It's About Time Dilation · 11:18pm Jan 10th, 2015
It is often said that every equation put into a popular science book will halve its sales. It is known that this is not true. But it's such a fun anecdote to put in the introduction, so we all keep repeating it.
Since all the equations in my math-heavy commentary on Rainbow Rocks did not seem to scare away too many readers, I reckon it's about time to turn my attention to the best chunk of mathematical physics we see in the entire show.
This:
[A quick recap: Season 2, episode 20, It's About Time. Twilight receives a frightening, but incomplete, warning of impending disaster from her future self. She proceeds to disaster-proof Equestria and return Cerberus to his post. But she then becomes increasing nervous about various portents of doom, and is not reassured by Pinkie's prophecy that she will get a really cool birthday present. She proceeds to “monitor everything”, which involves astronomical observations, notetaking, dressing-up as a pirate, and staring at an impressive equation-covered blackboard, before concluding that the only solution is to “stop time!”]
What's it all about?
Twilight's blackboard has been transcribed on derpibooru/406579, which seems to have come from the wikipedia page on time dilation, which in turn references “An Analytical Treatment of the Clock Paradox in the Framework of the Special and General Theories of Relativity”, Lorenzo Iorio. Foundations of Physics Letters 18 (2005) 1-19 (preprint available at arXiv:physics/0405038)
This is a calculation involving Einstein's theory of special relativity.
Special Relativity is a theory we need to understand things which move at close to the speed of light. It is regarded by theoretical physicists as one of the most beautiful theories ever, because despite the enormous complexity of its conclusions, it is all based on two simple postulates:
1. "The laws of physics are the same for all inertial observers". Inertial means either still, or moving at a steady speed. So if you're on a train running smoothly at constant speed, the laws of mechanics look just the same as if you are standing still. You can't actually tell that you are moving at all without looking out of the window.
2. "The speed of light is the same all inertial observers". Everyone measures the same speed of light, regardless how fast they are moving.
This is counterintuitive. We expect speed to be relative. If we are on a buffalo running at 29mph, parallel to a train running at 30mph, then the train is only doing 1mph relative the herd, so it's easy to hop on.
But for a light beam, the speed we measure is 299 792 458 m/s, whether we are on the ground, or on the train, or on another train running in the opposite directions. Not a bit more, nor a bit less. The speed of light in space is a universal constant, and nopony, not even Rainbow Dash can fly faster than light in the vacuum of space [1].
But how does that work? The speed is the same whether we are racing towards it, standing still, or running away? That isn't possible unless something funny is going on with time and space.
[1] It does go slower in air or water – see this post.
Exactly. By forcing us all to agree on the speed of light, relativity means we can't agree on measures of time and distance.
When we travel at close to light speed, our time slows down, relative to somepony staying still. Everything seems normal to us, but when we get home, we find our watch shows an earlier time than that recorded by a stationary clock. This is time dilation.
The equation at the top of Twilight's board is the so-called Lorentz factor, which sets the scale of how much time is dilated by [2].
[2] This gets a brief mention in Rainbooms and Rationality. It also sets the scale of other near-light-speed effects—the increase in mass and contraction of lengths.
Where v0 is the speed, c is the speed of light. For any reasonable speed, this is so close to one that it makes no difference. But as we creep close to light speed, it gets bigger. At the speed of sound it is 1.0000000000006, but at 99.9% the speed of light, it rises to 22. If we could reach the speed of light, it would be infinite.
The more you think about this, the weirder it gets. From one perspective, we are moving super-fast relative to everyone else. But from another, they are moving super-fast, and we are stationary. So if our time runs slower than theirs, and their time runs slower than ours, how does that add up?
This is the essence of the so called clock paradox or twin paradox. Let's explain it with the help of Celestia and Luna. Although they are not twins, they are much more cute than Flim and Flam, and they have the necessary longevity for this experiment.
By PluckyNinja
Suppose Celestia stays at home in Equestria, but Luna takes a trip to the Horsehead Nebula (1500 light years away). If she travels at close to the speed of light, then from Celestia's point of view, the round trip will take her just over 3000 years. But as Luna flies at a relativistic speed, her clock runs at a slower rate. Only a few years would pass for her. She would return to Equestria still young, to greet a much older sister.
Celestia by Fehlung, Luna by StarryOak
But from Luna's point of view, it is Celestia's clock which runs slow. So then the reverse situation would be true when they were reunited.
Celestia source, Luna by DraikJake.
This paradox can cause a lot of confusion for students, as the speed of Luna relative to Celestia, is the same as that of Celestia relative to Luna (just the in the other direction). So the maths seems to say that Luna is older than Celestia. But it also says Celestia is older than Luna. Which is not scientifically possible.
The resolution to this paradox is what Twilight was working out on her blackboard. The crucial fact is that Luna was not flying at near-light-speed for the full journey. She first had to accelerate to full speed, then slow down once she reached her destination, then (after stopping for lunch at a convenient interstellar restaurant) accelerate back to full speed in the opposite direction, then decelerate again on reaching home. This means her motion was not always inertial. To calculate the total time for Luna's journey from her perspective, we need to consider what happens as her speed changes during these four stages.
