Continuum, Time Travel and The Grandfather Paradox

The last episode to Continuum’s second season (“Second Time”) sees a grief stricken Alec Sadler (Erik Knudsen use the time travel device to go back in time to save the life of Emily (Magda Apanowicz). A shocked and distraught Keira (Rachel Nichols) looks on as Alec disappears from the time line in a flash of light. What is everyone’s favorite Protector to do now that the time travel device no longer exists? Is she forever trapped in the present unable to do anything about Alec’s betrayal?

If you were hoping to find out what Kiera would do now that she is stuck in the present then don’t hold your breath. Alec reemerges in the past–one week in the past to be exact–to create a new timeline while Kiera is captured by the Freelancers. The original timeline, as explained by Head Freelancer Catherine (Rachael Crawford) no longer exists. The intense roller-coaster ride and the jaw-dropping events of the last season, from the pinning of Agent Gardiner’s (Nicholas Lea) murder on Kiera to Carlos Fonnegra (Victor Webster) leaving the Vancouver Police Department to join Julian (Richard Harmon) and Liber8 never happened.

So what are Continuum fans to do? Are we simply to ignore the last episodes of Season Two as if they never happened? That remains an open question.
The Season Three premier certainly continues with the same action packed intensity from which the last season ended but it does something else–it explains the time travel rules upon which the show is based. It also appears that the show’s writers have put a lot of thought and effort into ensuring that these rules are as consistent as possible, especially where the issues of paradoxes are concerned.

Head Freelancer Catherine explains that time travel is not immutable and is like a brancing tree that needs to be pruned.

“Destiny is not set. Time is not immutable. The continuum is like a tree. It can grow wild or it can be cultivated.”

In the premier episode, head Freelancer Catherine explains the concept of timelines and the physics of time travel to Kiera. While there is only one timeline, it can be changed. Go back far enough in time and you can change everything to create an entirely new and different timeline.

This answers some of the paradoxial questions of the last two seasons, if Maddie (Olivia Ryan-Stern) was really Kellog’s (Stephen Lobo) grandmother, then why didn’t he pop out of existence in the current timeline. One idea put forward by Alec was that she may not have been Matthew’s grandmother. It turns out she could have been and to understand why, we must look into how Continuum’s physics of time travel resolves the Grandfather Paradox.

The grandfather paradox is one of the more well-known time travel paradoxes and was first described by the science fiction writer René Barjavel in his 1943 book “Le Voyageur Imprudent(Future Times Three). In this scenario, our time traveler goes back in time before his grandfather is married and kills him. The paradox is that the time traveler is never born and can not go back in time to kill his grandfather. His grandfather is free to meet the grandmother, get married, have kids and our time traveler is eventually born. When the time traveler gets older, he steps into his time-machine and goes back in time to kill his grandfather. This cycle continues ad infinitum and without resolution.

Recognizing this vulnerability, in the Season One episode “A Test of Time”, Kagame (Tony Amendola) decides to test this paradox and possibly get rid of their “Protector problem” once and for all. Liber8 start by killing young women with Kiera’s grandmother’s name–“Lily Jones”. During the course of the episode Liber8 take additional insurance by planning to kill Kellog’s grandmother; revenge against Kellog for helping Kiera. Travis (Roger Cross) shoots Kellog’s grandmother and everyone sees that Kellog does not disappear–the grandfather paradox does not apply to them. Or does it?

According to Catherine’s explanation of time travel, killing Maddie creates a new time line. In this new timeline Kellog will not be born and many of the events in Kellog’s life, from his meeting Kiera to his part in Liber8 will never happen. In a sense the very timeline that Kiera so desperately tries to protect no longer exists but this doesn’t mean that Kellog’s absence changes things so completely that the future Kiera and the Freelancers are trying to protect never happens. Kiera’s primary concern is her family but everything can still go according to plan even though Kellog no longer exists in the new timeline.

