What are the twin paradox and the garage paradox?
Everyone knows Special Relativity is hard to understand.
But...why? Well, that's because the universe is hard!
Even though it is hard to understand, people LOVE special relativity! I mean,
at least at the most superficial level... This is fine: there is a big difference
between getting a PhD in physics and reading an article in Discover, but the one thing
all we have in common is our... curiosity! So let's try to understand more of special
relativity today: I will try to explain to you the Twin Paradox and the Garage Paradox,
using some of the ideas that lie at the core of the theory of Special Relativity!
As we said, if there is one defect in the universe is that the universe is hard! Humans have a hard
time understanding it if they aren’t willing to study diligently. (And sometimes even if
they are.) There’s no way around this. In the case of science, no law says the universe has
to work in ways that the ape h. sapiens can make intuitive or visual sense of!
You can read an article in Discover and get a glimpse. A talented writer can give you a
slightly better glimpse. If you get a PhD in physics you’ll get an even better glimpse.
You’ll start to grasp simultaneity, light cones, stress-energy tensors, geodesics,
world lines, Riemannian geometries, and frame-dragging. But will you ever truly
understand? Will you ever truly be able to picture it? Probably not. You might eventually
be able to manipulate the algebra deftly, but at a visceral level, our brains evolved
to understand spear throwing and baby-raising, not differential equations or tensor analysis.
Special relativity is one of the most popularly famous ideas in physics – it's that thing that
Einstein figured out about the speed of light and space and time and E=mc^2! It changed our
understanding of the universe. And its core ideas are accessible in principle
to anyone who understands some basic algebra and geometry - you don't even need to know calculus!
Let's try to understand the Twin Paradox, starting from some core ideas.
According to special relativity, a moving clock runs slower, so a person in a high-speed moving
vehicle will get old...slower! This is due to the so-called “Time dilation”.
Time dilation is the “slowing down” of a clock as determined by an observer who is in relative
motion concerning that clock. In special relativity, an observer in inertial (i.e.,
nonaccelerating) motion has a well-defined means of determining which events occur simultaneously
with a given event. A second inertial observer, who is in relative motion with respect to the
first, however, will disagree with the first observer regarding which events are simultaneous
with that given event. (Neither observer is wrong in this determination; rather, their disagreement
merely reflects the fact that simultaneity is an observer-dependent notion in special relativity.)
A notion of simultaneity is required in order to make a comparison of the rates
of clocks carried by the two observers. If the first observer’s notion of simultaneity
is used, it is found that the second observer’s clock runs slower than the first observer’s by
a Lorentz Factor: (TO THE VIDEOMAKER: SHOW THE LORENTZ FACTOR FORMULA) where v is the relative
velocity of the observers and c equals 299,792 km (186,282 miles) per second—i.e., the speed
of light. Similarly, using the second observer’s notion of simultaneity, it is found that the first
observer’s clock runs slower by the same factor. Thus, each inertial observer determines that all
clocks in motion relative to that observer run slower than that observer’s own clock.
Let's see where this brings us. Suppose Albert and Isaac are twins
on Earth. At the age of 20, Albert decides to start a space trip in order to visit the
nearest-to-Earth extrasolar pub. The whole trip takes 10 years.
When he returns to Earth 10 years later, Albert is 30 and amazingly (and maybe
sadly?) he finds his twin brother much older than him. Let's say he finds Isaac is 40.
This is what the twin paradox says. I remember the first time I heard this: I
thought it was nonsense! How could it be possible? It's counter-nature! It couldn't be true!
But, as the years were passing by, I found myself having to do with
maths. I had to deal with it so much that I eventually started to believe in its power.
And the math behind the twin paradox was telling me it was true:
Albert would have been older than Isaac. The reason is that the high-speed motion
makes time pass slower in the spaceship, thus T can be much younger than E.
We could call this “the first twin paradox”. The first twin paradox isn't really a scientific
paradox, but a logical one: How old is Albert?
Albert has experienced 30 years of life, but he was also born at the same moment as Isaac,
which was 40 years ago. So is he 30 years old or 40 years old?
In this case, the answer is "both" ... depending on which way you're measuring age.
According to his driver's license, which measures Earth time (and is no doubt expired),
he's 40. According to his body, he's 30. Neither age is "right" or "wrong," although
the social security administration might take exception if he tries to claim benefits.
But the conclusion can also be the reverse.
The most important part of Special Relativity is the principle of relativity.
