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What is the most important concept in physics?
Quantum Mechanics?
I would argue that it’s spacetime.
Without it, we would have no stage for all the physics we know of, to act on.
It would be like an artist with all colors of paints, brushes in all shapes and sizes,
but nothing to paint on, and so there would be no masterpiece.
Just like an artist is nothing without a canvas, or an actor is just another person without
a stage, we must have some place for physics to unfold.
That place is spacetime.
At one time space was thought to be nothing, just a empty void, a vacuum with no matter
in it.
Then in 1908, Hermann Minkowski came up with the idea that time could be thought of as
a 4th dimension along with the three dimensions of space.
And Einstein later showed that this spacetime is a kind of geometry that can bend and contort,
affecting the trajectory and passage of time for objects.
We take it for granted because it’s all around us, ever present, like a fish takes
for granted the water it swims in.
But is thing all around us, this all-important, existential canvas without which the universe and we would
not exist?
And how can space and time be part of the same canvas?
Space is measured in meters and feet, while time is measured in seconds and minutes.
How are the two interchangeable?
This seems to make no sense.
And why do we only have one dimension of time, and three dimensions of space?
Why not 2, 4, or 5 spatial dimensions and more than 1 time dimension?
I am going to attempt to explain this intuitively.
Stay tuned because you won’t want to miss this.
That’s coming up right now…
The definition of spacetime is the set of points in space and time, located with 4 numbers.
This would be the location in 3 dimensional space, x, y, and z and a time.
You can also call these events.
So, if you had an event where you were meeting someone, you might say something like I will
meet you in the building on the corner of fifth avenue and Broadway, on the 6th floor.
And of course you would have to tell them what time, otherwise, the meeting might not
take place.
The problem is not in understanding what space and time are.
We can intuitively understand the three dimensions of space.
It’s all around us.
We can move front and back, left and right and up and down.
And we can understand time by the ticks of our clocks, and changes that we can observe
over time.
But the problem is in understanding how these two seemingly completely different concepts
can be combined into a 4 dimensional continuum called spacetime.
This is not intuitive at all.
Space is measured in meters and feet, time is measured in seconds and minutes etc.
These seem to be two completely different things.
How can distance and time be interchangeable?
And how can both be part of the same thing?
In order to answer this question this intuitively, let’s compare the geometry of spacetime
with the geometry of space, since space is something we experience every day and can
identify with.
For simplicity let’s look at only two spatial dimensions only.
Note that what I’ll show here works for three dimensions too, but the math is more
complex.
In ordinary Euclidean space, the distance between two point A and B is fairly simple
to figure out.
All we need to know is the coordinates in the X and Y direction.
Knowing this, we can use the Pythagorean theorem to do some simple math.
X^2 + y^2 = Z^2 The distance Z is the shortest path from A
to B. And it’s obvious also that any other path, from A to B will be longer.
So for example, if we take a curved path or a zigzag path from A to B, that path will be longer than Z.
That is simple enough.
Now, let’s look at spacetime.
In order to make this as similar to the 2D model as possible, lets again use two dimensions.
But this time we will use one spatial dimension, X, and the other dimension will be time, t.
This time, A and B are not just points in space, but also time, so you can think of
them as events rather than just simply points.
The math that we have to use here is not based on Euclidean geometry, but Minkowskian geometry.
There are some big differences.
The line between A and B does not represent distance but time elapsed between the two
events.
We will call this elapsed time E. A straight line represents traveling at a constant velocity
between the two events.
You might say, well, to figure out the time elapsed, all I have to do is subtract the
time I left at A from the time I arrived at B. Isn’t that the elapsed time?
This is where all the ideas of spacetime and special relativity take an unintuitive turn.
The answer is no, you can’t just subtract the time you left from the time you arrived
to find the elapsed time.
Why?
Because you have to remember that time is relative.
Unlike Newton who thought time was absolute, Einstein showed that it was not.
Newton was wrong about this.
Einstein was right.
Ticks of the clock change with velocity.
Time flows differently for anyone moving with a speed compared to someone standing still.
So this elapsed time will be different because traveling from A to B requires a velocity
in space.
If someone were simply standing at A, moving only in time, but not in space, then yes,
you could just subtract the time difference between A and B, and get the elapsed time.
But not in the case where there was traveling in space involved in order to get from A to
B. Given what I just said, you would be correct
to presume that the Pythagorean theorem could not work here.
So, x squared plus t squared is indeed not equal to E squared.
The formula that works here is actually only slightly different.
It is this: E, or the elapsed time (or duration) squared, is equal to t squared MINUS X squared.
The minus sign makes a huge difference.
E^2 = t^2 – x^2 Essentially, what this formula shows is that
more one travels in space, represented by X here, in any given amount of time, the shorter
the elapsed time.
Time and space seem to have a kind of inverse relationship.
So while in Euclidean space, a straight line is shortest distance.
In Minkowskian spacetime, the line between A and B is the LONGEST duration.
The more you move more in space, the less you move in time.
And the less you move in space, the more you move in time.
So for example, in spacetime, if you took a curved path from event A to event B, or
a zig zag path, then the elapsed time would be lower compared to the straight line between
A and B, because you will have traveled more in space than in time.
