Einstein’s light postulate saves Galileo

The amount of electromagnetic force on a charged particle moving in a magnetic field depends on the velocity of the particle with respect to the field.

Good evening (local time)! I’d like to share with you the answer I gave to one of my students earlier today. He asked how Einstein’s postulate of a constant light speed (independent from an observer’s own state of motion) can save Galilean relativity.

In Galilean relativity, the laws of physics are the same for all observers that have a constant speed relative to each other (read this post to be reminded of how that works). The key here is constant motion; as soon as the observers start accelerating, inertial forces start playing a role – and Galilean relativity no longer holds.

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De tweeling/postbode paradox

[Click here to read the English version of this blog post]

De tweeling paradox van Einstein staat minder ver van ons af dan we denken. Waarom is hij dan zo duizelingwekkend? Ik heb mijn studenten enige extra begeleiding beloofd, vandaar dit bericht op mijn blog.

De lichtklok van Einstein

Om na te gaan wat het zou betekenen als de lichtsnelheid voor iedereen hetzelfde zou zijn, bedacht Einstein zijn gedachtenexperiment over de lichtklok (zie figuur).

Het experiment, waarin een foton op-en-neer beweegt tussen twee parallelle spiegels, laat zien dat een bewegende klok langzamer moet lopen dan een stilstaande als het inderdaad zo is dat de lichtsnelheid voor alle waarnemers hetzelfde is (volgens Pythagoras is de afgelegde afstand immers groter).

Maar dat geldt niet alleen voor de lichtklok – íeder fysisch proces zal vertragen. Dit is een werkelijk effect: een persoon die met een bepaalde snelheid beweegt, veroudert langzamer dan iemand die niet in beweging is.

Tweeling paradox

Hieruit volgt de beroemde tweelingen paradox: stel dat er een tweeling is hier op aarde, en een van de twee maakt een ruimtereis met een snelheid die de snelheid van het licht benadert. De theorie van Einstein vertelt ons dat de achtergebleven persoon ouder is geworden dan de persoon die de ruimtereis heeft gemaakt. Voor de achterblijver is er dus meer tijd verstreken. Dit ervaren wij als een paradox omdat het gedachte experiment van Einstein symmetrisch is. Dat klinkt abstract, maar het is eigenlijk een eenvoudig idee: vanuit het perspectief van de bewegende klok is het juist ónze klok die langzamer lijkt te gaan. Als er deze symmetrie is, waarom is er dan een asymmetrie in het verouderingsproces? Als beide tweelingzussen elkaars klok zien vertragen, waarom is de een dan uiteindelijk meer verouderd dan de ander?

Een paradox is een schijnbare tegenstelling

Dat is nu juist waar het om gaat: er is een verschil in de verstreken tijd, terwijl beide tweelingzussen elkaars klok zien vertragen. We moeten ons realiseren dat dit een paradox is – een schijnbare tegenstelling – en geen echte tegenstelling. Het is volgens de speciale relativiteitstheorie nu eenmaal zo dat wanneer we sneller door de ruimte bewegen, we langzamer gaan door de tijd.

De postbode paradox

In de wereld om ons heen is er een vergelijkbaar effect, maar daar zijn we al zó aan gewend geraakt dat we het geen paradox noemen. Dit effect gaat niet over de snelheid van de tijd, maar over een snelheid waarmee we beter bekend zijn: de snelheid van beweging door de ruimte.
Stel dat twee personen (bijvoorbeeld postbodes) van A naar B lopen.

