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

Introduction

Einstein’s Theory of Relativity is a very useful model. It fits very well with a vast amount of observational data – such that any discrepancies are only found way down in the decimal places of experimental measurements.

There are currently only two areas in which the model is sometimes not a good “fit”. The first is the realm of the very small – where Quantum Theory prevails. Here the scientists have yet to find a way to make the two theories thoroughly compatible. The second relates to the very massive – such as black holes. In the interior of black holes, relativity theory predicts a mathematical “singularity” – a nonsensical result. Notwithstanding the fact that scientists can never observe the interior of a black hole, they still don’t feel comfortable with such singularities. Any scientist who manages to create a model which is a good fit even in these exotic realms is a surefire bet for a Nobel Prize…

But the most important thing to take from this preamble is that science is all about fitting data to models – it is about observations and measurements. These are the fundamentals. Even though Einstein himself referred to “Clocks and Rods”, what he really meant to imply was that it is the measurements we make with such tools that are the most important things to consider.

“Is the speed of light constant?”

This question requires clarification before it can be answered properly. Whilst “the speed of light” is a phrase in common usage in science-fiction movies, it is actually not a well defined notion in real science.

The question could be taken to be mean the speed of light in any circumstance – in which case the answer is definitely “no“. Light propagates in the form of a wave, and it does so at different speeds passing through different materials. Also, the speed of light through a material medium will be measured differently by observers who are moving relative to the medium.

You may have read some misleading statements about the reason why light propagates more slowly through a physical medium such as water or glass – for example:

• “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.”

I (and many others) find such explanations problematic. A better explanation might be:

“Light passing from a vacuum into and through a medium (for example 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 slowing of light through a medium was first demonstrated by Hippolyte Fizeau in 1851, (although he misinterpreted his results in terms of the prevailing theories of his time).

A modern demonstration of the variable speed of light in a material medium is the fact that a charged particle (for example, an electron) can travel through water faster than light-waves propagate through the same water. This is the origin of the observed blue glow effect called Cherenkov Radiation that is seen by people working with nuclear reactors.

However, we could alternatively assume that the question really being asked was “Is the speed at which light propagates through a vacuum constant?” – in which case the answer is yes – however, this was a fact well known long before Albert Einstein’s time, and – in and of itself – is not an axiom (critical assumption) of Special Relativity.

Understanding just why the speed of light in a vacuum must be a constant offers 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 empty space in a wave-like manner – as an electromagnetic field. His equations showed that in a vacuum, the unique and very particular velocity of that self-propagating wave depended upon only two things:

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

These two things are simply universal quantities that can be measured. 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 might be different – but we cannot investigate other universes in order to find out.

When he did the mathematics, Maxwell obtained a value for that all-important unique velocity – 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.

So, the constant speed of light through a vacuum is of historical interest, but should not be confused with the very different (and very important) axiom of Special Relativity; namely the assumption that the speed of light through a vacuum is invariant – which is to say, measured to be the same by all observers – regardless of the observer’s constant motion relative to the light source being measured.

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”, now defined to be exactly 299,792,458 metres per second).

That such a speed is also invariant is also 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.

“Is it true that moving fast through space causes your clock to slow down?”.

No, that is not true. The Special Theory of Relativity says that any clock that is alongside you (“co-moving” with you, or in your inertial frame) will always tick away the seconds steadily in the same way it has always done. Such a clock is said to measure your “proper time“.

The striking consequences of constant motion being relative are not about your movement or your clocks – rather, motion has a major effect on how you measure other things happening – things that are not alongside you in your inertial frame.

Also, please ignore anyone who who says “Even if you are standing still, you are still moving through time!” You are not moving 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 source of the light.

If you think about it, 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 also be invariant.

