to simultaneously take account of both clocks (measuring time) and sticks (measuring distance). To assist with this, they imagine a “3+1” dimensional spacetime, because doing so simplifies the associated mathematics down to an equivalent geometry in which we can gain useful insights about how objects are perceived to move in the real universe.

“Time Slows Down Around Heavy Objects”

No. Heaviness is a reflection of weight, which is due to matter exerting an electromagnetic force upon an object – a force which the object resists due to its inertia (Its inherent resistance to being “pushed”). Weight and time are only indirectly related.

When Einstein studied accelerating systems (General Theory of Relativity) he noticed that the force you feel pushing you backwards when a vehicle accelerates (caused by your inertia – your inherent resistance to that acceleration) feels very much like the force that holds you to the ground (caused by gravity). He proposed that in a small enough region of space, there would be no way to distinguish between inertia and gravity.

Imagine a twin-mirrored “light clock” (like the one we used in Special Relativity). This time we place it inside a rocket-ship that is accelerating in the same direction of the line of the to-and-fro light pulses.

As before, a “tick” of the clock is counted each time the pulse of light bounces off the topmost mirror. The lower mirror catches up to the pulse and receives it early. The pulse then reverses and chases the top-most mirror to catch up with it. But the rocket-ship’s motion is not constant – it is speeding up – so the second “leg” takes disproportionately longer to traverse. Overall, the light clock on the rocket counts fewer ticks than a stationary or constantly moving clock.

If it is true that acceleration and gravity are indistinguishable (the Equivalence Principle), that means that a clock that is located on the surface of a planet (experiencing gravity) should count fewer ticks than an identical clock located further away the planet (say, in outer space). This phenomenon is called gravitational time dilation. (Figure 5)

(As an aside, it is worth noting that over a large enough space, one can tell the difference between acceleration/inertia and gravity. The effect felt in the rocket ship is in one direction only. Over the surface of the Earth however, although the strength of the gravitational field is roughly the same everywhere, the direction of the field is everywhere different – always pointing towards the centre of the planet.)

That notwithstanding, the time dilation effect linked to acceleration/gravity is real, measurable, and (unlike Special Relativity effects) not reciprocal :

Figure 5
  • An observer on the ground sees light emanating from elevation as blue-shifted, meaning that the light gained energy as it lost height.
  • An observer positioned at elevation sees light emanating from a ground-based source as red-shifted, meaning that the light lost energy as it gained height.

In reality, what is known and confirmed by observations is that clocks that are distant from any concentration of mass/energy (such as a planet) actually do count more ticks between the same two events than clocks that are closer to that mass/energy. The presence of stress and momentum also have an effect, but for things like planets it is mostly mass that creates the effect. There is also an effect on measuring sticks, but the stick effect is small in comparison, so it is the clock effect which dominates the behaviour of moving objects as far as gravity is concerned.

Summary

Clocks closer to concentrations of mass/energy count fewer ticks between the same two events than clocks further away.

Any device operating a clock synchronized to a GPS satellite must allow for both the satellite’s constant motion (approx. 14,000 Km/hour) and its height (approx. 20,000 Km above the surface). If we call a millionth of a second a “tick”, then – per day – the satellite counts about 7 ticks fewer due to its velocity (described by Special Relativity), and about 45 ticks more due to its distance from the Earth (described by General Relativity). The net effect is a discrepancy versus the ground-based clock of about 38 extra ticks per day. The ISS on the other hand, is in a much lower orbit – where the velocity effect of Special Relativity dominates – and so its clocks count fewer ticks than a ground-based clock.

“Gravity is the Curvature of Spacetime”

No. Spacetime is not a thing, it cannot curve. It is the geometry of spacetime that has curvature. Gravity is caused (primarily) by what happens to clocks that are near to concentrations of mass/energy.

The clock effect has some interesting consequences for the geometry of spacetime. We can get some insight into this by dispensing with two of the space coordinates and showing a chart of just elevation and time. Below is “2-D” diagram (again courtesy of Physics Stack Exchange) :

In these diagrams (Figure 6), each vertical line represents the “tick” of a clock, progressing from left to right. Each horizontal line represent a distance in space. The red line is the history or world-line of an object.

