The Twin paradox, from both sides now (with apologies to Joni)

 I will give an exposition on the twin paradox, using an example I read in a book about 4 decades ago, so I’m relating this from memory.

Imagine that one of the twins goes to visit an extra-terrestrial world 20 light years away in a spaceship that can travel at 4/5 the speed of light. The figures are chosen because they are easy to work with and we assume that acceleration and stopping are instantaneous. We also assume that the twin starts the return journey as soon as they arrive at their destination.

 

From the perspective of the twin on Earth, the trip one-way takes 25 years because the duration is T = s/v, where s = 20 (light years) and v = 4/5c. 

So 5/4 x 20 = 25.

 

From the perspective of the twin on the spaceship, their time is determined by the Lorentz transformation (γ).

 

γ = 1/√(1 – v²/c²)

 

Note v²/c² = (4/5)² = 16/25

So √(1 – v²/c²) = √(9/25) = 3/5

 

Now true time for the space ship (τ) is given by τ = T/γ

So for the spaceship twin, the duration of the trip is 3/5 x 25 = 15

So the Earth twin has aged 25 years and the spaceship twin has aged 15 years.

 

But there is a relativistic Doppler effect, which can be worked out by considering what each twin sees when the spaceship arrives at its destination. 

 

Note that light, or any other signal, takes 20 years to come back from the destination. So the Earth twin will see the space ship arrive 25 + 20 = 45 years after it departed. But they will see that their twin is only 15 years older than when they left. So, from the Earth twin’s perspective, the Doppler effect is a factor of 3. (3 x 15 = 45). So the Doppler effect slowed time down by 3. Note: 45 years has passed but they see their twin has only aged 1/3 of that time.

 

What about the spaceship twin’s perspective? They took 15 years to get there, but the Doppler effect is a factor of 3 for them as well. They’ve been receiving signals from Earth ever since they left so they will see their twin only 5 years older because 15/3 = 5, which is consistent with what their twin saw. In other words, their Earth twin has aged 5 years in 15, or 1/3 of their travel time. 

 

If spaceship twin was to wait another 20 years for the signal to arrive then it would show Earth twin had aged 5 + 20 = 25 years at the time of their arrival. But, of course, they don’t wait, they immediately return home. Note that the twins would actually agree on each other’s age if they allowed for the time it takes light to arrive to their respective locations.

 

So what happens on the return trip? The Lorentz transformation is the same for the spaceship twin on the return trip, so they only age another 15 years, but according to the Earth twin the trip would take another 25 years, so they would have aged 50 years compared to the 30 years of their twin.

 

But what about the Doppler effect? Well, it’s still a factor of 3, only now it works in reverse, speeding time up. For the spaceship twin, their 15 years of observing their Earth twin is factored by 3 and 15 x 3 = 45. And 5 + 45 = 50, which is how much older their twin is when they arrive home.

 

For the Earth twin, their spaceship twin’s round trip is 50 years, so the return trip appears to only take 5 years. And allowing for the same Doppler effect, 3 x 5 = 15.

When the Earth twin adds 15 to the 15 years they saw after 45 years, they deduce the age of their spaceship twin is 30 years more (against their own 50). So both twins are in agreement.

 

Now, the elephant in the room is why do we only apply the Lorentz transformation to the spaceship twin? The usual answer to this question is that the spaceship twin had to accelerate and turn around to come back, so it’s obvious they did the travelling.

 

But I have another answer. The spaceship twin leaves the surface of Earth and even leaves the solar system. It’s obvious that the spaceship didn’t remain stationary while the solar system travelled through the cosmos at 4/5 the speed of light. There is an asymmetry to the scenario which is ultimately governed by the gravitational field created by everything, and dominated by the solar system in this particular case. In other words, the Lorentz transformation only applies to the spaceship twin, even when they only travel one way.

Relativity makes sense if everything is wavelike

 When I first encountered relativity theory, I took an unusual approach. The point is that c can always be constant while the wavelength (λ) and frequency (f ) can change accordingly, because c = λ x f. This is a direct consequence of v = s/t (where v is velocity, s distance and t time). We all know that velocity (or speed) is just distance divided by time. And λ represents distance while f represents 1/t. 