De eerste loopt in een rechte lijn, de zwarte lijn in de figuur hiernaast, terwijl de andere de rode lijn volgt. Net als in de paradox van Einstein is er hier een symmetrie in perspectieven, want beide personen zien de ander van zich af bewegen. Beide zouden ervan overtuigd kunnen zijn dat het de ander is die beweegt. Desondanks is er een verschil in de afstand die wordt afgelegd. Dit verschil in afstand kunnen wij gemakkelijk verklaren door in te zien dat de postbode die over de rode lijn loopt, een hogere snelheid moet hebben gehad (omdat deze – via een omweg – op hetzelfde punt B uitkomt. Op dezelfde manier is Einsteins paradox geen tegenstelling, maar moeten we inzien dat een klok die beweegt nu eenmaal langzamer loopt.


Soms lees je dat het verschil in de veroudering van de tweelingzussen wordt veroorzaakt door versnelling of door over die aanwezigheid van een zwaartekrachtsveld. Maar het effect dat de tweelingparadox paradoxaal maakt is de vertraging van een bewegende klok, en dat is een effect uit de speciale relativiteitstheorie.

Het effect bestaat natuurlijk ook in de algemene relativiteitstheorie, maar zwaartekracht of versnelling zijn niet nodig om het te kunnen afleiden. Dat kunnen we zien aan het gedachtenexperiment over de lichtklok. Daarin is er een vertragende klok (‘tijdsdilatatie’) maar geen zwaartekracht of versnelling.

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The twin/postmen paradox

Einstein’s twin paradox is more familiar to us than we think. Then why does it feel so strange? I promised my students some extra guidance, so I wrote this post.

[For a course I am currently teaching at a university in Amsterdam]

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Understanding the Quantum

My propoposal, “Quantum Theory Is Not As Strange As We Think (or is classical physics stranger than we think?)”, has been accepted by the organisers of the annual conference of the German Physics Society. I am proud of that!

atoms are quanta of matter; we can understand the quantum.

I argue that many of the concepts that make quantum theory so hard to swallow (entanglement, tunneling, and yes, even quantisation) can be understood from the perspective of Newton’s (17th century) physics.

Read the abstract of my presentation here

What are atoms other than quanta of matter?

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Lecture: Newton & Einstein – physics meets philosophy

This Wednesday I’m giving a “pub lecture”: the lecture is both public and in a pub. Drop by if you’re interested!

Location: “proeflokaal Faas (Zwaanshals 248, Rotterdam)” (google maps)

Attendance is free; no reservation required.

Time: 20.00

It is a common thought that classical, Newtonian physics presents us with an intuitive, easy-to-understand picture of what reality is really like, which follows unproblematically from the mathematical equations: point-like particles flying around in a rigid 3d container. If we know the container’s precise state at some point in time, Newton’s laws of motion and his universal gravitation tell us exactly where the particles will be at any other point in time. This intuitiveness and easy understanding come to an abrupt end early in the 20th century. Einstein’s relativity maims Newton’s neat concepts of space and time and we’re no longer sure which clocks are synchronous and which rulers are straight. Physics has become abstract and unintuitive so that it is now unintelligible for anyone untrained in higher mathematics.

In my lecture (based on my [Dutch] book) I argue that this is misrepresents the situation. I will show that the problems and concepts that make Einstein’s relativity so difficult to understand (the nature of space and time, locality and determinism, to name but a few) are also problems and concepts that underlie Newtonian physics: they must be faced by anyone who wishes to get from physics’ experimental outcomes to a view of reality.

On the bright side, the fact that the same philosophical issues play a role in both classical and modern physics means that once we really get to grips with the foundational issues underlying classical physics, we automatically gain a much better understanding of modern physics – physics without philosophy is blind.

This Wednesday I'm giving a lecture: Newton & Einstein - physics meets philosophy
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Quantum theory, free will and justice

A student of mine is writing his bachelor thesis on “free will, quantum mechanics and the system of justice”. I’m very happy about that, because it gives me an excuse to dive into matters I’ve been interested in since childhood. I want to use this blogpost as a noteblock on which I roughly sketch my thoughts on these matters. Sometimes the attempt to write down a coherent story about your point of view shows you that there are gaps in your argumentation you hadn’t expected.