Catherine and the Freelancers appear to subscribe to the Great Man Theory, a 19th-century idea in which history can be explained by the impact of “great men”, or heroes. As Kellog is not one of these supposed “great men” then many key events, such as Alec’s rise to power and discovery of time travel, will still happen. This means that Kiera’s family will still exist in the future. As they don’t know who Kellog is, they will also be completely unchanged by the ripple effect of Maddie’s death. The reason Catherine and the Freelancers aren’t too concerned of Maddie’s death is because her impact on the future is negligible–her life or death changes nothing. Talk about a major blow to one’s ego.

Continuum’s time-travel physics provides a logically consistent way to resolve the Grandfather Paradox. A time traveler who kills his own ancestor or whose ancestor is murdered won’t vanish from existence. Rather, the timeline he came from will disappear to be replaced by a new timeline in which he will never be born. As our time traveler is a refugee from the previous timeline that no longer exists, he won’t pop out of existence and is safe from the ramifications of the Grandfather Paradox.

This single and mutable timeline idea not only overcomes the logical paradox and inconsistencies of the Grandfather Paradox but others as well , such as the Bootstrap and Predestination Paradoxes–something we will examine in future blog posts. It also highlights how dangerous time travel is in the series. As time is not immutable, a nefarious time traveler can use a time machine as a weapon to mold the future to achieve for personal gain. Jason has hinted that the Freelancers have meddled with humanity’s history before. The extent of this meddling remains to be seen.

The Freelancers have the power to monitor the continuum and stop people from making these changes to the timeline but this doesn’t mean they are the good guys. Catherine has admitted they see themselves as guardians to the continuum, in essence, the ones who “prune” the tree. This makes you wonder whose interests they represent and whether those interests are the best for everyone. This is interesting because the series has yet to really identify who the good guys and bad guys really are.

While we may at times root for Kiera, as the series progresses we may discover we never should have. We have two Alecs in this timeline. In the last timeline we have seen hints that Alec could be turning to the “dark side” and become the man to usher in a totalitarian and dystopian future of 2077. All that may have been changed with Emily’s death. It is amazing the difference a week makes. Could we see Alec Sadler fight on both sides of this temporal war? Your guess is as good as mine.

The Discovery of the Higgs Boson: Week 2 Review Light Bulbs and Railroad Schedules

The second week of the FutureLearn course “The Discovery of the Higgs Boson” looks at physics of the 20th century–special relativity and quantum mechanics. These two branches of physics represented a fundamental shift in the way we view the world.

It may come as a surprise to some that these deep philosophical shifts have very unexpected origins. Our view of a quantized world came from the need to create a more efficient light bulb while the connection between space and time came from our need to run a more efficient railroad network and international time conventions.

Need for a more efficient light-bulb

Hungarian physicist Philipp Lenard, discoverer of the photoelectric effect in 1902.

In 1902, Hungarian physicist, Philipp Lenard, winner of the 1905 Nobel Prize in Physics for cathode rays, observed that the energy of individual emitted electrons increased with light frequency–the photoelectric effect. This appeared to be at odds with Maxwell’s theory of electromagnetism which predicted that an electron’s kinetic energy should be proportional to light intensity. In 1905, Albert Einstein published a paper that explained the experimental data from the photoelectric effect. Based on Max Plank’s theory of black body radiation Einstein postulated that light energy was being carried in discreet quantized packets.

In 1894, German theoretical physicist Max Planck was commissioned by the German Bureau of Standards with the task of creating more efficient light-bulbs. To do so, Planck needed to find one that would emit as much visible light as possible with very little to no infra-red and untra-violet light. Planck knew from experiments at when an object is heated, it emits radiation in the form of black-body radiation. Planck turned his attention to this problem.

Black body curves for various temperatures and comparison with classical theory of Rayleigh-Jeans. As the temperature decreases, the peak of the black-body radiation curve moves to lower intensities and longer wavelengths.

