# The “true magic hidden inside General Relativity”

“The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them.”

Sir William Lawrence Bragg

General relativity is conventionally summarized as follows:

There is no gravitational field. Gravity is merely the result of the curvature of spacetime. Mass and energy curve spacetime and as a result particles get attracted to them. Think of a heavy ball on a bed sheet. If we then let a smaller ball roll on the bed sheet it will roll down the depression created by the bigger ball.

This is the standard textbook story. Now the magic:

It also makes sense to turn the conventional “essential lesson” of General Relativity completely upside down!

When Einstein understood this, he referred to this insight as “beyond my wildest expectations.” Carlo Rovelli refers to this change of perspective as “the true magics hidden inside General Relativity“, which is where the title of this post comes from.

Now that you are hopefully interested, let’s discuss how we can achieve this change of perspective and what it means to turn the essential lesson of General Relativity upside down.

But first a short disclaimer: The following discussion and figures are heavily inspired by Section 2.2.5 in Rovelli’s “Quantum Gravity” book. The thing is that it is somewhat buried in a book about an advanced topic, although the whole argument is not hard to understand and should be much more widely known. Even otherwise great textbooks like Tony Zee’s “Einstein Gravity in a Nutshell” don’t discuss it – for whatever reason. This is why I wanted to rewrite the whole thing in less technical terms and promote it here. However, none of the ideas below are mine and all credit goes to Carlo Rovelli.

So now, let’s start. As usual in General Relativity, we need a “Gedankenexperiment” (= thought experiment).

## Problems inside an empty hole

The fundamental equation of General relativity – the Einstein equation –  famously stays the same under all sufficiently smooth transformations (diffeomorphisms). This means we can act with any sufficiently smooth transformation on a given solution of the Einstein equation and get another solution. The conventional name for this symmetry is “general covariance”.

While symmetries are usually great, too much of anything is harmful. The huge symmetry of General Relativity is no exception and leads to a huge problem.

We can understand the problem through the following Gedankenexperiment.

Consider a region of the universe that contains no matter – a hole. Inside this hole, we take a closer look at two specific points $A$ and $B$. At $A$ the gravitational field $e$ is flat, while at $B$ it is not.

Now, since the Einstein equation is generally covariant, we can find a transformation $\phi$ that maps $A$ to $B$ and leaves everything outside the hole unchanged. The gravitational field gets now changed to $\tilde e = \phi^\star e$. Since $A$ is mapped to $B$, the field $\tilde e$ is curved at $A$ and flat at $B$.

Both field configurations $e$ and $\tilde e$ are both physical solutions of the Einstein equation since our transformation $\phi$ is a symmetry.

This is a huge problem.

What is the real physics at the point $A$? Is the gravitational field flat or curved here? The Einstein equation seems to be meaningless since it doesn’t uniquely determine the gravitational field at the spacetime point $A$.

However, we know from experiments that gravity doesn’t behave strangely. We don’t need, for example, a superposition of all possible solutions. Gravitational physics is deterministic.

Now, there are only two ways out of this dilemma.

• Either we need different field equations that aren’t generally covariant and therefore determine the gravitational field at $A$ uniquely.
• Or we need to rethink the notion “spacetime point”.

Einstein actually searched for three years for non-generally covariant field equations. However nowadays, with the power of hindsight, we know that the Einstein equation is correct. Therefore, we need to talk about the second option.

## What is a spacetime point?

We usually accept that there is fixed, invisible background structure that we call spacetime. We can imagine it as some kind of lattice with infinitesimal lattice spacing that permeates everything. While the actual labels that we put on this lattice do not matter, we usually accept that this invisible structure exists.

This is how we ended up with our strange conclusion in the last section that the physics at point $A$ is not uniquely determined.

However is there actually any evidence for the existence of this invisible background structure? Sure, it’s a useful tool – but as argued above it leads to problematic conclusions.

So let’s leave the invisible stuff aside and talk about things that we can actually observe.

The situation described in the last section is rather meaningless. How could we ever notice that the gravitational field is curved or flat if there is nothing except the gravitational field inside the hole?

Therefore, to probe the physics inside the hole let’s send two test particles into the hole. We further assume that these particles meet at $B$.

Now, not only the gravitational field but also the worldlines, $x_a$ and $x_b$, of the two particles are determined by the Einstein equation.

