Category Archives: Sci-Fi Science

Articles on real science in science fiction TV shows and movies.

Time Loops and the Paradoxes of Continuum’s Time Travel Physics

The season finale to Continuum leaves the viewer with lots of surprises and questions that are sure to keep them engaged until Season Three begins in 2014. While Season Two doesn’t specifically answer questions regarding the rules to time travel used in the show, it does provide some hints and insights on what might be happening. If anything we can be certain that the rules are going to be as complex and interesting as the plot so far. To get up to speed you should read my first article, “The Wibbly-Wobbly of Continuum“.

Future Alec sends everyone back

We learn at the end of Season One and the start of Season Two that the future Alec (William B. Davis) is responsible for sending everyone back in time. It also turns out that there is a specific reason for sending Kiera back – to either prevent the future from happening or to prevent him from going down a certain path. It is apparent that future Alec regrets many of the things that have taken place when he tells his son, Jason (Ian Tracey), “Liber8 wouldn’t exist if it weren’t for me. Perhaps they are just a manifestation of my conscience.”

But Kiera’s presence isn’t without problems. Escher/Marc Sadler (Hugh Dillon) berates Kiera when he tells her

“You carry destruction in your wake, you’re the time bomb.”

Given that Escher is a former Freelancer, he may have some insight from his former occupation. Warren (Adrian Holmes), one of the enigmatic “Freelancers”, also hints that Kiera’s presence is an “anomaly” and a “glitch in the continuum” and it appears that Warren was ready to kill Kiera to “fix” things.

This raises several questions. If Kiera is a problem, why was she sent back in the first place? Did future Alec know of the trouble she would cause when he sent her back? In Season One, we witness the the “first” meeting between the future Alec and Kiera and it seems that he knew who she was. So while this was a first time meeting for Kiera, it certainly isn’t for Alec.

We Know Things can change

Freelancers and Continuum

Kiera meets with fellow time traveler Jason, who informs her that he’s being followed by other time travelers from the future he calls “Freelancers.”


One of the mysteries behind the show’s time travel physics is whether there is a single timeline or multiple ones. If there is a single timeline, for events to happen as they are supposed to, they must follow the Novikov self-consistency principle. This principle asserts if an event exists that would give rise to a paradox, or to any “change” to the past whatsoever, then the probability of that event is zero.

This means that for the events that leads up to Liber8’s trip back in time, then all the events in the past need to lead up to that one event. Alec can’t decide not to send Liber8 back in time as doing so ironically creates the corporate run future. In short, this makes it impossible to create time paradoxes which is different from a multiple timeline scenario where anything can happen. In this case, when Kiera and Liber8 arrive in 2012, they create a new timeline that has no impact on the timeline they came from.

In “Second Thoughts” (Season 2, Episode 3), Jason tells Kiera about the Freelancers and that Escher is also one of them. It’s not clear if Jason knows that Escher is his grand-father but it is likely. He seems to think that Escher is dangerous and not a person to be trusted. Jason also tells Kiera that:

“You know, you won’t be the same person when you left”.

This is one of the clues that might be telling us that we are dealing with a single timeline. If a person was to return to the 2077, they might take the place of their “double” and that person’s memories will eventually be replaced by new memories of the new timeline as if the old one never happened.

This gives credibility to the “time loop” theory of a single timeline. But the implication of a single timeline has always been that for the events that lead Liber8 to arrive in 2012, then everything must lead up to them being sent back in time in 2077. Jason could be saying this doesn’t have to be the case; everyone doesn’t have to follow a particular script.

The fact that things can be changed becomes all the more clear in “Second Truths” (Season 2, Episode 6). Kiera solves and stops a serial killer whose case file was unsolved in 2077 in future. Using information gleaned from the future is an example of the Bootstrap Paradox where information sent back from the future becomes the very information that was brought back in the first place.

The Bootstrap Paradox is problematic as it implies that the information was never created. It exists specifically because the loop occurs. This episode is important because it establishes one thing — paradoxes are possible in the Continuum Universe. This means that we don’t have to stick to Novikov even if we are dealing with a single timeline.

As her memories remain unchanged upon solving the case, then according to Jason, she won’t experience any changes to her memories until she returns to 2077. The events of her examining the unsolved cold case in the future have been wiped out and no longer exist. We don’t know what events have replaced of that day. All we know is that it didn’t happen. This has some profound repercussion for Kellog.