“Blackbody radiation” or “cavity radiation” is the characteristic radiation that a body emits when heated. This is seen in the form of a curve which peaks at a characteristic temperature where most of the radiation is emitted. Experiments showed that as the temperature changes, so too does the emitted radiation. When the wave picture of light was applied to this problem, it failed to predict the observed intensity for any given temperature.

Planck made several attempts to understand this problem. His first proposed solution in 1899 based on the entropy of an ideal oscillator, in what he called the “principle of elementary disorder”, failed to predict experimental observations. Planck revised his approach in 1900 using Boltzmann statistics to gain a more fundamental understanding of black-body radiation. This approach worked but Planck held an aversion towards statistical mechanics. He was also deeply suspicious of the philosophical and physical implications of its interpretation. His recourse was, as he later put it, “an act of despair… I was ready to sacrifice any of my previous convictions about physics.”

The central assumption behind his third attempt was the hypothesis, now known as the Planck postulate, that electromagnetic energy could only be emitted in quantized form. Planck didn’t think much of this method, regarding it as a mere trick. We know now that assumption is regarded as the birth of quantum mechanics. Try as he might, Planck struggled to grasp the meaning of energy quanta, going so far as to reject Einstein’s hypothesis and explanation of Lenard’s photoelectric effect. He was unwilling to completely discard Maxwell’s theory of electrodynamics.

Not everyone was convinced by Einstein’s hypothesis either, even after it was experimentally verified by Robert Millikan in 1914. Many physicists were reluctant to believe that electromagnetic radiation could be particulate in nature. Instead, it was believed that the observed energy quantization was the result of some constraint of matter and the way that it absorbs and emits radiation. It wasn’t until Compton’s experiments showed that light cannot be purely be explained as a wave that the idea of light quanta was accepted.

Train Schedules and Time Zones

The first passenger carriage in Europe, 1830, George Stephenson´s steam locomotive, Liverpool and Manchester Railway

The mid to late-19th century saw considerable and rapid improvements in transportation, communication and technology. One of these inventions, the steam locomotive, not only changed the way goods and people traveled but also the way we view time. Products could be moved more cheaply and much faster.

Before the invention of clocks, people marked the time of the day with apparent solar time or by noting the sun’s position in the sky. Local time was different for each town and settlement. With the invention of well regulated mechanical clocks, cities used local mean solar time. As clocks differed between towns by an amount corresponding to the difference in geographic longitude–a variation of four minutes for every degree of longitude–communication between towns and rail transport became awkward. The time difference between Bristol and London, for example, a difference of 2°35′ longitude, is about 10 minutes while the difference between New York and Boston is about two degrees or 8 minutes.

Time keeping on American railroads was even more confusing. Each railroad had their own standard of time time, usually based on the local time of its headquarters or main terminus. Each railroad schedule was published using the company’s own time and stations had a clock for each railroad, each showing a different time.

Non-uniform time zones weren’t just confusing. It was dangerous. The incidents of accidents and near-misses became more frequent as more people started using trains for travel. What was needed was a means to know exactly where trains were at all times. The use of time zones solves this problem and with it came the need to synchronize clocks at a distance.

It’s not surprising that many of Einstein’s though experiments concerns trains. As a young patent clerk, many of the inventions he reviewed focused on using light signals to synchronize clocks. Einstein took it a step further and realized that clocks moving with respect to each other would not tick at the same rate.

Physics students are familiar with Einstien’s Gedankenexperiments and that the power of abstract thought can allow one to fully visualize the consequences of an experiment without having to actually perform said experiment. Far from being esoteric examples, Einstein’s thought experiments are firmly grounded in reality and shares its origins in something as simple as a train schedule.