Next, we consider again the map $\phi$ from the last section.

The thing is that not only the gravitational field $e$ gets transformed, but also the worldlines $x_a$ and $x_b$! Again, $e$ becomes $\tilde e$ and in addition, $x_a$ and $x_b$ become $\tilde{x}_a$ and $\tilde{x}_b$. Since $\phi$ is a symmetry these are again physical solutions of the Einstein equation.

With this in mind, we can see how the paradoxical situation outlined in the last section can be resolved.

We previously asked: Is the gravitational field flat or curved at $A$?

Now, we can instead: Is the gravitational field flat or curved at the point where the two particles meet?

As discussed in the last section the first question is not uniquely answered by the Einstein equation. We can find a solution $\tilde e$ where the gravitational field is curved at $A$ and another one $e$ where it is flat at $A$.

In contrast, the second question is uniquely answered by the Einstein equation! The gravitational field is curved at the point where the two particles meet no matter what smooth transformation we apply. This is how the hole problem is solved. We were simply asking the wrong question.

## The fundamental lesson of General Relativity turned upside down

What we learn through the famous “hole Gedankenexperiment” is that it makes no sense to ask questions about spacetime points. Instead, we must ask questions about locations that are determined dynamically through elements in the theory, like the two particles in the example above.

In this sense, there is no invisible background structure like we usually assume it. Instead, spacetime emerges dynamically. I like the following analogy by Rovelli

“Objects are not immersed in space. Objects make up space. Like a marriage. It’s not a man and a women feel up marriage. They are the marriage. There is no marriage without a man and a women (or a man and a man or a women and a women. Whatever.). … Space is nothing that remains if you take away all the things. Instead it’s made up by things.  ”

To quote Einstein:

“All our space-time verifications invariably amount to a determination of space-time coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meeting of two or more of these points. Moreover, the results of our measurements are nothing but verifications of such meetings of the material points of our measuring instruments with other material points, coincidences between the hands of a clock and points on the clock-dial, and observed point-events happening at the same place at the same time. The introduction of a system of reference serves no other purpose than to facilitate the description of the totality of such coincidences. “

Rovelli concludes:

“The two solutions $(e,x_a,x_b)$ and $(\tilde e,\tilde {x}_a, \tilde{x}_b)$ are only distinguished by their localization on the manifold. They are different in the sense that they ascribe different properties to manifold points. However, if we demand that localization is defined only with respect to the fields and particles themselves, then there is nothing that distinguishes the two solutions physically. […] It follows that localization on the manifold has no physical meaning. […] Reality is not made by particles and field on a spacetime: it is made by particles and fields (including the gravitational field), that can only be localized with respect to one another. No more fields on spacetime: just field on fields.”

While the traditional summary of General Relativity states that gravity is merely an “illusion” caused by the curvature of spacetime, the line of thought outlined above suggests instead that there is no spacetime at all!

What we perceive as spacetime emerges dynamically through interactions like gravity. In this sense, we have turned the essential lesson of General Relativity completely upside down!

There is  a nice paragraph quite at the end of Rovelli’s “Quantum Gravity” book that summarizes everything said here much better than I ever could:

Einstein’s major discovery is that spacetime and the gravitational field are the same object. A common reading of this discovery is that there is no gravitational field: just a dynamical spacetime. In view of quantum theory, it is more illuminating and more useful to say that there is no spacetime, just the gravitational field. From this point of view, the gravitational field is very much a field like any other field. Einstein’s discovery is that the fictitious background spacetime introduced by Newton does not exist. Physical fields and the relations are the only components of reality.

( A short comment: the notion “dynamical spacetime” can be confusing. Spacetime is a static thing since time is already included. Hence, there is no wobbling spacetime etc. Spacetime is a fixed thing that doesn’t change.)

• Further information along these lines, can be found in the last chapter in the book Some Elementary Gauge Theory Concepts by Sheung Tsun Tsou and Hong-Mo Chan where they talk about the idea of a “pointless theory”.
• For some more philosophical thoughts, see Interpreting theories without a spacetime by De Haro and de Regt.

# How to Invent General Relativity

How exactly did Einstein come up with his theory of general relativity? Although I’ve read several books on general relativity and felt confident to say that I understand the fundamentals, I only recently understood where exactly it came from.