This can be seen in the equation we saw earlier, but we can also illustrate it with a simple
example.
Let’s say you had a twin brother or sister who stayed on earth, while you traveled to
far distance in space, at some speed, and them back to earth.
You will have aged less because you would have traveled more in space during the same
time that your twin stayed on earth.
Less time would have elapsed for you than your twin.
Now I’m guessing you have a lot of questions so let’s address some of them.
First, you might ask, where does this formula come from.
It comes from Minkowski.
For only two dimensions, the math is not that difficult, and I have a link in the description
if you want to see how it’s derived.
Another question you may have is what’s the different between the time duration “E”
and “t” the time coordinate.
So E, the time duration, would be what the person traveling from A to B would measure
on their wrist watch or clock.
Little "t" is the time coordinate of the system.
This is like a universal time.
Such a time really doesn’t really exist.
It’s a human invention, but if it helps, you can think of this as the time measured
by someone standing perfectly still in the universe.
In Newtonian physics, before relativity, the time coordinate t and E the elapsed time were
thought to be exactly the same.
This is the outdated concept of absolute time.
Einstein showed that there is no such thing as absolute time, and so that’s why we have
a new formulation.
Now, perhaps the biggest question you might have is how the heck do you add time and distance
together, since the units are completely different, meters vs seconds.
Well, this is a great question!
And it’s what I’ve been working to get to.
The key is that there in important conversion factor between time and space, that allows
us to convert one to the other.
And that conversion factor is, drum roll please, the maximum speed limit of the universe.
And that speed limit, as far as we know, is the speed of light.
The speed of light is the key to uniting space and time.
We call this maximum speed “c” in physics.
And c is 299,792,458 meters per second.
As you know speed is distance over time.
If we multiply this speed by time, we get a distance.
So now we can convert time in the same equation to distance – distance = c*t.
Thus, the equation works by using this conversion factor.
Note that this speed can also be more conveniently written as 1 light-second per second.
So converting all the distances to light-seconds instead of meters can makes it easier with
the math when using seconds and meters for our units.
Now let’s go back to the graph and look at some of its profound implications.
This simple formulation for a 2 dimensional spacetime can be extended to the real 4 dimensional
spacetime we live in.
And that 4 dimensional geometry is the foundation for understanding General Relativity, with
the addition that this spacetime is no longer flat, but can curve and contort.
So the math gets pretty complicated pretty quickly in General Relativity.
But one of the consequences of a curving spacetime is that this results in the phenomenon of
gravity.
Gravity also results in time dilation, that is, time runs slightly slower for objects
close to massive bodies, like the Earth where gravity is higher, compared to those far away
in space.
But this time dilation effect due to gravity is weaker on relatively low mass objects like
earth, then the time dilation effect due to high velocities.
Spacetime is a four-dimensional stage for physics to take place and it has existential
importance.
You might ask, well, why do we have only 3 dimensions, why not 4 or 5 or 6 or more spatial
dimensions?
And while we are at it, why not more than one time dimension?
Well, first of all, additional large spatial dimensions probably don’t exist because
we would have detected them if they did.
And more than one dimension of time could result in closed time-like loops, in other
words it would allow travel to the past.
This is considered an impossibility because it would break causality.
So for example it would allow you to travel to the past and kill your father such that
you would never have been born, so you could never have traveled to the past.
This would create paradoxes. So it is not considered possible.
There are also studies that show that any number of dimensions other than the 4 we live in
would either result in no life, or an unstable universe.
If we didn’t have at least one dimension of time, nothing would evolve, nothing would
change, so life could not exist.
So we know we need a time dimension.
But what if we had more than 3 large dimensions?
As it turns out, it would also likely have fatal consequences.
In a 1917 paper, Austrian physicist Paul Ehrenfest
showed that if we had even one extra dimension, gravity would fall by a factor of 8 when doubling the distance
from a planet to its star. This would cause even the tiniest variation in a planet's orbit to either spiral out of the solar system
or meet a fiery death by crashing into the star.
Other problems have also been identified, for example the orbit of the electron in atoms
would become unstable.
If we had fewer spatial dimensions, then spacetime would be too simple for life.
For example if we had only one spatial dimension, then orbits could not form.
Two dimension would also probably be too simple to result in life.
The chart here summarizes these ideas.
This leads us to the fact that, although this may sound like an anthropic idea, it appears that that
spacetime has to be composed of exactly 3 spatial plus 1 time dimension, because if
it were any different, we wouldn’t exist.
So count your lucky stars that we happen to live in precisely such a universe.
If you watched this video up to this point, you realize that I tried to simplify the concept
of spacetime by talking about it in the context of Special Relativity.
And as you might guess there is a lot more to special relativity.
If you want to understand this subject in detail, there is a wonderful course on Brilliant.org,
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The great thing about the special relativity course is that it starts with the history
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So before getting into advanced concepts like energy and dynamics, it starts with just trying
to build an intuition for the concept by reviewing the old concept of the luminiferous aether,
the propagation of light, and relative motion of objects.
As usual, Brilliant makes it fun to learn by using graphics, interactive quizzes and
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What's your idea of spacetime?
Let me know in the comments.
I’ll see you in the next video my friend.