The Higgs Boson Course

In one of the course’s lectures, Peter Higgs says that when he teaches undergraduates special relativity, he ignores the way that Einstein did it and asks, “how do you realize the principle of relativity, which was what was formulated by Henri Poincare?” To do this, you have to abandon Newton’s assumption of absolute time. Peter Higgs is correct, the development of special relativity need not have had anything to do with the Michelson-Morley experiment. In Einstein’s case, it came about from the practical need to synchronize clocks.

The second week of the course builds on the previous week. Though the concepts are quite literally mind-blowing, the ideas and mathematics were conveyed in a way that makes it easy for students to grasp. The third week looks even more exciting as we combine both special relativity and quantum mechanics to make much deeper predictions about our world.

The Discovery of the Higgs Boson: Week 1 Review Conservation Laws and Physical Revolutions

The Higgs boson has captured the world’s attention from the moment it was announced the Large Hardron Collider was being built to find it. The elementary particle’s discovery was announced by CERN on 4 July 2012 and is nothing short of monumental as it appears to confirm the existence of the Higgs field. This pervasive field is pivotal to understanding why some fundamental particles have mass. As interesting and exciting as this discovery may be, its consequences and implications remain out of reach for the general public. Future Learn, a privately company owned by Open University, along with the University of Edinburgh have started a seven week Massive Open Online Course (MOOC) “The Discovery of the Higgs Boson” to introduce the theoretical tools needed to appreciate this discovery.

The course starts with classical mechanics. While it may seem like a strange place to start, especially given the course’s goals, it is a good one. Rather than jumping straight into Quantum Mechanics, classical mechanics makes the mathematics is accessible to students, especially those who have completed A-Level Mathematics or done Calculus. The course doesn’t assume many of the fundamental conservation laws in physics are true but rather goes through the mathematically rigorous process of proving these concepts to students. This also allows students to see how the behavior of physical systems can be deduced. Though some of the physical principles will need to be revised as the course progresses, the mathematical tools students would have acquired remain the same. Student can thus build on what they have learned before.

An Evolutionary Revolution

One of the more interesting questions asked was, “Why is the Higgs boson discovery important?” Dr. Victoria Martin, a reader in particle physics at the University of Edinburgh, answers that in one of the course’s modules. In her video, she says the Sun is a massive burning ball of hydrogen and helium and it is a mystery why it all hasn’t burned up by now. She says the answer comes from Peter Higgs’ theory which predicts the Higgs boson. It predicts why the nuclear process in the Sun is slow enough that the Sun is still around after 4.6 billion years and provides just the right amount of light and heat to sustain life on Earth.

Science didn’t always believe that the Sun or the Earth, for that matter, was old. In the mid-nineteenth century, both Charles Darwin and Alfred Russel Wallace were making the case for biological evolution by natural selection. This theory described a process of slow, gradual changes over time and indicated the Earth had to be very old, at least hundreds of millions of years. This was supported by geologist observations of erosion rates.

Photograph of William Thomson, Lord Kelvin.

This posed a problem for the most prominent theoretical physicist of the time, William Thomson, 1st Baron Kelvin, who saw evidence that disagreed with Darwin. This guiding light of the Industrial Revolution, whose work in thermodynamics contributed to the steam engine, was a devout Christian who believed in a much younger Earth and with good reason. In 1862, using the thermodynamics of heat conduction, Thomson initial calculations showed that it would take between 20-400 million years for a molten Earth to completely solidify and cool.

This large uncertainty of the Earth’s age were due to uncertainties about the melting temperature of rock. This did not deter Thomson who then set out to calculate the Sun’s age in 1868 using what he knew of the Sun’s energy output. Kelvin rightly assumed the Sun formed from a giant gas cloud and gravity eventually caused the cloud to collapse into a ball. As with any falling mass, the cloud molecules’ potential energy would be converted into kinetic energy. This raise in kinetic energy would turn into heat, raising temperatures to result in star formation in a process known today as the Kelvin-Helmholtz Contraction.

