If you wanted to meet a friend at any possible location on earth, you would need to give
them a precise longitude, latitude, and altitude. Would this be enough information for two people
to meet? Well, if you showed up at that location, there is no guarantee that you would meet
your friend. Why? because you did not give him one more crucial piece of information
- the time. He or she would have to know what time they were to meet you at that precise
location.
These are the 4 dimensions of the universe – three spatial dimensions and one time
dimension. With these 4 coordinates, you could rendezvous with anyone anywhere in the universe.
In fact these 4 dimensions can describe any event in the universe.
But how did time become a dimension. It seems like an odd thing because time is not measured
in meters or inches, like the other spatial dimensions. It is measured in minutes and
seconds. It doesn’t seem to fit with the other three.
Yet it appears to be a fundamental component of the universe in which we live, as fundamental
as the other spatial dimensions. In fact, time is an inseparable component of space.
That’s why physicists refer to the geometry of the universe, not as space, but as spacetime.
How did the idea of time as a dimension come about? How can we best visualize these 4 dimensions?
And what really happens when space and time start doing seemingly weird things when two
objects move relative to each other?
That’s coming up right now…
In the late 1800’s, scientists had recognized that there was an inconsistency between two
theories – Newton’s laws of motion, and Maxwell’s equations describing electricity
and magnetism. The problem was the speed of light.
Maxwell had shown that light was a propagating wave – a disturbance in the electromagnetic
field - a changing electric field continually giving rise to a changing magnetic field which
in turn gives rise to a changing electric field again, leading to a self-propagating
electromagnetic wave.
And his theory predicted what the speed of this wave is, so it predicted the speed of
light, which is about 300,000 km/s or 300 million meters per second.
The question was what would be the measured speed of light, if the person measuring it was
moving, for example, a person moving with the spin of the earth. What would the speed
of light be then?
According to Newton, this moving observer should measure a different speed, than someone
who was not moving. The measured speed should be the speed of the person, PLUS the speed
of light.
But Maxwell’s equations seemed to indicate that light has just one speed, c. There was
no accounting for a speed of light that was any different than what his equation predicted.
Almost every scientist at the time, including Maxwell himself, presumed that Newton was
correct, and that Maxwell’s equations were incomplete and needed to be modified.
In 1887, two American scientists by the name of Michelson and Morley devised a highly sensitive
experiment that could measure the speed of light in the direction of motion of the earth.
So the expectation was that the speed would be measured to be c plus the speed of rotation
of the earth.
What they found shocked scientists of the time, because it was totally unexpected. They
found that the speed of light does not vary one iota, due to the motion of the earth.
This is one of the most consequential experiments of that last 200 years.
A resolution of this unexpected result remained a mystery for over a decade. Henri Poincare
came close to solving it,
but it was not until Albert Einstein came along and proposed a pretty radical idea at
the time, that we got a full resolution. Einstein suggested that it was not Maxwell’s equations
that needed to be changed, but Newton’s near-sacred laws of motion.
He determined that the speed of light does not change in any reference frame, and worked
out the crazy implications of this idea.
Einstein showed that observers moving at different speeds will disagree about the distance and time
between two events. In other words, they will experience space and time differently. This
is the meaning of the “Relative” in Relativity theory. It means that measurements of space
and time are relative to the person measuring them.
But the key to understanding this theory is that some things do not change. Namely, the
laws of physics, and the speed of light remain the same for everyone.
There is very important idea that came out of Einstein’s theory, that Einstein himself
did not first come up with. It came from German mathematician Hermann Minkowski, who was Einstein’s
former professor. He was astonished by his former student’s theory and thought about
it deeply, and realized that relativity is really a theory about the geometrical relationship
between space and time.
He famously said in 1908: “Henceforth, space by itself and time by itself are doomed to
fade away into mere shadows, and only a kind of union of the two will preserve an independent
Reality.” This union between space and time is what he called “spacetime.”
So we say there are 4 dimensions to specify an event in the universe, three dimensions
of space and one dimension of time. The problem is how the heck do you visualize these 4 dimensions.
This is not easy for our three dimensional brains to do. So to make it a little more visually
intuitive, we simplify it to 3 dimensions. Two spatial dimensions and one time dimension.
So we eliminate one of the spatial dimensions for simplicity. So what we get is a 3D looking
graph, called a spacetime diagram, where time is depicted in the vertical axis, and the
two dimensions of space are depicted on the horizontal axes.
The only problem is that time has completely different units than space.
So how do we put it on one diagram in a way that is consistent with the other coordinates.
Minkowski suggested that we can depict time as a length. How do we do this? We simply multiply
time by the speed of light.
If the speed of light is 300 million meters per second, then we can say that 300 millionths
of a second is one meter. It’s just the amount of time that it takes light to travel
one meter. So the time dimension can be expressed in terms of the speed of light times time, or
ct. So the vertical axis is labeled ct, which is a length. We can now we can have 3 consistent
coordinates.