Do we have free will? And can justice function if our will isn't free?
Do we have free will? And can our system of justice function if our will isn’t free?
Credit: francescoch/Getty Images

Do the theories of physics rule out the possibility of human free will? It is a popular thought that there is no room for free will in classical physics while in quantum mechanics there is room for free will (because of the uncertainty principle). It is very important that there is free will, so the popular story continues, because our system of justice presupposes the possibility of human choice – How else could a thief be held accountable for a misdeed?

As with any philosophical discussion, we need to be very clear on what we are talking about, so let’s start with some concepts:


I have a physicalist view of the human mind, in which consciousness emerges from the workings of the brain, and nothing else. So in my view ‘will’ is just matter (particles and energy) moving about in certain ways.


The word ‘free’ expresses a relation. The question whether something is free without further context has no meaning and therefore cannot be answered. Free is always understood as free with regard to something else. For example, something can be free in the context of a certain set of laws.

Using these definitions, we see that the question “does free will exists?” is incomplete. The question should be replaced by other questions:

  1. Is the human will free with respect to the laws of physics?
  2. Is the human will free with respect to the laws of criminal justice?

Question 1: Too early to tell

The first question, “Is the human will free with respect to the laws of physics?”, has a strange form. What are the laws of physics? Regardless of philosophical standpoints, the laws of physics describe what is going on on. Our will is one of the things that are going on, so the question whether our will can be free with respect to the laws of physics is circular: for our will to change the laws of physics would have to be different.

So, where are we? Is the human will free with respect to the laws of physics? No, it isn’t: the laws of physics are our will – but perhaps there is some other set of laws with respect to which the human will is free.

What the definitions tell us is that if the human will is completely described by physics it can not be free from the laws of physics. Whether perhaps there is an aspect of the human mind which is not described by current physics is a different question, which the future of scientific research will answer (what comes to mind is that Einstein himself did not believe that quantum theory is a final and complete description of reality).

I will explore the second question in my next blog post. Can our system of justice function if we have no free will with respect to the laws of physics?

“The laws of physics do not dictate our will; the laws of physics are our will”

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Alice in Numberland, to be continued…

winking professor: Alice in Numberland is to be continued in the Spring of 2023.

I am very happy to (re-)start teaching mathematics (calculus) at Amsterdam university in September, while teaching philosophy (the history of epistemology) at Utrecht University from November onwards. In November and December these courses overlap, so the end of the year is going to be a very busy period for me!

All this teaching leaves me with little time for my Alice-blog, but the plot is thickening, while some unexpected developments are on their way. But don’t hold your breath! The final three episodes of my Alice in Numberland story will appear on this blog in the Spring of 2023. [teaser: the queen is not at all what Alice expects her to be, and the Architect – himself a famous mathematician – has a surprising answer to the minotaur’s question (“do we need mathematics to build a labyrinth?”).

I would really appreciate it if you could let me know which of my Alice-episodes you liked best (and why you liked it). Please fill in the poll below and use the ‘feedback’ button on the left. Namaste!

Previous episodes of ‘Numbers in Wonderland’:

  1. Numbers in Wonderland
  2. Alice and the Friendly Minotaur
  3. Wonderland without Numbers?
  4. The White Rabbit’s Watch
  5. Believing Words
  6. Zeno the Zebra
  7. Zeno the Zebra and the truth in the middle

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Zeno the Zebra and the Truth in the Middle

“Is the Queen really such an unpleasant person, or did she simply run out of biscuits?” Alice asked Zeno the Zebra.

The coockie jar that Alice thinks about. Is the queen grumpy because her coockie jar is empty?

Last week, when aunt Caroline came over for tea, Alice’s mother had been rather embarrassed when she discovered that the coockie jar was empty, which had made her grumpy, so Alice thought that perhaps the Queen’s coockie jar was empty as well.

Even though Zeno had been smiling ever since he had introduced himself to Alice and the minotaur, the zebra snorted when he heard Alice’s question.