While we know today this is not the way stars generate all their energy, we know this is how the fusion process is started. Based on his assumption that the Sun built up all its heat as it was formed and radiating it away like a hot coal, Kelvin estimated the lifespan of the Sun to be about 30 million years.

The numbers posed a nagging contradiction–the Earth was older than the Sun. Thomson realized he needed to refine his calculations of the Earth’s age. In 1897 Thomson settled on an estimate that the Earth was somewhere between 20-40 million years old. This fit in nicely with his estimation of the Sun’s age.

Darwin, aged 45 in 1854, by then working towards publication of On the Origin of Species.

The age of the Earth was an important part to Darwin’s theory of evolution. As a geologist, he had conducted his own studies and concluded that the time needed to wash away the Weald, a valley in south-east England formed of the eroded remains of an anticline, would require 300 million years. Thomson believed that geologists were wrong to assume a steady rate of erosion. Floods and other natural disasters could accelerate this process. Thompson though the geologist’s thinking could use a dose of mathematical rigor. Though Darwin’s observations supported an old Earth, he was so bowled away by Thomson’s analysis that he removed any reference to time scales in later editions of his Origin of the Species.

A Quantum World

Thomson’s opponents argued that his time scales were too short for life to develop. He ignored them. With hindsight, it is easy to see that the brilliant Thomson was wrong and Darwin was right. But should we judge Thomson harshly for not listening to his opponents?

We must be careful when we judge the past and not look through the lens of our own experiences or biases. Thomson lived to 1907, to a time when radioactivity had firmly been established. The observation then by geologists that the Earth could be heated from within by radioactive decay meant that the Earth could be a lot older than Thomson thought. In fact, it was widely believed that the discovery of radioactivity invalidated Thomson’s estimates of the age of the Earth.

This image shows the Sun as viewed by the Soft X-Ray Telescope (SXT) onboard the orbiting Yohkoh satellite

Despite the discovery of radioactivity, Thomson refused to acknowledge this. He had strong reason to believe that the Sun was younger than 20 million years. Even with an old Earth, without sunlight, there could be no explanation for the sediment record on the Earth’s surface. It wasn’t until the discovery of fusion in the 1930s that this paradox was resolved.

While history has proven Thomson wrong in this debate, we must remember one thing. Given what we knew at the time, Thomson’s calculations and conclusions were correct–his science was sound. We can not fault Thomson for this.

The Higgs Boson Course

Data from the CMS experiment, one of the main Higgs-searching experiments at the Large Hadron Collider. Image: CERN

The Higgs boson is the latest addition to our understanding of what happens in our Sun. Just like Thomson demanded a certain mathematical rigor, so too does “The Discovery of the Higgs Boson” course. The course recommends that the week’s module should take about two hours. Though it has been some time since I last sat in a Physics class and while the concepts and proofs were not entirely new, it does take some time to view the lectures and complete the exercises. I think students will realistically have to dedicate more than two hours to complete the week’s exercises.

Overall, it is a strong course that demands a lot of its students. I look forward to the rest of the course.

Video

Aerial Photography Demonstration at ScienceWriters2013

Some scenes at the aerial camera drone demonstration at the ScienceWriters2013 convention in Gainesville, FL.

Pinning Butterflies

University of Florida, Gainesville student, Lucasz, shows us the process of pinning a butterfly. The butterflies are taken from their moist environment and pinned to where they are dried and can be displayed.

University of Florida Gainesville student Emma shows us butterflies wrapped before pinning.

Collection of butterflies wrapped and stored in a container before they can be pinned.

University of Florida student Emma shows us what a butterfly looks like before she pins it. Butterflies can last for a long time under these conditions.

University University of Florida student, Lucasz, demonstrates the process of pinning a butterfly at the Florida Museum of Natural History. This was part of the welcome reception of ScienceWriters2013. The process, from start to finish, takes about five minutes. Lucasz does it in three. Video shot by David Latchman.