So, here’s the thing: it occurred to me that while wavelength and frequency would change according to the observer’s frame of reference (meaning relative velocity to the source), the number of waves over a specific distance would be the same for both, even though it’s impossible to measure the number of waves. And a logical consequence of the change in wavelength and frequency is that the observers would ‘measure’ different distances and different periods of time.

 

One of the first confirmations of relativity theory was to measure the half-lives of cosmic rays travelling through the Earth’s atmosphere to reach a detector at ground level. Measurements showed that more particles arrived than predicted by their half-life when stationary. However, allowing for relativistic effects (as the particles travelled at high fractional lightspeeds), the number of particles detected corresponded to time dilation (half-life longer, so more particles arrived). This means from the perspective of the observers on the ground, if the particles were waves, then the frequency slowed, which equates to time dilation - clocks slowing down. It also means that the wavelength was longer so the distance they travelled was further. 

 

If the particles travelled slower (or faster), then wavelength and frequency would change accordingly, but the number of waves would be the same. Of course, no one takes this approach - why would you calculate the Lorentz transformation on wavelength and frequency and multiply by the number of waves, when you could just do the same calculation on the overall distance and time.

 

Of course, when it comes to signals of communication, they all travel at c, and changes in frequency and wavelength also occur as a consequence of the Doppler effect. This can create confusion in that some people naively believe that relativity can be explained by the Doppler effect. However, the Doppler effect changes according to the direction something or someone is travelling while relativistic effects are independent of direction. If you come across a decent mathematical analysis of the famous ‘twin paradox’, you’ll find it allows for both the Doppler effect and relativistic effects, so don’t get them confused.

 

Back to the cosmic particles: from their inertial perspective, they are stationary and the Earth with its atmosphere is travelling at high fractional lightspeed relative to them. So the frequency of their internal clock would be the same as if they were stationary, which is higher than what the observers on the ground would have deduced. Using the wave analogy, higher frequency means shorter wavelength, so the particles would ‘experience’ the distance to the Earth’s surface as shorter, but again, the number of waves would be the same for all observers.

 

I’m not saying we should think of all objects as behaving like waves - despite the allusion in the title - but Einstein always referred to clocks and rulers. If one thinks of these clocks and rulers in terms of frequencies and wavelengths, then the mathematical analogy of a constant number of waves is an extension of that. It’s really just a mathematical trick, which allows one to visualise what’s happening.

What is scientism?

 I’m currently reading a book (almost finished, actually) by Hugh Mackay, The Inner Self; The joy of discovering who we really are. Mackay is a psychologist but he writes very philosophically, and the only other book of his I’ve read is Right & Wrong: How to decide for yourself, which I’d recommend. In particular, I liked his chapter titled, The most damaging lies are the ones we tell ourselves.

You may wonder what this has to do with the topic, but I need to provide context. In The Inner Self, Mackay describes 20 ‘hiding places’ (his term) where we hide from our ‘true selves’. It’s all about living a more ‘authentic’ life, which I’d endorse. To give a flavour, hiding places include work, perfectionism, projection, narcissism, victimhood – you get the picture. Another term one could use is ‘obsession’. I recently watched a panel of elite athletes (all Australian) answering a series of public-sourced questions (for an ABC series called, You Can’t Ask That), and one of the take-home messages was that to excel in any field, internationally, you have to be obsessed to the point of self-sacrifice. But this also applies to other fields, like performing arts and scientific research. I’d even say that writing a novel requires an element of obsession. So, obsession is often a necessity for success.

 

With that caveat, I found Mackay’s book very insightful and thought-provoking – It will make you examine yourself, which is no bad thing. I didn’t find any of it terribly contentious until I reached his third-last ‘hiding place’, which was Religion and Science. The fact that he would put them together, in the same category, immediately evoked my dissent. In fact, his elaboration on the topic bordered on relativism, which has led me to write this post in response.