Let’s look at how something would be depicted on this graph. A non-moving particle would
be depicted as a vertical line because it would not be moving in any of the spatial
dimensions, but it would be moving forward in time. Mikowski called this a world line
for the particle.
A uniformly moving point would be depicted as a diagonal line on this graph because it
would be moving in at least one of the spatial coordinates as it is moving forward in time.
An accelerating particle would be a curved line.
Now, what would happen if we flashed a light somewhere in this 2D space. Let’s call this
event A. The light would spread in all directions with time. This forms the shape of a cone.
So Minkowski called this a light cone.
Now if a person was standing still and observing the light. There would be a point on this
cone where his world line would intersect with the light cone. That would be an event,
which we can call event B. So the light cone really represents all the future events in
spacetime that the light reaches from its initial event A.
Similarly, a light cone can be built for any event A in spacetime to show all other future
events that are reached by event A.
In diagrams like this, you will sometimes see an upside down cone. This is the past
light cone, and represents all the past events in spacetime that reach Event A.
Event A can be you here and now. If you were in space at any moment in time, then all the
starlight reaching you from all directions would be the past light cone of events reaching
you at a particular moment of time, the here and now. And the further out you look, the
further back in time you would be looking as well.
The points outside these two light cones are causally disconnected from event A, meaning
they cannot reach or be reached by event A.
Through the ideas of Events, world lines and light cones, Minkowski proposed that Einstein’s
relativity is really about geometry - the geometry of spacetime. Believe it or not,
Einstein himself was not keen on this idea initially. He came around later and realized
his old professor was indeed on the right track.
At this point you might be asking, ok, this if fine. I kind of get it. But how does relativity
enter into the picture? How is the idea of different observers in different reference
frames, moving at different speeds, affect this light cone?
So let’s look again at our light cone and see what happens when the light is not moving
from the perspective of a static observer. If I place two light detectors an equal distance
apart from the source of the light, then these detectors will detect the light simultaneously.
This is true as long as the objects are not moving. The world lines of the two detectors
and the light source are perfectly vertical because they are not moving.
However, If the objects are moving from the perspective of the static observer, and we
perform the same experiment, then the detection events are no longer simultaneous. This is
because the light reaches the two detectors at different times. One detector is moving
towards the light, so it detects the light first. The second detector is moving away
from the light, so it detects the light later.
Note that the light cone remains the same because the speed of light does not change.
However, the world lines of these three objects change from the perspective of the static
observer. They are tilted towards the direction of movement.
Since one detector is moving closer to the light, it’s detection event will be lower
on its worldline than the detector that is moving away from the light, so its detection
event will be higher on its world line.
Now here is the crucial part, what I have shown you is from the perspective of an observer,
Observer A, who is watching from a distance does not move in any of the reference frames.
However, from the perspective of the objects, things are quite different.
In other words, if there was an observer B, attached to the objects, such that he was
not moving with respect to the objects in any of the reference frames, he would see
simultaneous detection by the two light detectors, even though he’s moving with respect to
the first observer, observer A.
From observer B’s perspective, he and the objects are static, and it’s observer A
that is moving to the left. So the light cone and world lines that Observer B experiences
will be straight up, identical to the one seen by observer A when he was not moving
with respect to the objects, that is, he sees simultaneous detection.
Observer A, however sees the tilted world lines and no simultaneity.
So here’s the seeming paradox, how can Observer A and Observer B experience the same event
in the same spacetime, but have two completely different conclusions. That is, how is it
that Observer A sees no simultaneity, but Observer B sees simultaneity? Who is right?
Well, they are both right.
If observer A is a physicist, he’ll explain that from his perspective, Observer B’s
light cone was tilted such that he experienced simultaneity.
Observer B, on the other hand, also being a physicist, will say that the reason Observer
A did not see simultaneity was because his light cone was tilted such that the two detection
events happened at different times. Neither is incorrect.
So what this shows is that simultaneity is relative. It is relative to the observer.
There is no absolute simultaneity in the universe.
But each observer sees and experiences exactly the same spacetime. The events are the same.
So things can be relative but they are all part of one spacetime. These observers divide
space and time in different ways depending on their frame of reference. But ultimately
it is one universe, and although simultaneity is relative, they will both agree on causality.
Causality is preserved. I'll have details on why that's the case in an upcoming video.
For now, I wanted to give you an overview of what light cones and world lines are all
about, so you can refer to this video to understand the basics. In a future video, I will show
you what happens to these light cones are in extreme conditions, such as infinite speed of light,
inside a black hole, through a worm hole, and how one could build a potential time traveling
machine. So stay tuned for that.
In the meantime, if you want to study this topic in greater detail, there’s a wonderful
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Brilliant has a special offer right now for Arvin Ash viewers right now. If you are among the first
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And if you have a question, leave it in the comments because I try to answer all of them.
Be sure to check out my other videos on this topic, to which you will see a link shortly.
I will see you in the next video my friend.