What this means if or when she returns, remains to be seen. The multiple timeline scenario posed one problem. If 2077 Alec sends Liber8 back in time, he can never achieve his plan to stop the dark path the world has taken as he will only change the history of an alternate Universe.

The time loop scenario poses a similar problem. If all the events must happen in a way that must lead up to Liber8’s “execution”, then there is no way Alec can change the past. Allowing paradoxes to be a part of the show’s time travel physics solves that problem and allows Alec to hatch the plan he needs to save the world.

Garza kidnaps a young Alec

In “Second Wave” (Season 2, Episode 10), Jasmine Garza(Luvia Petersen) kidnaps young Alec on the orders of his future self. It seems this was future Alec’s contingency plan in case the corporate controlled future appears imminent. Unfortunately, it seems that in an act of desperation and frustration, Garza jumped the gun and acted prematurely.

Continuum Time Travel Alec and Jason

In 2077, the older Alec tells his son Jason of his plans to send Liber8 back in time to stop the corporate controlled future.


It seems that this isn’t the only plan the future Alec has put in motion. In the season finale, “Second Time”, we learn that Jason is his son and not his father as previously believed. In this scene, future Alec says:

“Because you will inherit my failure if I don’t succeed and I don’t wish that on anyone”

Could it be that he means that if he fails and is killed, Jason will be wiped out of existence when he returns to the future?

It also seems that a time traveler won’t feel the effects of his actions until he returns to the future timeline. Jason tells Kiera that:

One day, you’re going to wake up and wonder if any of it really happened.

This may mean that once they return to the future, they not only replace their future selves but eventually take on the experiences and memories of that time line as if they never went back in time.

This provides some insights to the “A Test of Time” (Season 1, Episode 5) episode when the young girl believed to be Kellog’s (Stephen Lobo) grandmother is killed by Travis. Whether this girl is really Kellog’s grandmother or not is never answered but we always assumed that as Kellog didn’t pop out of existence then Kellog was mistaken. It turns out that if Maddie (Olivia Ryan Stern) really is Kellog’s grandmother, he may pop out of existence upon his return to the future.

This also has some repercussions for Kiera herself. In the last few episodes of Season 2, Kiera has become almost obsessed with returning to 2077 and to be with her family despite Alec explaining that so many changes have happened that the future she knows may not even exist. Her best bet at ensuring the existence of her family is to stay in 2013 and see her predicament through.

What happens in the future doesn’t stay there

It would seem that whether there is one universal time-line or multiple ones lies somewhere in the middle. Escher calling Kiera the “time bomb” and Warren referring to her as a “glitch” indicates this. It seems that Kiera’s very presence is problematic.

Kiera’s presence also makes you wonder who the Freelancers are and their true goals and mission. According to Jason, humanity’s history has been guided by this enigmatic group as they seed the past with the technology that enables mankind to make the next leap froward; another example of the Bootstrap Paradox. When Warren describes his mission to Jason, we are lead to believe that the Freelancers might be the good guys — the equivalent of time cops.

But Jason believes this group is bad and might comprise individuals who manipulate history for their own benefit. It’s not clear whether Jason knew of the Freelancers before his meeting with Warren in 2077 or he picked up information about the group when he arrived in 1990. We were lead to believe that Jason’s arrival in 1990 was accidental but could his father have sent him back to investigate the circumstances of his birth and find out more about the Freelancers? Could he have discovered the existence of rogue Freelancers like Escher in the process and hence the reason why he believes his grand-father is dangerous?

As future Alec knows of the Freelancers, this could mean that this isn’t the first time any of this is happening. Not only does he know of the Freelancers from past experience but it may have happened several times in the space-time continuum — he just doesn’t “remember” it. Escher may have gone back in time to start Pyron, thereby creating the catalyst that will lead to the eventual creating of Sadtech and the dystopian corporate controlled future. This also leads young Alec to create the time-travel device and start the “family business”.

As the Freelancers are known to “interfere” in humanity’s development, it could mean that the Freelancers meant for the corporate controlled future to happen. This isn’t without consequences. Alec, realizing the horrible mistake he has made, sends Liber8 back in time to prevent this. But if the future is changed by events in the past, how would he know he has succeeded or that he even made an attempt in the future? The best way is to send a message in a bottle and this is where Kiera come in. He uses Kiera’s Mark 4 polymeric nano-composite body armor to save that message and this is where the time loop theory becomes intriguing. If this has happened before then he can send the information he needs to change the future. He can send his past and future selves the information they need, telling them what worked and what didn’t.