 

Many years ago (over 2 decades) when I studied philosophy, I took a unit that literally taught relativism, though that term was never used. I’m talking epistemological relativism as opposed to moral relativism. It’s effectively the view that no particular discipline or system of knowledge has a privileged or superior position. Yes, that viewpoint can be found in academia (at least, back then).

 

Mackay’s chapter on the topic has the same flavour, which allows him to include ‘scientism’ as effectively a religion. He starts off by pointing out that science has been used to commit atrocities the same as religion has, which is true. Science, at base, is all about knowledge, and knowledge can be used for good or evil, as we all know. But the ethics involved has more to do with politicians, lawmakers and board appointees. There are, of course, ethical arguments about GM foods, vaccinations and the use of animals in research. Regarding the last one, I couldn’t personally do research involving the harming of animals, not that I’ve ever done any form of research.

 

But this isn’t my main contention. He makes an offhand reference at one point about the ‘incompatibility’ of science and religion, as if it’s a pejorative remark that reflects an unjustified prejudice on the part of someone who’d make that comment. Well, to the extent that many religions are mythologically based, including religious texts (like the Bible), I’d say the prejudice is justified. It’s what the evolution versus creation debate is all about in the wealthiest and most technologically advanced nation in the world.

 

I’ve long argued that science is neutral on whether God exists or not. So let me talk about God before I talk about science. I contend that there are 2 different ideas of God that are commonly conflated. One is God as demiurge, and on that I’m an atheist. By which I mean, I don’t believe there is an anthropomorphic super-being who created a universe just for us. So I’m not even agnostic on that, though I’m agnostic about an after-life, because we simply don’t know.

 

The other idea of God is a personal subjective experience which is individually unique, and most likely a projection of an ‘ideal self’, yet feels external. This is very common, across cultures, and on this, I’m a theist. The best example I can think of is the famous mathematician, Srinivasa Ramanujan, who believed that all his mathematical insights and discoveries came directly from the Hindu Goddess, Namagiri Thayar. Ramanujan (pronounced rama-nu-jan) was both a genius and a mystic. His famous ‘notebooks’ are still providing fertile material 100 years later. He traversed cultures in a way that probably wouldn’t happen today.

 

Speaking of mathematics, I wrote a post called Mathematics as religion, based on John Barrow’s book, Pi in the Sky. According to Marcus du Sautoy, Barrow is Christian, though you wouldn’t know it from his popular science books. Einstein claimed he was religious, ‘but not in the conventional sense’. Schrodinger studied the Hindu Upanishads, which he revealed in his short tome, Mind and Matter (compiled with What is Life?).

 

Many scientists have religious beliefs, but the pursuit of science is atheistic by necessity. Once you bring God into science as an explanation for something, you are effectively saying, we can’t explain this and we’ve come to the end of science. It’s commonly called the God-of-the-gaps, but I call it the God of ignorance, because that’s exactly what it represents.

 

I have 2 equations tattooed on my arms, which I describe in detail elsewhere, but they effectively encapsulate my 3 worlds philosophy: the physical, the mental and the mathematical. Mackay doesn’t talk about mathematics specifically, which is not surprising, but it has a special place in epistemology. He does compare science to religion in that scientific theories incorporate ‘beliefs', and religious beliefs are 'the religious equivalent of theories'. However, you can’t compare scientific beliefs with religious faith, because one is contingent on future discoveries and the other is dogma. All scientists worthy of the name know how ignorant we are, but the same can’t be said for religious fundamentalists.

 

However, he's right that scientific theories are regularly superseded, though not in the way he infers. All scientific theories have epistemological limits, and new theories, like quantum mechanics and relativity (as examples), extend old theories, like Newtonian mechanics, into new fields without proving them wrong in the fields they already described. And that’s a major difference to just superseding them outright.

 

But mathematics is different. As Freeman Dyson once pointed out, a mathematical theorem is true for all time. New mathematical discoveries don’t prove old mathematical discoveries untrue. Mathematics has a special place in our system of knowledge.

 

So what is scientism? It’s a pejorative term that trivialises and diminishes science as an epistemological success story.