Time travel is nothing special. You’re time traveling right now…into the future, one
tick of the clock at a time. But relativity theory shows that the time that ticks for
you is not absolute. The flow of time is relative, and it can be molded and changed. It turns
out that higher gravity and higher speed can slow time down when compared to a reference
frame that has lower gravity or lower speed. This really does happen, and has been proven
in many experiments. So for example, when scientists placed one atomic clock at sea
level and another synchronized atomic clock on the top of a mountain, they found that
the clock at sea level ticks slightly slower than the one on the mountain. Similarly, atomic
clocks placed on a fast moving jet were found to tick slightly slower than synchronized
clocks at rest on the ground. So the idea of traveling to the future is
not exotic at all, you are doing that right now. And if you could speed yourself up to
a significant fraction of the speed of light, or put yourself near an extreme gravitational
source, you could slow down time enough to travel decades into the future earth in a
matter of months or weeks or less. This was depicted in the movie Interstellar for example.
But, the question is, how would you come back to the time you started with. It’s no fun
going to the future, if you can never come back? But to do this, you would have to
travel back in time. This is not so straight forward. Is it theoretically possible?
Let’s take a look…coming up right now!
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to click the link in the description. In order to better understand time travel,
it’s important to understand the concept of light cones. We discussed this in a prior
video, but let’s do a quick review. Spacetime consists of 4 dimensions, 3 dimensions
of space, and one dimension of time. But in order to visualize it in our 3 dimensional
brain, we eliminate one of the spatial dimensions and create a 3D-looking graph. The horizontal
axes represent movement in space, and the vertical axis represents movement in time.
In order to have consistent units on all the axes, we convert the time axis to length by
multiplying time by the speed of light – C times t.
This results in representing time in terms of lengths. For example, one meter of time
is about one three hundred millionth of a second, the time it takes light to travel
one meter. All points on this graph are called events.
So using this graph, an object not moving spatially will create a line going straight
up because it will be moving in time only. This is called the object’s world line.
An object moving in one or more of the spatial dimensions will create a diagonal line because
it will be moving both in time and space. If we flash a light, this point will be an
event. We can call it event A, and it will create a cone because the light will propagate
in both spatial dimensions and the time dimension. The volume of the cone represents all future
events that can be causally connected to event A. Event A also has a past light cone which
represent all events in the past that are causally connected to event A. Points outside
the cone are events that are causally disconnected from the event A.
In other words, they can't have en effect on Event A.
The coordinates are made so that anything moving at the speed of light will always create
a 45 degree angle from vertical. So the world line of any moving object will always be somewhere
within or on the cone. Any world line moving inside the light cone
is called "time-like." A particle moving at the speed of light would move at a 45-degree
angle between the time and space component. A world line like this is called "light-like"
because it is how light moves. Now what happens if an object has a world
line close to the space component? We get what’s called "space-like" world line.
Since this is outside the light cone, it would mean that the object is traveling faster than
the speed of light. In principle there isn’t any problem with such world lines, as long
as no information is transferred, because this would ruin causality as discussed in
the previous video. But as we’ve discussed before, moving anything
to the speed of light would require an infinite amount of energy to accelerate. And this is
not possible. In addition, moving an object close to the
speed of light will alter time, in that it will be slower compared to a stationary object.
But if we want to do really complex stuff like traveling back in time, then we would have
to not only slow time down, but make it go in the opposite direction. How can we do that?
Is it even possible? As we showed in the prior video, going beyond
the speed of light can create scenarios that allow you to travel back in time. But since
this is not physically possible, we need to figure out some other clever manipulation
of space time. And anything that involves manipulations of space and time is going to
be described by the Einstein’s equations of General relativity.
The simplest spacetime is a flat spacetime. The equation for such a spacetime looks like
this: where “ds” describes how a line moves or evolves in this spacetime.
And all the terms with "d" and some component squared tell us what happens to these components
of space time at different coordinates - dt describes the time component, and x, y, and
z describes the 3 spatial components. This is expressed in Cartesian coordinates,
but the exact same equation can be expressed with spherical coordinates like this:
Now we have a metric in terms of the radius from the center, r, and the two angles theta
and phi – as shown in this graphic.
But to have any hope for traveling back in time we need to come up with something much more
exotic than just flat spacetime like we have here. It turns out that the first solution ever
presented to Einstein’s field equations was done by Karl Schwarzschild. Essentially
he formulated a simple non-flat spacetime. And it happened to describe a black hole,
at a time when no one had ever heard of it. The first dt squared component describes what
happens at different times, the next dr squared component describes what happens at different
radii. The third component does not change compared to flat spacetime because a black
hole is spherically symmetrical. The r_s term represents the smallest possible
radius of an object where light can still escape. It is called the Schwarzschild radius.