When there is no nearby concentration of mass/energy (top chart), the object stays put and the geometry of the spacetime chart is flat and homogeneous in the vicinity. This is the chart of an object in deep space far from any stars or planets.

When mass/energy (e.g. a planet) is present (lower chart) the distance grid represents distance from the source of mass/energy. We could say that the top edge of the chart represents some elevation, and the the lower edge represents the ground.

In order to accurately reflect the fact that the clock at ground level counts fewer ticks that the clock at elevation, we must increase the spacing between ticks at the lower edge of the chart.

Consequently, the geometry of the lower chart becomes non-homogeneous – it now displays a pronounced curvature.

You can see from the red world-line in the lower chart that the object (over time) moves ever faster from the higher elevation to the ground. The chart reflects the pseudo-force of gravity.

Nothing has changed concerning the object. The red line – its “world-line” – is still straight. We just factored in the clock effect, which distorted the grid lines (the geometry).

Figure 6

These diagrams are very simplistic. The correct mathematics in the imagined 4-dimensional space is complex. The stress-energy-momentum that affects clocks (and therefore creates a gravitational potential) can only be mathematically expressed using tensors – a kind of multidimensional array of numbers that can represent both space and time and all of the many properties present, and additionally render certain quantities invariant – even from the point of view of accelerated observers.

So is the force of gravity real?

The force due to gravity is real – but only when looking from a certain perspective. Change the frame of reference, and the force disappears. For this reason, the force of gravity is more correctly called a pseudo-force.

You can easily demonstrate to yourself that gravity is a pseudo force.

In the the screenshot (Figure 7), I held my phone approximately one metre above a soft pillow on the ground. at the start of recording, it records sensing a gravitational force of 1-g (ignore the little wiggles, that’s my hand shaking).

Then I dropped the phone (somewhere between 28 and 29 seconds on the chart.)

You can clearly see that as it falls, the g-force felt by the phone drops almost immediately to nothing (zero-g). While it is falling, the phone records no gravitational force acting on it.

Of course the phone quickly collides with the pillow, and the large spike (up to 4-g) and what follows shows the phone bouncing before coming to rest, at which time it resumes recording that it feels a steady force of 1-g.

So what is the origin of gravity?

Time dilation causes gravity. If you imagine any object being composed of millions of very tiny little light clocks, then when the object is far away from any significant mass, all those clocks are synchronous. The object’s “world-line” in 4-D spacetime is aligned entirely in the time direction.

But if we add in a large mass like a planet nearby, then the clocks on one side of the object will count fewer ticks than the clocks on the opposite side of the object. Although the discrepancy may be small, the object’s world-line is skewed slightly. (Imagine a cart with one wheel binding and dragging against the ground…)

Since the object was previously aligned exactly with the time direction, then any skew will divert its world-line into the space direction. Just like our cart, the object veers towards the source of the “dragging” – which is the planet.

Figure 7

In the geometry of the imagined 4-D spacetime of General Relativity, the “straight line” is called a geodesic. It is important to avoid talking of objects “moving” along a geodesic. The spacetime construct is only a tool, and it is a static picture – time is already present in the picture. Rather, one should imagine a geodesic as the shape of the history of an object traced out through both space and time.

Geodesic paths often manifest in the real 3-dimensional universe as curved or even elliptical paths.

The differently shaped paths of freely falling objects such as:

  • the downward straight line of an apple falling from a tree.
  • the ballistic curve of a cannon ball in the air.
  • the eliptical orbit of the International Space Station.

are all due to those objects tracing out a shape through 3-D space that is “straight line” through space and time, towards a place where their co-moving clock will count fewer ticks.

Summary

The behavior of clocks in the vicinity of large masses makes objects move so as to maximize their proper time. From the point of view of we observers who are not free-falling, and who live in our 3-dimensional universe, such motion appears to be driven by a force. We call that psuedo-force Gravity.

Einstein’s General Theory of Relativity equates the strength and direction of the gravitational field with the distribution of stress/energy/momentum of everything in the universe, in the curved geometry of a imagined spacetime.