So here’s a hopefully coherent story of how we can invent Einstein’s theory from scratch.

## Why should accelerating frames be special?

The first thing we need to wonder about is the notion “acceleration”. While no one doubts that there is no difference between frames of reference that move with constant speed relative to each other, accelerating frames are special. In a soundproof, perfectly smooth moving train without windows, there is no way to tell if the train moves at all. However, we notice immediately when the train accelerates. A glass of water is indistinguishable in a perfectly smooth moving train from a glass of water in a standing train. However if the train accelerates rapidly, the water spills over.

The equivalence of frames that move with constant velocity relative to each other is known as Galilean relativity and was successfully extended by Einstein to his theory of special relativity. The additional thing that special relativity takes into account is that the speed of light has the same value for all observers.

Now, Einstein was dissatisfied by the special role of accelerating frames. Why should they be special? What makes them special? Is there any way to put them on an equal footing to all other frames?

Galilean relativity and Einstein’s relativity put all frames that move with constant velocity relative to each other on an equal footing. That’s a big unification. However, the unification is not complete as long as accelerating frames are special.

To understand what defines acceleration after all and what makes it special, we need to consider some extreme situations.

For example, let’s imagine an infinite, completely empty universe with just one observer somewhere inside. Is he stationary? What if he starts spinning? Would he feel dizzy? How could he tell that he spins at all? What if the universe starts spinning around him? Would he feel dizzy? is there any way to distinguish these scenarios?

There are only two answers. Either there is an absolute frame of reference, let’s call it “God’s frame”, or there is no way to distinguish these situations. While many people preferred for a long time the first possibility, the second one is Einstein’s perspective. He was a huge fan of the philosopher Ernst Mach, who argued using many thought experiments that the idea of absolute motion makes no sense.

In special relativity, what one observer calls a moving object, is an object at rest for another observer. There is no way to make an absolute statement of the form “this object is moving!”. In contrast, we can always state absolutely “this object is accelerating”.

However, Einstein’s hope was that it would be equally possible to make acceleration relative somehow. What one observer would call an accelerating object, another would call an object at rest. If this would be possible somehow, the laws of nature would be exactly the same for accelerating observers and observers at rest.

As already mentioned above, there are many good reasons to believe that this is simply not possible. Just take the next train and you’ll see how different accelerating frames are! Einstein was well aware of these obstacles, but he was obsessed by the idea that no frame should be special.

To move forward, we need to find out what makes accelerated frames special. Einstein, as any student of physics, learned in his mechanics lectures that in accelerating frames we need to take care of additional forces, called “fictitious forces”. These additional forces take care of the anomalous movement of objects in accelerating frames. A special feature of fictitious forces is that they are proportional to the mass of the object in question. While thousands of students learned these things every year, no one paid special attention to them. This is no wonder. Calculations in accelerated frames are extremely complicated and you can always simply choose another observer that isn’t accelerating.

However, Einstein was obsessed with his idea that no frame should be special and somehow at the right moment, these distant memories of what he learned in his mechanics’ course started popping up.

Another thing every student of physics learns is Newton’s formula $m M /d^2$ that describes the gravitational force between two objects of mass $m$ and $M$ that are separated by a distance $d$. This law is completely analogous to Coulomb’s law $qQ/d^2$ that describes the electric force between two charged objects. The only difference is that gravity is always attractive, while the electric force can also be repelling. (Like charges repel each other, while unlike charges attract each other)

Remembering these two facts about accelerating frames and gravity, Einstein’s thought process could have been as follows:

“Fictitious forces are proportional to the mass of the object in question … The gravitational force is equally proportional to the mass of the object in question … Gravity reminds me of a fictitious force… Maybe gravity is a fictitious force!”

To explore this possibility, let’s imagine a spacetime somewhere in the universe far away from anything else. Usually, the astronauts in a spaceship float around of the spaceship aren’t accelerating. There is no way to call one of the walls the floor and another one the ceiling.

Now, what happens if another spaceship starts pulling the original spaceship? Immediately there is an “up” and a “down”. The passengers of the spaceship get pushed towards one of the walls. This wall suddenly becomes the floor of the spaceship. If one of the passengers drop an apple it falls to the floor.