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

Figure 2 (courtesy of Physics Stack Exchange) represents a device called a light-clock that counts a “tick” each time a light pulse bounces off the topmost mirror of two mirrored surfaces mounted facing one another. The image represents two such clocks, one non-moving, and one moving relative to the other :

Watch the little wiggly arrows. These light pulses are moving at exactly the same speed, but the clock moving to the right counts just one bounce (“tick”) off the top mirror in the same time that the stationary clock at the left counts two bounces.

So, invariance of the speed of light in a vacuum has the consequence that if you can 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. It is important to always remember that what you measure is real.

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 (or “rod”) 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 they expected between the departure event and the arrival event, then they can only conclude that the distance to their destination was shorter than expected by the observer who did not travel. 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. This is the essence of the Special Theory of Relativity.

All this is 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.

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 very fast 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 surprising consequences of relative motion.

Summary

Having made 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

then 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.

“But what about the so-called Twins Paradox?”

If we agree that that constant motion is relative, and that there is no privileged state of motion (inertial frame), then two observers in constant relative motion both see the other observer’s clock as counting fewer ticks. An obvious question to ask is based on identical twins. The Twins Paradox is usually stated something like this:

“Imagine two twins, one of whom takes a space voyage at relativistic speeds and returns home to find that the twin who remained on Earth has aged more. At first, this result appears puzzling; because according to Special Relativity, each twin sees the other twin as moving, and so each should – paradoxically – find the other to have aged less.”

The paradox is resolved when we realize that any two clocks can only be checked for synchronisation by bringing them back together at the same place. (Aside: As we will see later – strictly speaking – bringing the physical clocks to the same place for comparison is not essential. It is only necessary for the information recorded by different observer’s clocks to be brought to a place where the data can be compared.)

But for now, if we are considering the reunion of the twins after one of them took a journey and returned, we observe that the experiences that the two twins had are not equivalent. One twin (G) remained on Earth and stayed in the same inertial frame for the whole time. The second twin (R) spent time in four different inertial frames:

1. Initially standing next to their twin on Earth
2. Traveling outward.
3. Traveling inward.
4. Standing back on Earth

The travelling twin is younger than the stay-at-home twin because the traveling twin changed inertial frames.

Now, some scientists (including Einstein!) initially tried to resolve the paradox by invoking the effects of acceleration/gravity. They argued that the travelling twin in the example accelerates (4 times), and the Earth-bound twin does not. To this day, you may still read this explanation in books or articles popularizing science. However, it is fairly easy to show that the time dilation would occur even if no acceleration is involved – as follows:

The Triplets Paradox:

This thought experiment is set up in deep space, far away from any effects of gravity. We imagine that we have established a set of five “space beacons” laid out in a line with equal spacing between them.

Three observers (triplets) then take up their initial positions. Gail will remain stationary throughout, positioned at one of the beacons (second from right). She has two identical atomic clocks. Her siblings Rachel and Betty each have a rocket-ship that is also fitted with the same type of atomic clock. Rachel and Betty travel outwards beyond the beacon line, and come to a standstill with respect to Gail:

Gail has two clocks. One of the clocks did not travel out of Gail’s inertial frame. The other clock shows data recorded by the clocks that traveled on the two rocket ships. Gail finds that her two clocks are out of syncronisation. Between the start and the end of the experiment, the traveling clocks recorded fewer ticks than Gail’s own clock – but importantly: at no time during the experiment did any of the observers undergo acceleration.

Just to put anyone’s mind at rest concerning “Ah! But you have not factored in the travel time of the laser signals!“. Let us assume that in the experiment setup stages, the rockets take around 4 days accelerating at 1 gravity to reach around 1% of the speed of light. Let us also assume that the rockets closest approach during signalling is 100 km (Trust me, you don’t want to be closer than that to something travelling at 3,000 km/s !! ) If the travel time between beacons is approximately 10 days, then the experiment lasts 20 days. The travel time for signalling one-way is around 300 millionths of a second. Any errors introduced by signaling are trivial compared to the discrepancy between Gail’s two clocks due to Special Relativity time dilation – which is of the order of 2 minutes. Gail knows that her triplet siblings are now approximately 2 minutes younger than herself.