From the standpoint of T, E is in a high-speed motion relative to him, thus the clock on Earth
should run slower, and E should be much younger. This is what we'll call “the twin paradox two”.
The twin paradox two is a bit more technical, and really comes to the heart of what physicists mean
when they talk about relativity. The scenario is based on the idea that Albert was travelling
very fast, so time slowed down for him. The problem is that in relativity, only the
relative motion is involved. So the question now is: what if you considered things from Albert's
point of view, then he stayed stationary the whole time, and it was Isaac who was moving
away at rapid speeds. Shouldn't calculations performed in this way mean that Isaac is the
one who ages more slowly? Doesn't relativity imply that these situations are symmetrical?
Now, if Albert and Isaac were on spaceships travelling at constant speeds in opposite
directions, this argument would be perfectly true. The rules of special relativity, which govern
constant speed (inertial) frames of reference, indicate that only the relative motion between
the two is what matters. In fact, if you're moving at a constant speed, there's not even an
experiment that you can perform within your frame of reference which would distinguish you from
being at rest. (Even if you looked outside the ship and compared yourself to some other constant
frame of reference, you could only determine that one of you is moving, but not which one.)
But there's one very important distinction here: Albert is accelerating during this
process. Isaac is on the Earth, which for the purposes of this is basically "at rest"
(even though in reality the Earth moves, rotates, and accelerates in various ways).
Albert is on a spaceship that undergoes intensive acceleration to read near lightspeed. This means,
according to general relativity, that there are actually physical experiments that could be
performed by Albert which would reveal to him that he's accelerating ... and the same experiments
would show Isaac that he's not accelerating (or at least accelerating much less than Albert is).
The key feature is that while Isaac is in one frame of reference the entire time,
Albert is actually in two frames of reference - the one where he's
travelling away from the Earth and the one where he's coming back to the Earth.
So Albert's situation and Isaac's situation are not actually symmetrical in our scenario.
Albert is absolutely the one undergoing the more significant acceleration,
and therefore he's the one who undergoes the least amount of time passage.
Ok, let's take a break now because we are really tired: here's some koala pics that
will give you the strength to go through the second paradox of the video: THE GARAGE PARADOX!
To explain the garage paradox, we have to
introduce first the well-known concept of Length contraction:
“The length of any object in a moving frame will appear foreshortened in the direction
of motion or contracted. The amount of contraction can be calculated from the
Lorentz transformation. The length is maximum in the frame in which the object is at rest.”
If you think about it, it's almost intuitive: since the speed of light is constant,
if the time is dilated in one reference frame, then space will be...contracted. This gives
you a sort of intuition about the process. For the sake of the argument, we will call into
action our Superhero Batman and its Batmobile. Let's suppose Batman has a bat-mobile
and he wants to put it into his bat-garage. The batmobile enters the garage and stops in it.
The size of the mobile and garage are perfectly alike,
and the batmobile enters just inside the garage. Now let's suppose Batman has invented a new super
engine that allows reaching comparable speeds with those of light. Then let's suppose the
batmobile has an automatic pilot and Batman wants to see how the automatic pilot parks
the car in the bat garage. So he is out of the car, standing next to the garage.
The car is going at a constant speed comparable with c.
Then batman will see the batmobile shorter than it actually is. We will notice that there will
be a certain moment where the batmobile will be a size smaller than that of the bat garage. Then
the batmobile will stop, and it will be seen from the garage reference frame with its proper length,
that is, the same length as the garage. Now let's suppose Batman itself wants to
ride the batmobile into the garage, in order to park it properly.
This time Batman sees that the garage is coming towards the batmobile,
so what he sees is that the garage is shorter than it is when it's stationary, that is, the
batmobile is now longer than the garage. So there is a problem: how can the batmobile enter the box?
Believe me or not, the car, in some ways, manages to enter the garage. In fact,
whatever result we get by correct reasoning in any one inertial frame, must be true.
This means that in particular, it must be true when viewed from any other inertial frame.
(and in our case both are inertial frames: the batmobile frame and the garage frame!).
In particular, if one is willing to do some math and physics,
as long as the set of physical laws we are using is self-consistent and Lorentz-invariant,
there must be an explanation of the result in every other inertial frame,
although it may be quite a different explanation from that in the first frame.
And the explanation does exist! However, it's a bit complicated, and luckily we
don't really need it for the sake of the video! This was the so-called “garage paradox!”
" Do you think special relativity is mindblowing? Was this vide useful to
you? Let us know in the comments below! Drop a line below and let us know what you
think. and remember to get ice cream before it's too late!