The 2013 Ig Nobel Physics Prize: How to Run on Water

Check out my latest Decoded Science post on the 2013 Ig Nobel Prize on “How to run on Water“.

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

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.

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

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 Dolbear’s Law

Season 03, Episode 02: “The Jiminy Conjecture”

In the “The Jiminy Conjecture” episode, Raj (Kunal Nayyar), Howard (Simon Helberg) and Sheldon (Jim Parsons) are having dinner when they hear the sound of a cricket chirping. Sheldon claims to know the species from the cricket’s chirping speed and the room’s ambient temperature. Not to be outdone and still angry from losing a bet in the comic book store, Howard wagers his rare Fantastic Four #48 (“The Coming of Galactus”) against Sheldon’s The Flash #123 “Flash of Two Worlds”.

Dolbear’s Law

Sheldon is able to determine the room’s temperature by using Dolbear’s Law, an equation that states the relationship between air temperature and the rate at which a cricket chirps. This relationship was formulated by an American physicist, Amos Dolbear in 1897 in his article “The Cricket as a Thermometer”. While Dolbear didn’t specify the species of cricket in his article, it is generally believed to the snowy tree cricket.

Amos Emerson Dolbear (November 10, 1837 – February 23, 1910) was an American physicist and inventor. He is noted for his pioneering work in wireless telegraphy — several years before Guglielmo Marconi. He is also known for finding a relatoinship between a cricket’s chirp rate and ambient temperature.

If we look closely we can see Dolbear’s equations written on the whiteboard in the background. They all describe the relationship between temperature in Fahrenheit to the number of chirps different species of cricket make in one minute.

\begin{align*}
\mathrm{Field~Cricket:} \; T_{f} &= 50 + \frac{(N-40)}{4} \\
\mathrm{Snowy~Tree~Cricket:} \; T_{f} &= 50 + \left[\frac{(N – 92)}{4.7}\right] \\
\mathrm{Katydid:} \; T_{f} &= 60 + \left[\frac{(N – 19)}{3}\right]
\end{align*}

It is a popular myth that crickets chirp by rubbing their legs together but that couldn’t be further from the truth. Only the males chirp and they produce sounds through the process of stridulation i.e by rubbing certain body parts together. In the case of crickets, there is a stridulatory organ, a large vein running along the bottom of each wing, that is covered with “teeth” or serrations like a comb. The chirping sound is created when the cricket runs the top of its wing along the bottom serrated edge of the other. As the cricket does this, it holds its wings up and open so they act as acoustical sails.

Insects are cold-blooded and take the temperature of their surroundings. As a result, a cricket’s chirp rate depends on its metabolism. This can be determined by the Arrhenius equation which describes the activation energy needed to induce a chemical reaction. The less energy there is, the slower a chemical reaction and hence metabolism. As temperatures rise, more energy is available for muscle contractions. This accounts for the relation of observed by Dolbear.

Measuring Dolbear’s Law

We know by the end of the show that Howard was right and Sheldon was wrong. We can calculate the number of chirps with temperature for the various cricket species seen on the whiteboard.

Table showing Temperature and Number of Cricket Chirps per Minute
Temperature(°F) Temperature(°C) Field Cricket Snowy Tree Cricket Katydid
60 16 80 139 19
65 18 100 163 34
70 21 120 186 49
75 24 140 210 64
80 27 160 233 79
85 29 180 257 94
90 32 200 280 109

There are noticeable differences in chirp rates between the snowy tree cricket and the field cricket–the snowy tree cricket has a faster chirp rate. But how could Sheldon been wrong? According to the roommate agreement, the apartment is supposed to be 71°F(22°C) which translates to a chirp rate of 191 for the snowy tree cricket. Assuming Sheldon counted the chirps correctly, that would translate to a much warmer temperature of 88°F(31°C) for the field cricket. Could someone have changed the temperature in violation of the roommate agreement? Surely this would have been noticed without looking at the thermostat. Your guess is as good as mine.