For an outside observer, this isn’t surprising. Through the pulling of the second spaceship, the floor is moving towards the floating pencil. This leads to the illusion for the passengers inside the original spaceship that the pencil falls to the floor.

If there is no window in the original spaceship, there is no possibility for the astronauts to tell if they are sitting still on some planet or if they are accelerating. If their spaceship sits on a planet the pencil and the passengers themselves would be equally pulled to the floor, however in this case through gravity.

Even if we try to exploit some special possibility of gravity, there is no way to distinguish these situations. For example, a bowling ball and a pencil that is released from the same height would hit the floor at the same moment. This is what Galileo demonstrated for gravity, by throwing things down the Tower of Pisa. For an observer outside of the original spacetime, this fact would be by no means mysterious. The floor simply moves constantly towards the floating bowling ball and pencil. Hence, the floor touches the pencil and the bowling ball at exactly the same moment.

An outside observer would call the force that pushes thing to the floor in the original spacetime a fictitious force. It is merely a result of the floor moving towards the floating objects. However, for the passengers inside the spaceship, the force would be very real. They experience a real floor and a real ceiling and things really fall down if you let them drop. Without getting an outside view, it would be impossible for them to distinguish this fictitious force caused by the acceleration of their spaceship, from the force we call gravity. They can’t distinguish between acceleration and sitting at rest in an appropriate gravitational field.

This is crazy. Consider where we started. We were certain that accelerating frames are easily distinguishable. But now we find ourselves in a situation where we can’t find any difference between an accelerating frame and a stationary frame in a gravitational field.

Of course, the situation is only indistinguishable if the acceleration has a precise value that mimics the effect of the gravitational field. If you want to mimic the earth’s gravitational field, you need to accelerate quicker, than if you want to mimic the weaker gravitational field of the moon.

By remembering a simple fact about fictitious forces, Einstein was able to expand his principle of relativity even further. Now, even accelerating frames aren’t that special anymore.

This observation that accelerated frames are indistinguishable from a resting frame immersed in a gravitational field is called, well, the principle of equivalence.

## The Principle of Equivalence

To bring this point really home, let’s imagine another situation. Instead of a spaceship somewhere in the middle of nowhere, let’s consider a spaceship floating 100 kilometers above the earth. The spaceship is pulled down by the earth’s gravitational field, but for the moment let’s imagine the spaceship is stationary. In this situation, the astronauts in the spaceship are able to distinguish “up” and “down” without problems. A pencil falls down, thanks to the earth’s gravitational field.

Then suddenly the spaceship is released from whatever holds it still 100 kilometers above the earth. What happens now? Of course, the spaceship starts falling down, i.e. moves towards the earth. At the same time the notions “up” and “down” start losing their meaning for the astronauts inside the spaceship. Everything inside the spaceship falls down towards the earth with exactly the same speed. This property of gravity was demonstrated by Galileo through his famous experiments at the Tower of Pisa. Thus, everything inside the spaceship starts floating. They experience zero gravity. For them, without the possibility to look outside of their spaceship, there is no gravitational field and nothing is falling down.

Therefore, gravity is nothing absolute. While for some observers there is a gravitational field, for the free-falling observers inside the spaceship there is none. If we want, we can therefore always consider some frame where there is no gravity at all! The gravitational force vanishes completely inside the free-falling spaceship. In contrast, an observer standing on earth would describe the spaceship by taking the earth’s gravitational field into account. To such an observer everything falls down because of this gravitational field. However, for the astronauts inside the spaceship, nothing is falling.

This situation is exactly the reversed situation to our first imaginary scenario. There we considered a spaceship somewhere in the middle of nowhere, where there was no gravitational field. Then the spaceship got pulled by another spaceship and suddenly the situation inside the original spaceship was as if they were immersed in a gravitational field. In our second imaginary situation, we started with a spaceship immersed in a gravitational field. However, all effects of this gravitational field vanish immediately when the spaceship starts falling freely towards the earth. Gravity has no absolute meaning. For some frames of reference there is gravity, for others, there isn’t.

## The final Punchline

So far, we only considered linear acceleration. In both examples above the spaceship moved in a fixed direction with varying speed. However, not only when the magnitude of speed changes we have acceleration but also when an object changes direction. Another kind of accelerated frame is rotating frames.