Even little girls know that animals’ snorts don’t mean anything, so Alice waited politely for Zeno to respond to her question.

Zeno snorted again, more loudly this time.

All this snorting made Alice feel quite uncomfortable, so she cast an uncertain look at her companion, the Friendly Minotaur, who was still standing next to her. Contrary to popular belief, minotaurs speak many languages. Besides Minoan and English, most minotaurs are quite good at understanding the snorts of other animals.

“I believe that the zebra means that you are wrong, Alice.” The minotaur said with his deep, bombastic voice

The zebra’s wide smile returned. “Indeed I do”, he replied. “The Queen is not merely upset, she is downright unkind!”

Alice thought about this for a moment. How could the zebra be so sure? But then she remembered that she and the Friendly Minotaur were guests, while Zeno lived here. Always be polite to your hosts, mamma always said. “I guess,” Alice said while she did her best to sound like a grown-up, “…that the truth lies somewhere in the middle.”

When he heard Alice’s words, Zeno’s eyes became huge with excitement, while his smile, which had already been wide, became even wider, so that now it seemed to reach his ears. “That means I am right!”

“No,” Alice said while she frowned, “it means that we are both wrong, while the truth is somewhere in the middle of our views.”

While Alice didn’t quite know what it meant for the truth to be ‘somewhere’, she was proud of herself: It had felt so wise and mature to admit that she herself was wrong.

But Zeno did not give up so easily. “I say A, you say B, and you admit that the truth is in the middle, so your view shifts halfway towards mine.” Zeno paused to see if Alice was still listening. “But you already admitted that the truth is in the middle, so again your view moves closer to mine.”

“Need I go on?” The zebra asked self-assuredly.

Image showing the distant hill beside the path where Alice is to find a small cave.

“Not really,” Alice had grown tired of the zebra’s talkativeness. She didn’t care anymore who was right, she just wanted to know where the queen was. To hurry things up, Alice decided to give the zebra what he wanted. “I see what you mean”, she said, “my view shifts ever closer to yours, so that we end up having the same view.”

“Yes,” the zebra said triumphantly, “and that’s the view I started with!”

Alice nodded indifferently. “And do you know where we may find this unkind queen?”

The zebra now pointed one of his muddy hoofs towards a distant hill that lay beside the path. “In the side of that hill over there, there is a cave with directions on one of its walls.” As the zebra paused for a moment, his smile seemed to become smaller. After a while he said, now with a sad face: “But the cave is too small to enter, you simply won’t fit in.”

“I suppose we will go there anyway.” Alice said, glad to get away from the gloating zebra. “After all, I am a very small girl.”

Alice in Numberland will continue on this blog in the Spring of 2023

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Previous episodes of ‘Numbers in Wonderland’:

  1. Numbers in Wonderland
  2. Alice and the Friendly Minotaur
  3. Wonderland without Numbers?
  4. The White Rabbit’s Watch
  5. Believing Words
  6. Zeno the Zebra
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Zeno the Zebra

Image showing Zeno the Zebra with his radiant, welcoming, smile.

“I am Zeno, Zeno the Zebra!” the animal shouted cheerfully when he had come to the spot where Alice and her travel-companion stood. When the Zebra offered one of his muddy hoofs for a handshake, Alice remembered what her dear uncle always said: “It is bad-mannered to refuse someone’s handshake”

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Believing Words

\Leftarrow Previously in ‘Numbers in Wonderland’

I don’t believe a word you say!” Alice’s mother yelled at the little girl.

Yesterday, before Alice’s adventure in the rabbit hole began, her mother had been very angry with her for eating all the tea biscuits. Alice remembered that her eyes filled with tears at seeing her mother so angry, but it had also made her wonder: “How can anyone believe words? Surely, words don’t say things, like people do, so why would you ever believe them? It seemed the oddest thing to do…

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