Time Travel in Continuum

Kiera is held prisoner in Continuum

In an unknown location and time, Kiera is held prisoner by the Freelancers with her fellow time travelers, their fates uncertain.


It seems that Contonuum’s time travel physics utilizes a single timeline. Travel back in time creates a new timeline that wipes out the old one. In several episodes, we see Kiera being dragged to a futuristic holding cell. At first, we are lead to believe that this might be a hallucinatory effect of future Alec uploading a file into Kiera’s memory but it seems that these hallucinations are very real and might be a Freelancer controlled prison.

We are never told where or even when this prison cell is. All we know is that the Freelancers are kidnapping other time-travelers. What they do with them after they are captured is anyone’s guess. If they return them to their proper place in the time line, could Kiera’s nightmares be a result of her remembering her traumatic capture from a previous time line? Or maybe she is remembering some event that is supposed to happen. The revived Curtis Chen (Terry Chen) offers no clues.

In the season’s finale, Warren and Curtis talk about “this timeline”. This indicates that the events with Liber8 and Kiera have happened several times. Exactly what the Freelancers do with their prisoners is unanswered. Do they “reinsert” them or do they keep them captive? Can we expect several prison cells all holding different versions of everyone? Are these doppelgangers connected in any way?

Fans of the show know that there is nothing easy when it comes to understanding the show. The seemingly basic premise of the “evil” Liber8 fighting to stop the distopian future hides a much deeper plot. By allowing paradoxes to be a part of the show’s physics, it appears that a much deeper plot exists — a battle between Alec and the Freelancers and where their battle is a game of chess played across space and time. The question is, are everyone just pawns or is there a bigger eventual battle where everyone plays a part to come?

The Big Bang Theory of the Inclined Plane

Season 01, Episode 02: “The Big Bran Hypothesis”
Sheldon and Leonard solve a problem using an inclined plane

Sheldon and Leonard contemplate using the stairs as an inclined plane to move Penny’s furniture to her apartment.

In Season 1 Episode 2 of The Big Bang Theory, “The Big Bran Hypothesis”, Penny (Kaley Cuoco) asks Leonard (Johnny Galecki) to sign for a furniture delivery if she isn’t home. Unfortunately for Leonard and Sheldon, they are left with the task of getting a huge (and heavy) box up to Penny’s apartment.

To solve this problem, Leonard suggest using the stairs as an inclined plane, one of the six classical simple machines defined by Renaissance scientists. Both Leonard and Sheldon have the right idea here. Not only are inclined planes used to raise heavy loads but they require less effort to do so. Though this may make moving a heavy load easier the tradeoff is that the load must now be moved over a greater distance. So while, as Leonard correctly calculates, the effort required to move Penny’s furniture is reduced by half, the distance he and Sheldon must move Penny’s furniture twice the distance to raise it directly.

Mathematics of the Inclined Plane

Effort to lift block on Inclined Plane

Now we got an inclined plane. Force required to lift is reduced by the sine of the angle of the stairs… call it 30 degrees, so about half.

Steps being used as an inclined plane to raise a block

Free-body Diagram of a block on an inclined plane. It shows the forces acting on a block and the force needed to keep it stationary and not let it slip down.


To analyze the forces acting on a body, physicists and engineers use rough sketches or free body diagrams. This diagram can help physicists model a problem on paper and to determine how forces act on an object. We can resolve the forces to see the effort needed to move the block up the stairs.

If the weight of Penny’s furniture is \(W\) and the angle of the stairs is \(\theta\) then
\[\angle_{\mathrm{stairs}}\equiv\theta \approx 30^\circ\]
and
\[\Rightarrow\sin 30^\circ = \frac{1}{2}\]
So the effort needed to keep the box in place is about half the weight of the furniture box or \(\frac{1}{2}W\), just as Leonard says.

Distance moved along Inclined Plane

Inclined plane and steps

The relationship between the height \(h\) the block is raised and the distance it moves \(d\) is the sine of the angle \(\theta\).

While the inclined plane allows Leonard and Sheldon to push the box with less effort, the tradeoff is that the distance they move along the incline is twice the height to raise the box vertically. Geometry shows us that
\[\sin \theta = \frac{h}{d}\]
We again assume that the angle of the stairs is approximately \(30^\circ\) and \(\sin 30^{\circ} = 1/2\) then we have \(d=2h\).