The simplest kind of system we can consider is a disk that spins with a fixed number of revolutions per second. Each point on the disk undergoes a change in direction at every instant and is, therefore, accelerating all the time.

To understand a spinning disk we need to remember one of the most curious properties of special relativity: length contraction. The length of an object is smaller for some observer who moves with some velocity relative to the object than the length measured by some observer for whom the object is at rest.

Each point on our spinning disk moves round and round, but not in and out. Thus, according to special relativity, there is length contraction along the circumference, but none along the radius. Thus, when we measure the circumference of a spinning disk, we measure a different value than an observer who sits on the spinning disk. For the observer sitting on the spinning disk, the disk is at rest and hence no length contraction happens. However, we agree with this observer on the diameter of the disk, since even for us there is no radial movement of the disk.

The formula everyone learns in school for the relation of the radius $r$ to the circumference $C$ of a circle is:

$$C = 2 \pi r .$$

Thus, the ratio of the circumference and radius that the observer on the disk, for whom the disk is at rest measures is $C/r = 2 \pi$. For us outside observers, the disk spins and therefore there is length contraction along the circumference. Therefore, what we measure is not the same: $C/r \neq 2 \pi$! This crazy result was known for some time as “Ehrenfest’s paradox”.

Once more Einstein understood what was going on here by remembering something he learned in a seminar. In a seminar about the geometry of two-dimensional curved surfaces, he learned that in non-Euclidean geometry the ratio $C/r$ needs not necessarily be $2 \pi$. Depending on the surface we are considering the ratio can be any value.

The simplest example to understand this is a sphere. Let’s consider the ratio $C/r$ for the equator of a sphere, say the earth. The circumference is some number $C= C_E$. The radius of this circle is the distance from any point on the equator all the way up to the north pole. For a perfect sphere, the length of such a line is exactly one-half equator: $r= C_E /2$. Therefore, the ratio $C/r$ for a circle on a sphere is not $2 \pi$, but $C/r= C_E /(C_E / 2) = 2$!

Einstein remembered this property of curved surfaces and connected it with the seeming paradox situation concerning the $C/r$ ratio of a spinning disk. The mathematics of how to describe things on curved surfaces was at this time already developed by Gauss and Riemann. Einstein’s idea was to borrow these mathematical tools to describe what is going on in accelerating frames. Thanks to his equivalence principle accelerating frames are equivalent to resting frames in a gravitational field. Hence he noted that he could use the mathematics of Gauss and Riemann to describe gravity.

## The Einstein Equation

Now, all we have to do is write down the correct equation that describes the idea “gravity = curved spacetime” mathematically. Einstein needed 6 years to discover the correct formula, but nowadays with the power of hindsight, the derivation is relatively straight-forward.

What causes gravity is mass, and thanks to Einstein’s famous formula $E = m c^2$, we know that equally energy causes gravity. Thus, at one side of our equation, we must have something describe the “charge” of gravity, i.e. energy. In mathematical terms, the “charge” of gravity is described by the energy-momentum tensor $T_{\mu \nu}$.

As a first step towards our equation, we must now remember one of the most important laws of physics: the conservation of energy and momentum. In mathematical terms this conservation law is expressed as

$$\partial^\mu T_{\mu \nu} = 0.$$

Next, we need something to describe curvature mathematically. This is what makes general relativity computationally very demanding. The most important object in this context is the metric. Metrics are mathematical objects that enable us to compute the distance between two points. In a curved space, the distance between two points is different than in a flat space as illustrated the following figure: (Geodesic)

Therefore metrics will play a very important role when thinking about curvature in mathematical terms.

Having talked about this, we are ready to “derive” the Einstein equation. It turns out that there is exactly one mathematical object that we can put on the left-hand side: the Einstein tensor $G_{\mu \nu}$. The Einstein tensor is the only divergence-free ($\partial^\mu G_{\mu \nu}=0$) function of the metric $g_{\mu\nu}$ and at most its first and second partial derivative. Therefore, the Einstein tensor may be very complicated, but it’s the only object we are allowed to write on the left-hand side describing curvature. This follows, because we can conclude from
$$T_{\mu \nu} = C G_{\mu \nu} \quad \text{ that } \quad \partial^\mu T_{\mu \nu} = 0 \rightarrow \partial^\mu G_{\mu \nu} = 0$$
must hold, too. The Einstein tensor is a second rank tensor and has exactly this property. Second rank tensor means two indices $\mu \nu$, which is a necessary requirement, because $T_{\mu \nu}$ on the right-hand side has two indices, too.