It is the point beyond which nothing can escape the black hole, because in order to escape,
you would have to go faster than the speed of light, which you can't do. It’s like
an information barrier between the inside and outside of the black hole.
The important thing to note in this equation is that when r is equal to r_s in the dr squared
term, we get a zero in the denominator. This makes the term undefinable. Its physical
representation and meaning is the event horizon,
The zero in the denominator.
signifies the point of no escape.
Let’s now look at what happens to the light cone of an object falling into a black hole.
If the object is far away, then it’s not falling anywhere and it’s just like an upright
static light cone. As it starts falling into the black hole, we would see the light cone
starting to tilt. The reason it tilts is because gravity is a warping of spacetime.
The strong gravitation of the black hole attracts light and other objects towards it. Future world lines
point increasingly towards the black hole. Only world lines moving away from the black hole
at higher and higher speeds will escape it, as objects get closer to it.
And exactly at the event horizon
the light cone lies tilted between the time and space axes at 45 degrees. All future events
point to inside of the event horizon. This means that just prior to entering the event
horizon, a particle could escape away by traveling close to the speed of light. However, once
it enters the event horizon, there is no timeline which leads to outside of the event horizon.
There is no escape even at the speed of light. The light cone of the object will no longer
have any future world line to the outside, and it thus loses connection to anything outside
the black hole. The light cone will continue to tilt inside
the black hole as it falls towards the singularity. The singularity at the center of the black hole represents
a point of infinite mass density, but a finite mass. Eventually the light cone will point completely
towards the singularity at the center. This means that all future events will lie at the
singularity, there is no escaping it. The singularity is a future moment in time rather
than a point in space. All future world lines points to the singularity at the center.
It turns out that the spacetime inside black holes has the potential to allow travel back
in time. I’ll show how in a minute. But the important thing to remember is that even
if we can go back in time inside the black hole, the event horizon prevents us from escaping
the black hole. So even if we can travel back in time, what good is it we are trapped inside
the black hole, not being able to escape into our time in the real universe. So we have
to figure out a way to get rid of the event horizon. And it turns out there is a way.
In 1965 the Kerr-Newman metric was described by Ezra Newman. This metric is even more complicated,
because it describes a rotating black hole. We can write the metric of this space time
as shown here. The angles theta and phi are the same as in the Schwarzschild metric.
- “a” is angular momentum. Rho is a function of the radius,
angular momentum, and the angle theta. The delta is a function of the radius, the Schwarzschild
radius, angular momentum, and Newton’s constant of gravity. Q is the charge – black
holes can have a charge. There are ways we can remove the event horizon
in this metric. What we have to do is make sure that the delta on the dr squared term
does not equal zero, similar to what it does in the Schwarzschild metric, signifying the
event horizon. In other words, if we can avoid getting a zero in the dr squared term, there
will be no event horizon. It turns out, when you do the math, that if
the angular momentum divided by the mass of the black hole, times speed of light is greater
than half the Schwarzschild radius, then delta can never equal zero. In other words, if the
black hole is spinning fast enough, the event horizon disappears. It is then no longer a
black hole, but a naked singularity. A naked singularity, is just a singularity with no
event horizon. Now why is this important? Well the reason
this is important is because when you don’t have an event horizon, you can go near the
singularity in the center, but come right back out. There is no event horizon, that
otherwise prevents you from coming back out of a black hole.
And what we can do by going near a singularity and coming back out is theoretically travel
back in time. How does this happen? Well, we can construct what’s called a "closed
time-like curve," which allows world lines from the future cone to loop around into the
past light cone. Let’s look more closely at how that happens.
If you get close enough to the singularity, the spinning singularity warps the space time
around it such that we can have world lines that loop around the singularity in such a
way that the future world line ends up in the past light cone.
Looked another way, we can loop our light cone around the singularity such that our
future light cone ends up in the past light cone of where you started. And now since we
are not bound inside the black hole by the boundary of the event horizon, we can come
out of this spacetime back to about where we started, but at a time BEFORE we started.
We went back in time. This is a real solution to Einstein’s field equations.
So now you might think, this is too good to be true, what's is the catch? Well, there are
a few problems, first of all just because something works in the theory of general relativity
doesn’t mean it works in reality. There’s good reason to believe that black
holes can’t physically rotate fast enough for the event horizon to disappear.
So scientists don’t think naked singularities can exist. In addition, the solution shown
here just considers a test particle with little to no gravitation, and not a human. If you
tried the mathematics with a human, the higher mass and gravity does not seem to work the
same way. So this concept of going close to a naked
singularity and traveling back in time, may not be possible. But I’ll have you know
that no concrete proof of this has been made. So what am I saying?
I’m saying there might still be a chance!
I’ll see you in the next video my friend.