**What does it run on, Doc? Critical density!**

*From astronomy.com*

**Keywords: **Big Crunch, Special Relativity, General Relativity, time dilation, cyclic universe

Welcome to this blog post! Here we will see how time travel could be possible at all.

“We all have time machines of our own. Those that take us back are memories, and those that carry us forward are dreams.”-Über Morlock inThe Time Machine

To me also, we all have time machines of our own. But to those BTTF fanatics who are currently yammering at this screen, “WHERE IS MY DELOREAN? WHERE IS MY DINOSAUR SAFARI? WHERE????” I reply that first we look to the future, then we dwell on the past.

**To the Future!**

In Einstein’s theories of Relativity, the theories of Special Relativity and General Relativity discuss how ultra-high speeds and proximity to matter affect the speed of time respectively in a certain reference frame. First, let’s look at Special Relativity.

**Special Relativity.**

First, let’s set some ground rules:

1. Nobody can ascertain whether they are stationary or in motion. In other words, the rules and fundamental constants (speed of light, gravitational coefficient etc.) of physics to you would be the same whether you are in a train at 100 mph or sitting down on a rock.

2. The speed of light is therefore measured to be the same in all reference frames: 300000000 m/s (299792458 m/s actually).

Why must this second rule be true?

Well, if a stationary person measures the speed of a light-beam on a train (the train is moving at 30 m/s) to be 299792458 m/s (assuming that he has remarkably amazing equipment), and the speed of light was not measured as the same in all reference frames, a person on the train could measure the speed of the light-beam to be **299792428** m/s and could deduce that he was travelling at 30 m/s. But this cannot be true! No-one should be able to discern their actual speed by observing changes in the fundamental constants of physics! Great Scott! Aahh!

To stop this calamity, we say that lightspeed (denoted as *c*) must measure the same in all reference frames.

What is a reference frame?

The wikipedia definition is: A frame of reference consists of an abstract coordinate system and the set of physical reference points that uniquely fix the coordinate system and standardize measurements within that frame... *YAWN* <<SNORE>> <<<VERYBIGSNORE>>>, Whaat? Whawassat? In Non-Scientific English it could be defined simply as a point of view where one considers themselves stationary and the surroundings (or all other reference frames) to be in motion.

For example, a person sitting on the grass and a Londoner travelling on the Eurostar at 300mph have different reference frames. To the person on the grass, the earth is stationary, and the train is in motion. To the Londoner on the train, the train is stationary, and the world is whizzing past him.

This might seem to be a silly assumption – why would the earth be moving around him? My double comeback is that a) we are all kind of in our own “personality” reference frames (If two people agree, they will say that they are both right, but if they disagree, each of them will say that they themselves are right and the other is wrong) and b) it is also quite absurd to say that the Earth is stationary.

Because in absolute terms, the Earth is moving in its orbit around the Sun at 30km/s, the Sun is orbiting around the Supermassive Black Hole, Sagittarius A, at 20 km/s and the Milky Way itself is moving at 230km/s. That means that the person on the grass might be, on average, travelling at 230km/s!!! This is quite useless in physics terms, so we talk about relative motion instead (the car is moving 50 mph relative to the ground, I am moving at 500 m/s relative to you etc.).

N.B. Traffic cops often say “You were speeding at 70 mph” instead of “You were speeding at 70 mph relative to the ground” because they are just used to taking the earth as the reference frame in that instance. Even with your best intentions, do **NOT** correct them. Don't speed anyway.

Okay, now onto light.

If a ship with a certain velocity emits a light pulse to reflect off a mirror and to return (straight up and down to a person on the ship), the light path would be (to a stationary observer):

Note that the light pulse has to travel a longer distance than one on a still one. This means that the time recorded as 1 second is more than 1 second to the stationary observer (s.o.). But by how much?

To do this, I am going to decompose a light pulse.

If T seconds are counted in the ship’s reference frame, t seconds are counted by the s.o., c is the speed of light and v is the velocity of the ship:

cT = ctsin(theta)

ct = cT/sin(theta)

t = T/sin(theta)

t = T/sqrt(1-cos^2(theta))

t = T/sqrt(1-(vt/ct)^2)

t = T/sqrt(1-(v/c)^2)

This is my own proof for Einstein’s equation of time dilation (slowing down of time).

What does this equation mean?

So, 1 second in a vehicle with a speed v would be 1/sqrt(1-(v/c)^2) seconds.

It also means that for everyday speeds (1 m/s), the effect of time dilation is negligible but for significant fractions of the speed of light, the 'slowdown' factor slowly creeps up:

*A graph of the 'slowdown' factor from desmos.com*

Here, 1 unit on the x axis is the speed of light. Imagine where our everyday speeds of 1 m/s would be on the graph (#Hint: it would be very close to the y-axis, and so the slowdown factor would be relatively (#joke) close to 1). In the equation:

t = T/sqrt(1-(v/c)^2)

v/c is incredibly small and so (v/c)^2 is almost 0

t = T/sqrt(1 - almost 0)

t = T/sqrt(almost 1)

t = T/almost 1

t = ~ T

The level of approximation here is higher than that of the refractive index of air (actual: 1.0003, approximation: 1)

Now imagine what value 0.5 c would have:

t = T/sqrt(1-(v/c)^2)

t = T/sqrt(1-(0.5 c/c)^2)

t = T/sqrt(1-(1/2)^2)

t = T/sqrt(1-1/4)

t = T/sqrt(3/4)

t = T/(sqrt(3)/2)

t = (T * 2) / sqrt(3)

t = T * 1.155 (4 s.f.)