The Einstein tensor is defined as a sum of the Ricci Tensor $R_{\mu\nu}$ and the trace of the Ricci tensor, called Ricci scalar $R =R_{\nu}^\nu$
$$G_{\mu \nu} = R_{\mu\nu}-\frac{1}{2}Rg_{\mu \nu}$$
where the Ricci Tensor $R_{\mu\nu}$ is defined in terms of the Christoffel symbols $\Gamma^\mu_{\nu \rho}$

R_{\alpha\beta} = \partial_{\rho}{\Gamma^\rho_{\beta\alpha}} – \partial_{\beta}\Gamma^\rho_{\rho\alpha} + \Gamma^\rho_{\rho\lambda} \Gamma^\lambda_{\beta\alpha} – \Gamma^\rho_{\beta\lambda}\Gamma^\lambda_{\rho\alpha}
and the Christoffel Symbols are defined in terms of the metric

\Gamma_{\alpha \beta \rho} =\frac12 \left(\frac{\partial g_{\alpha \beta}}{\partial x^\rho} + \frac{\partial g_{\alpha \rho}}{\partial x^\beta} – \frac{\partial g_{\beta \rho}}{\partial x^\alpha} \right) = \frac12\, \left(\partial_{\rho}g_{\alpha \beta} + \partial_{\beta}g_{\alpha \rho} – \partial_{\alpha}g_{\beta \rho}\right).
This can be quite intimidating and shows why computations in general relativity very often need massive computational efforts.

Next, we need to know how things react to such a curved spacetime. What’s the path of an object from A to B in curved spacetime? The first guess is the correct one: An object follows the shortest path between two points in curved spacetime. We can start with a given distribution of energy and mass, which means some $T_{\mu \nu}$, compute the metric or Christoffel symbols with the Einstein equation and then get the trajectory through the \textbf{geodesic equation}
\label{eq:geodesics}
\frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu }\frac{dx^\mu }{dt}\frac{dx^\nu }{dt} = 0.

The geodesic is the locally shortest curve between two points on a manifold. (This is a bit oversimplified, but the correct definition needs some terms from differential geometry we haven’t introduced here.)

That’s it.

My plan is to write next about why this idea “gravity = curvature” is not the really the essence of general relativity. This somewhat controversial statement is something that even Einstein himself only understood after several decades. However, this post is already incredibly long and thus I’ll write another post about this.

To finish this post, here are some recommendations where to read more about Einstein’s theory:

My favorite GR textbooks are

# One Electron and The Egg

In the preface to my book I wrote: “To me, the most beautiful thing in physics is when something incomprehensible, suddenly becomes comprehensible, because of a deep explanation.”

Here is one such example, although most experts would argue not a good one. More about that later.

We know there are electrons. Many, many electrons. They all have exactly the same quantum numbers, such as electric charge. In fact, nowadays phycisist go as far as saying they are all indistinguishable. This means if we perform an experiment with many electrons it’s impossible to say which electron is which at the end of the experiment.

Another thing we know is that there is to every elementary particle an antiparticle with exactly the opposite quantum numbers. The antiparticle of the electron is the positron.

These are facts and the standard model of modern physics works perfectly if we put these things into the theory by hand. Unfortunately this does not answer the question why this is the case? Why are there antiparticles? Why are all electrons exactly the same? The same holds true for every other fundamental particle. For example, all up-quarks are exactly the same and there exists an anti-up-quark. (Unfortunately there isn’t a better name.)

There is a beautiful answer to both questions that was popularized by nobel price laureate Richard Feynman. In his Nobel lecture he told the story that Princeton proffessor John Wheeler called him and said :  “Feynman, I know why all electrons have the same charge and the same mass” “Why?” “Because, they are all the same electron!”

What? To understand why a sane person would suggest something like this we need some background.

The standard model describes interactions of fundamental particles with quantum field theory. The result of computations in quantum field theory are the same if we replace positrons with electrons that move backwards in time.