“Is time the 4th Dimension?”

The red line therefore represents a “history” of the price of gold changing over time. You might wish to call such a 2-D space “Goldtime”.

The goldtime 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 moves.

Mathematicians also use imaginary spaces to help them understand how physical objects in the real world move. The “spacetime” of Relativity Theory is one example of such a thing. By using 3 axes representing space, and 1 axis representing time, we can imagine a 4-dimensional “chart” (even if we can’t easily visualize it) in which each point represents a single event happening at a place and at a time. In this way, a line of connected points in such a chart could represent the history or “world-line” of an object moving. This not dissimilar to a flat map of the world on which someone has drawn a line to show the route taken by a ship travelling from Southampton to New York. Even though the geometry of the map is much distorted (the real world is not flat!), nevertheless, the line as a representation of the journey can be a useful construct to work with, and helps us to understand how the real ship moved.

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” (i.e. the map) 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 “Goldtime” 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 construct suggested by spacetime is somehow more real than the actual universe we experience, and in which we measure the distances between events happening at certain places, and record the elapsed time between those events.

When understanding gravity (see later) 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.

“Does time slow down near massive objects?”

Yes. Clocks are affected by proximity to energy density, and mass is just energy that is confined to one place.

When Einstein studied accelerating systems (the 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.

But the rocket-ship’s motion is not constant – it is speeding up – so the second “leg” of the tick is disproportionately shorter than the first leg is longer.

Overall, the light-clock on an accelerating rocket counts fewer ticks than a stationary or constantly moving clock.

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

• 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 rods (distance), but the rod 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.

“Is gravity caused by curved spacetime?”

The (unfortunately) common phrase “the curved fabric of spacetime” confuses many people into believing that spacetime is something real – something more than an imagined 4D space. Spacetime is not a thing. You cannot take a lump of spacetime and stick pins in it, then measure the distance between pins. Spacetime cannot curve. It is the geometry of spacetime (the map!) 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 an imagined spacetime. We can get some insight into this by dispensing with two of the space axes and showing a chart of just elevation and time. Below is “2-D” diagram (again courtesy of Physics Stack Exchange) :

These diagrams are very simplistic. The correct mathematics in the imagined 4D 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.

“Is the force due to gravity gravity real?”

However, it is very important to remember that the above demonstration is about forces, and we cannot – we must not – on that basis dismiss gravity itself! Gravity is very real, there exists a gravitational field which is measurable at every point in space, and which can transfer energy from one place to another. Scientists/astronomers have detected gravitational waves originating from cataclysmic collisions of very far away super-dense objects. If the gravitational field were not real, such waves could not be observed.

“So what ultimately is the source 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/energy – all those clocks are synchronous. The object’s “world-line” on 4-D spacetime chart is aligned entirely in geometrical time axis.

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 very slightly as a consequence. (Imagine a cart with its left wheel binding and dragging against the ground – the cart develops a tendency to veer to the left.)

Since the object was (in the imagined 4D geometrical chart) previously aligned exactly with the time axis, then any skew will divert its history (world-line) into the space axis of the chart. Just like our cart, the object veers towards the source of the “dragging” – which is the planet.

In the geometry of the imagined 4-D spacetime of General Relativity, a world-line that is “straight” 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 – for example:

• the downward straight line of an apple falling from a tree.
• the ballistic curve of a cannon ball in the air.
• the elliptical 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.

If you can forgive a rather fanciful metaphor, then one might say that in the presence of gravity, all objects display a sort of “fundamental laziness“. Every object moving in the presence of gravity:

1. Wants to move towards the nearest place where its clocks will count the fewest ticks (where it will age slowest), and
2. Always chooses a route along which its clocks will count the fewest ticks for the journey.

“Does gravity really travel in the form of waves?”

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 in a specific situation.

However, disturbances or changes in the gravitational field do indeed 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?

While 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.