Uses of the Inclined Plane

We see inclined planes daily without realizing it. They are used as loading ramps to load and unload goods. Wheelchair ramps also allow wheelchair users, as well as users of strollers and carts, to access buildings easily. Roads sometimes have inclined planes to form a gradual slope to allow vehicles to move over hills without losing traction. Inclined planes have also played an important part in history and were used to build the Egyptian pyramids and possibly used to move the heavy stones to build Stonehenge.

Lombard Street (San Francisco)

Lombard Street, San Francisco Inclined Plane

Lombard Street is one of the most visited street in San Francisco as seen from Coit Tower. It is best known for the one-way section on Russian Hill between Hyde and Leavenworth Streets, in which the roadway has eight sharp turns (or switchbacks) that have earned the street the distinction of being the crookedest “most winding “street in the world, though this title is contested. (Photo by David Yu).

Lombard Street in San Francisco is famous for its eight tight hairpin turns (or switchbacks) that have earned it the distinction of being the crookedest street in the world (though this title is contested). These eight switchbacks are crucial to the street’s design as the reduce the hills natural 27° grade which is too steep for most vehicles. It is also a hazard to pedestrians, who are more accustomed to a more reasonable 4.86° incline due to wheel chair navigability concerns.

Technically speaking, the “zigzag” path doesn’t make climbing or coming down the hill any easier. As we have seen, all it does is change how various forces are applied. It just requires less effort to move up or down but the tradeoff is that you travel a longer distance. This has several advantages. Car engines have to be less powerful to climb the hill and in the case of descent, less force needs to be applied on the brakes. There are also safety considerations. A car will not accelerate down the switch back path as fast than if it was driven straight down, making speeds safer and more manageable for motorists.

This idea of using zigzagging paths to climb steep hills and mountains is also used by hikers and rock climbers for very much the same reason Lombard Street zigszags. The tradeoff is that the distance traveled along the path is greater than if a climber goes straight up.

The Descendants of Archimedes

We don’t need strength, we’re physicists. We are the intellectual descendants of Archimedes. Give me a fulcrum and a lever and I can move the Earth. It’s just a matter of… I don’t have this, I don’t have this!

We see that Leonard had the right idea. If we were to assume are to assume — based on the size of the box — that the furniture is approximately 150 lbs (65kg) and the effort is reduced by half, then they need to push with at least 75 lbs of force. This is equivalent to moving a 34kg mass. If they both push equally, they are each left pushing a very manageable 37.5 lbs, the equivalent of pushing a 17kg mass.

Penny’s apartment is on the fourth floor and we if we assume a standard US building design of ten feet per floor, this means a 30 foot vertical rise. The boys are left with the choice of lifting 150 lbs vertically 30 feet or moving 75lbs a distance of 60 feet. The latter is more manageable but then again, neither of our heroes have any upper body strength.

The Big Bang Theory of Trans-Neptunian objects

Season 02, Episode 04: “The Griffin Equivalency”
The Big Bang Theory astrophysicist Rajesh Koothrappali looks pleased as he talks about trans-Neptunian object 2008 NQ17

Rajesh Koothrappali (Kunal Nayyar) looks pleased as he talks about his inclusion into People magazine’s “30 Under 30 to watch” list for his discovery of the fictional trans-Neptunian object, 2008 NQ17 which exists beyond the Kuiper belt

In “The Griffin Equivalency” episode of The Big Bang Theory, Raj (Kunal Nayyar) announces that he has been chosen as one of People Magazine’s “30 visionaries under 30 years to watch” list for his discovery of the trans-Neptunian object 2008 NQ17. This fictional astronomical body exists just beyond the Kuiper Belt, a region of the Solar System that extends from the orbit of Neptune (about 30 Astronomical units(AU) away), to a distance of 50AU from the Sun. (1AU is roughly the distance of the Earth from the Sun).

Trans-Neptunian Objects in the Solar System

The distribution and classification of the trans-Neptunian Objects in our solar system

The first trans-Neptunian object discovered was Pluto in 1930 and is the second most massive known dwarf planet in the Solar System. These objects are so small and distant that it wasn’t until 1992 that the second trans-Neptunian object orbiting the Sun, (15760) 1992 QB1, was discovered. As cryptic as these names sound, i.e 2008 NQ17 and 1992 QB1, the nomenclature comes from a provisional designation in astronomy that names objects immediately following their discovery. Once a proper orbit has been calculated, a permanent designation consisting of a number-name combination is given by the Minor Planet Center. Unfortunately for Raj, so many of these minor planets are discovered that most won’t be named for their discoverers.