To complete Twilight's work:
This answer gives the time recorded by Luna, if she was pushed by an acceleration g. If that is equal to the Earth's gravity (and if I've got my numbers right) then the answer for total duration of the trip to the Horsehead Nebula is 3004 years for Celestia, and 28.5 for Luna. So Luna is indeed the younger sister.
So what was Twilight thinking in this scene? She wanted to stop time, leaving all Equestria in a suspended state, to give herself time to figure out how to avert the impending disaster. That would mean staying still herself, but accelerating the rest of Equestria to near-light-speed, which would need a near-infinite amount of energy, and crush everypony with an excessive force. So veto that idea, and go to look for another solution in the Stawswirl the Bearded wing of the Canterlot achieves.
The next thing to talk about would be the speculative science of time travel. But I've now written over 1000 words, and I expect an excessive amount of text would deter far more readers than a set of equations. So I'll leave that for next time.
Ah, light, the primadonna of physics. None may be as swift as she, and space and time will tie themselves into knots before offending her.
Light is a lot like Rainbow Dash.
In any case, this is one of the cooler Easter Eggs in the show. Unlike Maxwell's rainbow, I already knew about this one, but it's still neat to see relevant relativistic equations in a cartoon aimed at 6-year old girls.
Ah, this takes me back to Physics 3, one of the best and hardest classes I ever took.
Four hours per homework problem - hard.
Watching the prof's board full of derivations suddenly spit out the ultraviolet catastrophe - glee.
I found out a way to avoid the ultraviolet catastrophe, but I dont know anything about teh last hundred years or so of fundamental physics in teh actual maths to see where the idea I was hoping for quick breaks down.
The piano tuners correction. If you assume mass follows the law of relativity in space time, then applying relativity To spacetime, gives you a self limiting function. That is, things quantise. which is where I cant get anywhere near.
then again, two articles I saw in 2014 approached teh same problem from opposite directions and came up with the same answer. Black hole guys decided to use Octonions to calculate a 1 dimentional quantum box, and got equivalence. Quantum guys took an Octonion quantum box, and compared to a 1 dimentional representation of a black hole, event horizon, and got equivalence. Somewhere in tadding the extra dimentions on either side, theres a confusion as to what does which.
Simple thing, Compare space time to atmosphere, and speed of light to speed of sound. when approaching the speed of sound, you compress the air up, this increses the speed of sound locally, So you are still going less than speed of sound locally, but faster universally. Be nice if the gravity field means space time is sretching so that your local speed remains less than llight, whe your universal speed is greater than light. Also means that blackholes are hyperbolic inside , in higher dimentions. Possibly.
I thought that it wasn't that nothing was faster than light, but light was only going the maximum the universe allowed it to? If it wasn't held back by c, would it go faster?
Also, I'm glad I now know that since time behaves wonky as you approach c, light and other things that go at or near c have their own wonky time things happening all the time :)
Now, this is way outside my field, but I thought that a fundamental part of the story was that the speed of light is derivable from fundamental laws of physics, specifically Maxwell's electromagnetism. I haven't seen any other plausible a priori reason why postulate 2 should be true, since taken in isolation it seems quite arbitrary.
2714178 Indeed, I've seen several explanations of special relativity where 2. is explained as a natural consequence of 1, and thus not really a postulate.
¡I love these BlogPosts about Physics!
Currently, I work on a BlogPost about Physics, but it is not ready for Prime Time yet.
2714178
Maxwell's theory lets you determine the speed of light from measured properties of electric and magnetic fields, but it's not fundamental. After Maxwell it was thought light travelled as an electromagnetic wave in the 'luminous aether'. This lead to the Michelson-Morley experiment, which tried to measure variations in the speed of light in different directions due to the movement of the Earth through the Aether. It failed to do so, and this lead on to Einstein's work.
2714600 The more formal way of writing it is to say the laws of physics are invariant under Lorentz transformations. It's certainly easier to explain using Einstein's postulates. It seems to me that these are just different ways of saying the same thing, although maybe a mathematician or philosopher would disagree with me. I'm not aware of a truly fundamental reason why this is true. It is tied up in so many theories that if it isn't true, the consequences would be big. But who knows? Theories are not always complete.
Well, given that the 300,000 km/s figure is only a round approximation, yeah it actually is a bit less.
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Good point...
2714153 Light is the fastest possible thing. Although we don't actually know whether light actually does move at the observed speed. For all we know it could travel instantly, or its speed could vary. We aren't completely sure about that.
2759334 ...What? I'm so at a loss for how to respond to this that I can't think of anything to do but paraphrase Wolfgang Pauli and say not only is that not right, it's not even wrong.
Yeah I realize this is old but I just found this post.
2759334
4216081
All that we can say is that we can measure the round-trip speed of light.
We have no idea if the two halves are at the same speed or not.