This means for 1 second in a ship at half lightspeed relative to s.o. X, 1.155 seconds pass in X’s reference frame. Not so different from 1 itself but at least not as close as everyday speeds.

What about 0.75 c?

t = T/sqrt(1-(3/4)^2)

t = T/sqrt(1-9/16)

t = T/sqrt(7/16)

t = T/(sqrt(7)/4)

t = (T * 4) / sqrt(7)

t = T * 1.512 (4 s.f.)

For 7/8 c:

t = T/sqrt(1-(7/8)^2)

t = T/sqrt(1-49/64)

t = T/sqrt(15/64)

t = T/(sqrt(15)/8)

t = (T * 8) / sqrt(15)

t = T * 2.066 (4 s.f.)

For 15/16 c:

t = T/sqrt(1-(15/16)^2)

t = (T * 16)/sqrt(31)

t = T * 2.874 (4 s.f.)

We can see that even if the speed is converging to c, the slowdown factor is increasing rapidly. What happens at c itself? I will probably have pi more years in maths prison, but here goes:

t = T/sqrt(1-(c/c)^2)

t = T/sqrt(1-(1)^2)

t = T/sqrt(1-1)

t = T/sqrt(0)

t = T * ♾

Whaa? This means that if 1 second passes in X’s reference frame, 0 seconds would pass in the ship’s reference frame. This means that time stops in the ship’s reference frame! This seems either preposterous or exciting, but sadly it is impossible since the momentum of something at the speed of light would be infinite and so an infinite energy would be required to accelerate the ship to lightspeed. Faster than light, well, that would have an *imaginary* slowdown factor. This is too weird for the universe, as we know it, to allow.

But for speeds *close* to the speed of light, the slowdown factor would be huge. If the ship travels fast enough, a whole century in X’s time could be squashed into one second in the ship’s time.

The exact speed would be:

t = T/sqrt(1-(v/c)^2)

t^2 = T^2/1-(v/c)^2

1-(v/c)^2 = (T/t)^2

1-(T/t)^2 = (v/c)^2

v = c*sqrt(1-(T/t)^2)

Therefore:

v = 299792458*sqrt(1-(365.25*100*24*3600)^-2) =299792457.9999.... m/s

But also, if we go close to highly dense objects (like black holes), time slows down also. If one went on the event horizon of a black hole, time would stop. Therefore, by travelling at high enough speeds around highly dense objects we can create a high enough slowdown factor and travel years into the future in what seems an instant to us. This is exactly what happens in the film *Interstellar*, where Joseph Cooper (Matthew McConaughey) speeds forward 76 years by travelling at 0.55c around Gargantua (a black hole) in what seems like a few hours to him.

Okay, so we understand how to travel into the future. What about the past?

**The Ghost of Christmas ... is he knocking on the wormhole "door"?**

So let’s cut to the chase: To travel back in time, you need to go faster than light, which as we said before, can’t be done (at least we don't think so, yet). A popular theory suggests that wormholes, 4D tunnels that transport matter through space almost instantly, faster than light, might be the answer. This is a good theory since it allows travel faster than light. The Achilles’ heel of this, though, is that one could not travel to before the wormhole was built, as it did not exist then.

My personal theory is somewhat dependent on the structure of the universe itself and here it is:

We do not know the fate of the universe. But we do know that the universe has three possible fates: a) A *Big Crunch* where the universe’s internal gravitational forces overpower the dark energy that pushes the universe out and so the universe implodes and shrinks to a singularity just before it explodes in a *Big Bang* again; b) A *Big Rip* where the universe’s internal gravitational forces are overpowered by the dark energy and so the universe keeps on expanding over time at an ever-increasing rate; c) A *Big Nothing* where the gravity cancels the effect of dark energy and so the universe expands but at a constant rate.

But what determines the gravity in the universe? Well, that is the density of the universe. If the density of the universe is greater than a critical density (Ω), we will get a Big Crunch. Otherwise, we will get a Big Rip or a Big Nothing.

What has this got to do with time travel?

Well, if the universe is cyclic (Big Bang **-->** Big Crunch **-->** Big Bang **-->** Big Crunch …) then we only have to keep going into the future to return back to the past. How is this? Well, imagine 2 eskimos. One of them stays at the North Pole whilst the other walks on the Prime Meridian and then the International Date Line. Even without changing his course by one degree, he ends up back at the North Pole, to the surprise of the eskimo who stayed. This is because the Earth is curved and is *spatially cyclic*. Because the eskimo walked around on the circle on the diameter of the sphere, he was bound to get back to where he started. Similarly, if the universe is cyclic, we can travel from 2021 to the future, to the Big Crunch and simultaneously to the Big Bang (year 0 of the universe) and then back to 2021. But instead of fully completing the journey, we could stop at 1985 or 1990 or 0 BC. So, in effect, we have travelled to the past. But all of that is dependent on the universe’s density. And that is dependent on the Hubble Constant (the constant that determines the separation of objects in the universe). Despite the name, Hubble’s Constant changes over time! However, if we take the average of the Hubble Constant over all time, we can ascertain whether the universe is cyclic or not.

But for now, until we have figured all this out, I think we should not mind staying put and try to enhance the present!

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