Provisional designation of Minor Planets

The provisional naming system for minor planets, of which dwarf planets, asteroids, trojans, centaurs, Kuiper belt objects, and other trans-Neptunian objects are a part of, have been in place since 1925. The first element in a minor planet’s provisional designation is the year of discovery, followed by two letters and sometimes a number.

The first letter indicates the “half-month” of the object’s discovery. “A” denotes a discovery in the first half of January (between January 1st and January 15th) while “B” denotes a discovery in the second half of January (between January 16th and January 31st). The letter “I” is not used so the letters go all the way up to “Y”. The first half is always the 1st through to the 15th of the month, regardless of the numbers of days in the second “half”.

First Letter (A-G)
A B C D E F G
Jan 1 Jan 16 Feb 1 Feb 16 Mar 1 Mar 16 Apr 1

First Letter (H-O)
H J K L M N O
Apr 16 May 1 May 16 Jun 1 Jun 16 Jul 1 Jul 16

First Letter (P-V)
P Q R S T U V
Aug 1 Aug 16 Sep 1 Sep 16 Oct 1 Oct 16 Nov 1

First Letter (W-Y)
W X Y
Nov 16 Dec 1 Dec 16

The second letter and the number indicate the order or discovery within that half-month.

Second Letter (A-K)
A B C D E F G H J K
1 2 3 4 5 6 7 8 9 10

Second Letter (L-U)
L M N O P Q R S T U
11 12 13 14 15 16 17 18 19 20

Second Letter (V-Z)
V W X Y Z
21 22 23 24 25

So the 14th minor planet discovered in the first half of June 2013 will get the provisional planetary designation 2013 LO. As modern techniques has made it (relatively) easier to detect these objects, there can be more than 25 discoveries in a given half-month. To solve this a subscript number is added to indicate the number of times that the letters have cycled through. So the 40th minor planet to be discovered in the first half of June 2013 will gain the designation 2013 LP1.
\[\begin{aligned}
40 & = 25 + 15 \\
\therefore P & = 15
\end{aligned}\]

Planet Bollywood

Using this information we can calculate when Raj discovered “Planet Bollywood”. It was discovered some time between July 1-15 and it was the 441st minor planet to be discovered in that half month.
\[\begin{aligned}
Q & = 16 \\
25 \times 17 & = 425
\end{aligned}\]
And \(425 + 16 = 441\).

It must have been a particularly busy month in the Big Bang Theory Universe. A quick check at the Minor Planet Center’s Provisional Designations webpage indicates that there were 137 astronomical bodies discovered in that half-month. That would make that last trans-Neptunian object to be discovered to receive the provisional designation 2008 NM4.

Long live Planet Bollywood!

The Big Bang Theory of the Rolling Ball Problem

Big Bang Theory Season 2, Episode 5: “The Euclid Alternative”

In the opening scene of the “The Euclid Alternative”, we see Sheldon (Jim Parsons) demanding that Leonard (Johnny Galecki)needs to drive him around to run various errands. Leonard, after spending a night in the lab using the new Free Electron Laser to perform X-ray diffraction experiments. In the background, we can see equations that describe a rolling ball problem on the whiteboard in the background.

Equations of a rolling ball

Scene with Sheldon Cooper. The background shows a whiteboard with the physics equations of a rolling ball.

Rolling motion plays an important role in many familiar situations so this type of motion is paid considerable attention in many introductory mechanics courses in physics and engineering. One of the more challenging aspects to grasp is that rolling (without slipping) is a combination of both translation and rotation where the point of contact is instantaneously at rest.The equations on the white board describe the velocity at the point of contact on the ground, the center of the object and at the top of the object.

Pure Translational Motion

Pure Translatoinal Motion

An object undergoing pure translational motion

When an object undergoes pure translational motion, all of its points move with the same velocity as the center of mass– it moves in the same speed and direction or \(v_{\textrm{cm}}\).

Pure Rotational Motion

Pure Rotation

An object undergoing pure rotational motion. The velocity of each point of the object depends on the distance away from the center. The further away the faster it moves as can be seen by the length of the arrows.