Einstein’s equations correctly explain and predict the motion of (almost) all bodies in the universe with exquisite accuracy and precision. The only situations where this approach fails is (for example) near black holes, where there is such a huge concentration of mass/energy that clocks (as viewed by a distant observer) count fewer and fewer ticks – eventually to such an extent that any change in the clock situated near the black hole is in the distant observer’s infinite future.

Einstein had hoped that his theory would also explain the origin of inertia (often referred to as Mach’s Principle), thereby justifying Newton’s 1st law, creating a satisfying harmony between:

[1] The gravitational field being determined by the nature and distribution of all the stuff in the universe.

and

[2] The inertia of a body being determined by the nature and distribution of all the stuff in the universe.

But he failed in this. The observed effects of inertia such as occur in the accelerating rocket do not fall out of the theory, they have to be taken for granted. The Equivalence Principle is an a-priori axiom of General Relativity – nobody really knows why the inertia of a body in free-fall exactly cancels out the effects of gravity such that a free-falling body feels no force.

“Gravity Travels in the Form of Waves”

No it doesn’t. Gravity itself does not travel – it simply exists as the effect of a non-homogeneous field which permeates the whole of space. The strength of the gravitational field at any particular place can be calculated by taking into account stress-energy-momentum – all of the mass/energy in all of space, how it is arranged and how it is moving. It is impossible to know all those things for the whole universe, but Einstein’s General Relativity equations can be solved by making certain simplifying assumptions (for example about the average distribution of matter in the universe), and this allows scientists to make very accurate and precise predictions about how objects move.

However, disturbances or changes in the gravitational field do propagate like waves. Compared to other forces, gravity is very, very, weak, so such gravitational waves are difficult to detect via apparatus. It also requires tremendously energetic circumstances (such as the collision of two black holes) to send out gravity waves that are detectable.

Summary

Changes in the gravitational field propagate as wave-like disturbances, and (in a vacuum) travel at the vacuum speed of light – the maximum possible speed for any phenomenon that can carry information.

CODA: So why does an apple fall from a tree?

Whilst an apple is attached to the branch of a tree, it is experiencing a force due to the molecular attraction between its stem and the branch, which is electromagnetic in origin. That constant force is accelerating the apple away from its geodesic path, and so the apple is in an accelerated frame of reference. A clock attached to the apple would be counting more “ticks” than an identical clock positioned on the ground.

In the Autumn, the tree withdraws nutrients from the stem and it withers, eventually breaking the electromagnetic attraction. There is now no force acting on the apple. If a tiny camera were attached to the apple, the video would show the branch moving suddenly and rapidly upwards.

Meanwhile, the ground of planet Earth (which is an accelerated frame) rushes up towards the apple. During that time, the apple is quite still – it feels no force, and no acceleration, it has no weight – it is in an inertial frame or free-falling. (Note that for this thought experiment we ignore the resistance of the air.)

From the point of view of the accelerated frame of the tree (and the philosopher sitting under it) the apple is observed to move. It moves towards the nearest place where it will experience the maximum proper time. Such a place is the nearby high concentration of mass/energy known as the centre of the Earth, so the path of the moving apple is directly downwards towards the centre of the Earth. As it falls, the apple exchanges gravitational potential for kinetic energy of motion with respect to the ground.

A short time later (from the apple’s point of view), the surface of planet Earth collides with the it. Electromagnetic repulsion forces between the molecules of the ground and the molecules of the apple cause a very rapid acceleration – forcing the apple away from its free-falling geodesic path. Electromagnetism (not gravity!) causes a high rate of change of acceleration (from zero to 9.8 metres per second per second – all achieved in a fraction of a second). This third differential – the rate of change of acceleration – is called jerk, and it is this that causes damage, bruising the flesh of the apple as the ground hits it.

The apple now feels the Earth pushing it via electromagnetic repulsion forces, and its inertia (its resistance to acceleration) pushes back, creating what we call the “weight” of the apple on the ground. The apple is now returned to an accelerated frame of reference which is the same as the clock on the ground, so from now on, the apple is stationary on the ground, and its “clock” counts the same number of ticks of the “clock” on the ground.

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