This is just a mathematical observation and, of course, it’s up to you which one is you favorite:

• two particles with opposite charge, the positron and the electron, or
• just one particle, the electron, going forward and backward in time.

To give some credibility to the second possibility let me tell you that it’s called Feynman–Stueckelberg interpretation and quote another Nobel laureate, Yoichiro Nambu. He applied this idea to all production and annihilation of particle-antiparticle pairs. In “The Use of the Proper Time in Quantum Electrodynamics” he wrote: “the eventual creation and annihilation of pairs that may occur now and then is no creation or annihilation, but only a change of direction of moving particles, from past to future, or from future to past.”

So… do we agree that the second choice is not complete non-sense?

Electron-positron annihilation by Joel Holdsworth (Joelholdsworth) [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/) or CC BY 2.5 (http://creativecommons.org/licenses/by/2.5)], via Wikimedia Commons

Now we can understand John Wheeler’s answer to the two questions at the beginning of this post. An electron moving forward and backward in time is enough for all interactions that involve electrons. For example, an electron can move backwards in time and come then back to the same point in time several times. Therfore all interacting electrons at one point in time may be one and the same electron. The famous electron-positron annihilation process is then nothing other than an electron that moves forward and then vanishes by moving backwards in time. The backwards moving electron is then what we call a positron.

To summarize: All electrons are exactly the same, because they are all one and the same electron. This is possible, because electrons are able to move backwards in time. An electron moving backwards in time is a positron.

Now for some “Woah Dude” read “The Egg” by Andy Weir and then come back and read the epilog. I’ll wait.

#### Epilog

So, where’s the catch?  The one-electron theory is experimentally disfavoured. It would mean that there is exactly the same number of electrons and positrons in the universe. All observations show that there are almost only particles in the universe.

If particles and antiparticles meet they annihilate, which results, for example, in photons. Thus if anywhere in the unvierse large amounts of antiparticles where present, we would see them. They would annihilate with ordinary particles and result in a distinct signal: large amounts of photons with energy equal to two times the particle mass, because of energy conservation. We don’t see such a singal anywhere and thus there is no reason to believe that there are antimatter galaxies somewhere in the universe.

Maybe there is a catch to the catch. It turns out that we don’t know if antiparticles have the same massas particles or if they differ by a sign. In other words: we don’t know if antiparticles have negative mass.

For normal particles the gravitational force is attractive, although the masses are the same. This is what causes the earth to revolve around the sun. The mass of the earth and the mass of the sun are positive and the gravitational force between them is attractive. This feature of gravity is in contrast to the other forces. For example, the electric force is attractive for opposite charges (+ and -) and repulsive for equal charges (+ and +).

Thus if antiparticles have negative mass the gravitational force between them and particles is repulsive. This could be the reason that particle and antiparticle galaxies never collide. (On cosmological scales the gravitational force is dominant, because galaxies consist of atoms that are mostly neutral.)

Negative mass for antiparticles is not a crackpot theory. There are arguments that if the Feynman-Stueckelberg interpretation is correct, antiparticles should have negative mass.

Threre are several experiments at CERN that test this right now. They use an old collider to produce stable, neutral, atoms that consist only of antiparticles. Then they check if they fly up or down. Think about Newton and his apple. He observed that it falls down on earth because of gravity. An apple with negative mass would fly up instead.

Such an experiment is not possible for a single antiparticle like a positron. The other forces are much stronger than the gravitational force and the effect of a negative mass is not visible for charged particles. This is the reason that we still don’t know if antiparticles have negative mass, although colliders like the LHC produce billions of them everyday.

The standard point of view is that the masses of particles and antiparticles are the same. This is the result if you apply the mathematical charge conjugation operator on the mathematical object that we use to describe a particle. This computation yields the mathematical object that we use to describe antiparticles and all quantum numbers get flipped, except for the mass.

Still, despite the Higgs mechanism, we don’t understand mass. We have no quantum theory of gravity. The standard model is not the end of the story. At a first glance it seems strange that all quantum numbers get flipped, except for mass – but who knows? In the end the experiments have to decide and I’m looking forward to the results of these experiments.

A last comment: the matter-antimatter asymmetry is one of the biggest unsolved problems of modern physics. It’s not possible to explain with the standard model why there is only matter in the universe and not equal amounts of antimatter.