In the case of a rotating body, the speed of any point on the object depends on how far away it is from the axis of rotation; in this case, the center. We know that the body’s speed is \(v_{\textrm{cm}}\) and that the speed at the edge must be the same. We may think that all these points moving at different speeds poses a problem but we know something else — the object’s angular velocity.
The angular speed tells us how fast an object rotates. In this case, we know that all points along the object’s surface completes a revolution in the same time. In physics, we define this by the equation:
\begin{equation}
\omega=\frac{v}{r}
\end{equation}
where \(\omega\) is the angular speed. We can use this to rewrite this equation to tell us the speed of any point from the center:
\begin{equation}
v(r)=\omega r
\end{equation}
If we look at the center, where \(r=0\), we expect the speed to be zero. When we plug zero into the above equation that is exactly what we get:
\begin{equation}
v(0)= \omega \times 0 = 0
\label{eq:zero}
\end{equation}
If we know the object’s speed, \(v_{\textrm{cm}}\) and the object’s radius, \(R\), using a little algebra we can define \(\omega\) as:
\[\omega=\frac{v_{\textrm{cm}}}{R}\]
or the speed at the edge, \(v_{\textrm{cm}}\) to be \(v(R)\) to be:
\begin{equation}
v_{\textrm{cm}}=v(R) = \omega R
\label{eq:R}
\end{equation}

Putting it all Together

Rotating Object

An object that rolls undergoes both translational and rotational motion. To determine the speed at any point we must add both of these speeds. We see the translational speed show in red and the rotational speed show in in blue.

To determine the absolute speed of any point of a rolling object we must add both the translational and rotational speeds together. We see that some of the rotational velocities point in the opposite direction from the translational velocity and must be subtracted. As horrifying as this looks to do, we can reduce the problem somewhat to what we see on the whiteboard. Here we see the boys reduce the problem and look at three key areas, the point of contact with the ground (\(P)\), the center of the object, (\(C\)) and the top of the object (\(Q\)).

Whiteboard of the Rolling Ball Problem

Rendering copy of the whiteboard showing the rolling ball problem as seen on the Big Bang Theory.

We have done most of the legwork at this point and now the rolling ball problem is easier to solve.

At point \(Q\)

At point \(Q\), we know the translational speed to be \(v_{\textrm{cm}}\) and the rotational speed to be \(v(R)\). So the total speed at that point is
\begin{equation}
v = v_{\textrm{cm}} + v(R)
\label{eq:Q1}
\end{equation}
Looking at equation \eqref{eq:R}, we can write \(v(R)\) as
\begin{equation}
v(R) = \omega R
\end{equation}
Putting this into \eqref{eq:Q1} and we get,
\begin{aligned}
v & = v_{\textrm{cm}} + v(R) \\
& = v_{\textrm{cm}} + \omega R \\
& = v_{\textrm{cm}} + \frac{v_{\textrm{cm}}}{R}\cdot R \\
& = v_{\textrm{cm}} + v_{\textrm{cm}} = 2v_{\textrm{cm}}
\end{aligned}
which looks almost exactly like Leonard’s board, so we must be doing something right.

At point \(C\)

At point \(C\) we know the rotational speed to be zero (see equation \eqref{eq:zero}).
Putting this back into equation \eqref{eq:Q1}, we get
\begin{aligned}
v & = v_{\textrm{cm}} + v(r) \\
& = v_{\textrm{cm}} + v(0) \\
& = v_{\textrm{cm}} + \omega \cdot 0 \\
& = v_{\textrm{cm}} + 0 \\
& = v_{\textrm{cm}}
\end{aligned}
Again we get the same result as the board.

At point \(P\)

At the point of contact with the ground, \(P\), we don’t expect a wheel to be moving (unless it skids or slips). If we look at our diagrams, we see that the rotational speed is in the opposite direction to the translational speed and its magnitude is
\begin{aligned}
v(R) & = -\omega R \\
& = -\frac{v_{\textrm{cm}}}{R}\cdot R \\
& = -v_{\textrm{cm}}
\end{aligned}
It is negative because the speed is in the opposite direction. Equation \eqref{eq:Q1}, becomes
\begin{aligned}
v & = v_{\textrm{cm}} + v(r) \\
& = v_{\textrm{cm}} – \omega R \\
& = v_{\textrm{cm}} – \frac{v_{\textrm{cm}}}{R}\cdot R \\
& = v_{\textrm{cm}} – v_{\textrm{cm}} = 0
\end{aligned}
Not only do we get the same result for the rolling ball problem we see on the whiteboard but it is what we expect. When a rolling ball, wheel or object doesn’t slip or skid, the point of contact is stationary.

Cycloid and the Rolling ball

Cycloid drawn by a Rolling Ball

The path a point on a rolling ball draws is called a cycloid.

If we were to trace the path drawn by a point on the ball we get something known as a cycloid. The rolling ball problem is an interesting one and the reason it is studied is because the body undergoes two types of motion at the same time — pure translation and pure rotation. This means that the point that touches the ground, the contact point, is stationary while the top of the ball moves twice as fast as the center. It seems somewhat counter-intuitive which is why we don’t often think about it but imagine if at the point of contact our car’s tires wasn’t stationary but moved. We’d slip and slide and not go anywhere fast. But that is another problem entirely.

The Wibbly Wobbly of Continuum

The series Continuum is a time travel science fiction TV show that follows the adventures of City Protective Services (CPS) law enforcement officer Kiera Cameron (Rachel Nichols) as she attempts to stop the self-proclaimed freedom fighters known as “Liber8”. It is easy to dismiss many science fiction shows dealing with time travel for the simple reason so many of them are done badly. Continuum is different in that there is an underlying mystery. A big part of that mystery is the motivation of an older Alec Sadler (William B. Davis) and the principles on which Continuum’s time travel is based.

The Continuum World of 2077

Continuum LogoIn 2077, the future of this world is both a distopian and Orwellian. World governments have somehow failed and, by implication, we assume so too has democracy. This has given way to a corporate controlled government that monitors and records every moment of a person’s life. People are born indebted and indentured to  the Global Corporate Congress until they pay off their life debts.

Among the largest corporations in the North American Union and possibly the world is SadTech, owned by Alec Sadler — a genius responsible for inventing and developing much of the technology we see in the world of Continuum. This hasn’t only made Sadler one of the richest men in the world but possibly one of the most powerful.

In an attempt to end the corporation’s rule over the people, Liber8 leader, Edouard Kagame (Tony Amendola) attempts to bring down the Corporate Congress by blowing up the building where a scheduled meeting is to take place thereby killing all 20 members. To Kagame’s surprise Sadler is the sole survivor. Sadler gives Kagame a time travel device to allow Liber8 to escape into the past; Sadler’s full intentions are unknown.

Upon arriving in 2012, Kiera meets and befriends a younger Alec Sadler (Erik Knudsen) who is coming to terms with the loss of his father and has decided to follow his father’s work and research. The beginnings of much of the “tech” that Keira uses was developed by Alec’s father. The young Alec posits two likely time traveling scenarios, both of which have their roots in physics. Evidence for both scenarios has been presented over the course of the first season and they both seem equally valid. The first is a “time loop” where conditions can not be altered. Everything that is happening has already happened and will give way to the future that Kiera knows. The second is an alternate time line and the very presence of both Kiera and Liber8 have altered Continuum’s 2077 future.

Time Loop

The first possibility is the “time loop” where the older Alec, knowing of his interactions in 2012 with Kiera, deliberately sends her back in time. This fulfills the events in the past and allow the future to progress in the way it has. All the events we see in 2012 have already happened and none of the Continuum characters can deviate from that. In a sense, everything is preordained. One of the most tantalizing clues, though unrelated, clues is the presence of Mr. Escher, the unknown head of the high security government agency, Section 6, who vouches for Kiera to the Vancouver Police Department.

Time Loop hypothesis for Continuum

“Ascending and Descending” is a lithograph print by the Dutch artist M. C. Escher which was first printed in March 1960. This perpetual loop illustrates the concept of the time loop used in Continuum.

This name may be a hidden clue that supports the the Time Loop theory. It may  Continuum’s writers way of telling viewers which of the two theories are correct. The name may have been taken from the Dutch artist, Maurits Cornelis Escher or M.C. Escher, who is best known for his mathematically inspired artwork. Escher was known for his depictions of impossible realities, especially those that related to infinite loops in some way. The “Ascending and Descending” litographic print is an artistic depiction of the “Penrose stairs“, an impossible object created by Lionel and Roger Penrose . The reality bending idea of the Penrose Stairs was used in the 2010 movie “Inception”.

Visual Time Loop in Continuum using a Mobius strip and ants

The ants crawl along a Möbius strip. This is another illustration of the principle of a time loop in the TV series Continuum.

In addition to impossible objects, Escher’s art also features insects. One of his paintings, Möbius Strip II (Red Ants), shows an ant walking along a surface with only one side — a Möbius strip. This strip looks very much like the infinity symbol. If an ant was to start at one point on the strip it will eventually return to that same point. We can create a Möbius strip of our own by twisting one end of a strip of paper by one half-turn and gluing the ends together. This object was dicovered independently by August Ferdinand Möbius and Johann Benedict Listing in 1858.

If this name is really a clue, then it add credibility to the time loop hypothesis. It could also mean that the reason that the older Sadler sends Keira back into time isn’t just to preserve the future but because he has to; he knows and recalls his interactions with this time traveling cop in his past.

This raises the question, can Kiera or Liber8 do anything to change the future? Does it mean that every little action, from the coffee they drink in the morning to the time they go to bed at night, has already been scripted? Can they do anything that might create a time paradox?

To resolve the problem of time paradoxes, which is permitted in some solutions of Einstein’s Theory of General Relativity, Russian physicist Igor Dmitriyevich Novikov came up with his self-consistency principle . In simplest terms, the Novikov self-consistency principle says that if an event exists that would give rise to a paradox, or to any “change” to the past whatsoever, then the probability of that event is zero. This means that it is impossible to create time paradoxes.

Some may find this idea unpalatable as it interferes with our ideas of free will and destiny. Does it mean that Kiera and Alec and all of Liber8 are doing nothing more than following a script from which they can never deviate? Well, not entirely. The Novikov self-consistency principle does not imply that time traveler have no free-will. It just says that the results of their actions can not produce inconsistent results. Kiera can choose what to have for lunch but not stop the bombing of the building even if she tried. If she succeeded, her actions would be inconsistent with how events unfold in the future.

Alternate Timelines & the Many Worlds of Continuum

Alec also considers the possibility that Kiera’s and Liber8’s presence may have altered the time line making the fate of 2077 is uncertain. But what does that mean for the inhabitants of 2077 if Liber8 or even Kiera make a change in the present? Do they “blink” out of existence just like old Biff Tannen in Back to the Future II? No, they don’t have to and quantum mechanics may provide an answer.

To resolve some of the strange results observed in quantum mechanics, physicist Hugh Everett came up with his Many-Worlds Interpretation in 1957 . In this version of quantum mechanics, at every single instant of the tiniest portion of a second the Universe is splitting into countless billions of parallel universes. Thus our Universe branches out in which every possibility exists. The arrival of Kiera and Liber8 in the past results in a a new branch and a new alternate reality. This idea was seen in the 2009 Star Trek reboot. This spells bad news for Kiera as she can only travel to the future of the new time-line she is already in. Getting home or to the timeline she came from is an impossibility.

So which is it?

It’s difficult to say which of the two hypotheses are true as the series has given evidence to support either hypothesis. In “A Test of Time”, Kagame decides to test these theories of time travel. If Liber8 were to kill Kiera’s grandmother, then the threat posed by this future cop would literally cease to exist. While Kiera manages to save her grand-mother, Matthew Kellog’s (a former Liber8 member) teenage grandmother is killed in a confrontation between Kiera and Liber8.

Matthew’s continued existence provides no clear answers as Alec explains at the end of the episode. While we are certain who Kiera’s grandmother was, we are not quite sure in Kellog’s case. For all we know, Kellog could have been mistaken about the young girl’s identity and she was never his grandmother. The young girl could also have been Kellog’s grandmother and though she died thus preventing Matthew’s future birth, Kellog’s journey through time might protect him from this causality paradox. It does mean that the future can be changed which gives Liber8 even more incentive to succeed.

The good and bad guys in Continuum

If the future is a bad place to live and Kiera is trying to preserve the future, does that mean she is the bad guy? Liber8’s methods and motivations, while they are extreme, are to free the world from a tyrannical and oppressive system from ever taking place. Both sides are fighting to save the future they believe in. Liber8 seeks to create a better world while Kiera is trying to preserve her family’s existence. Neither knows if that is even possible.

It all comes down to one man, the older Alec Sadler and his ultimate goal and motivations. Young Alec discovers at the end of the season finale that he is the reason that Kiera is now living in the past. Did he send Kiera and Liber8 to save the future or to preserve it? Whatever the answer, we are going to have to wait for the answer in future episodes.

Further Reading