The Big Bang Theory of Trojan Asteroids

Season 06, Episode 03: “The Higgs Boson Observation”

In “The Higgs Boson Observation”, we learn that the subject of graduate student and Sheldon’s (Jim Parsons) new assistant, Alex Jensen (Margo Harshman), is on Trojan asteroids at the Earth-Sun L5 Lagrange point. This also happens to be Raj’s (Kunal Nayyar) area of expertise. But what exactly are Trojan asteroids and why are they of interest to astronomers and astrophysicists?

Trojans are minor planets or satellites that share an orbit with a planet but does not collide with it because it orbits around one of the two stable Lagrangian points (trojan points). The L4 and L5 lie approximately 60° ahead of and behind the larger body, respectively .

Area of Research

These are the five Lagrangian points in a two-body system with one body far more massive than the other (e.g. the Sun and the Earth)

There are five points on or near Earth’s orbit known as Lagrange points where an asteroid will remain stationary with respect to Earth as it orbits around the Sun. These points mark the positions where the combined gravitational pull of two large bodies, such as the Earth and the Sun, provides the centripetal force needed to orbit with them.

The Lagrangian points are approximate solutions to what is known as the three-body problem, a famous problem that attempts to model the motion of three bodies as they gravitationally affect each other. In essence, this problem dates back to 1687 when Sir Isaac Newton first published his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), often referred to as simply the “Principia”. Several scientists and mathematicians have all attempted to find an analytical solution to the problem or a simple equation that would describe the motions of all three bodies.

The three-body problem is difficult to solve because the three bodies tug against each other in chaotic and unpredictable ways. Only by simulating the problem on a computer can we see the objects paths. In comparison, the quantum three-body problem is relatively easy to solve and understand. However, approximate solutions can be useful. If the third mass is taken to be small enough that it does not affect the other two masses some interesting solutions crop up.

Celestial Mechanics

To better understand how celestial bodies move in our solar system, we turn to Newton’s law of universal gravitation. This law states that every mass in the Universe attracts every other mass with a force that is proportional to the product of their masses. If we had two masses, $$M_{1}$$ and $$M_{2}$$, then the force between them would be proportional to the product of the two masses.
$F \propto M_{1}M_{2}$
This law also states that the force decreases with the square of the distance between them.
$F \propto \frac{1}{r^{2}}$
We can combine these into one equation that will help is analyze and solve the movement of celestial bodies anywhere in the Universe.
$F=G\frac{M_{1}M_{2}}{r^{2}}$
where $$G$$ is the gravitational constant. This law was deduced by Newton from observations and was formulated in the Principia.

If we were to look at the Sun and Earth system, we would have what is known as the two-body problem. While we may think this is a simple problem but in reality it isn’t. The Earth does not revolve around the Sun as if the more massive Sun was a stationary body but rather both the Earth and Sun revolve around a common point — the center of mass or barycenter. Though makes the problem a little more difficult, as this point is very close to the center of the Sun and we can make an approximation where the Sun is in the center and doesn’t move while the Earth moves around it.

If we use this technique, we can derive an equation that describes the Earth’s motion around the Sun. This is a classic “two-body” problem and its solutions describe the familiar elliptical orbits of the planets known since the time of Kepler. Unfortunately, when we add a third body to our equations of motion, such as a spacecraft lost in space between the Earth and the sun, we can no longer find an analytical solution; the equations become unsolvable.

While the three-body problem has no analytical solution, Joseph-Louis Lagrange in 1772 showed that if we restricted the mass of the third body to be so small that it couldn’t affect the other two, there were some solutions to be found. The solutions to this “restricted three-body problem” finds that the three bodies move in unison and always maintain the same position relative to each other.

These five points, where the third body stays in the same relative point in space, are known as Lagrange points and mark the positions where the combined gravitational pull of the two large masses precisely provides the force to orbit with them. They are labelled L1, L2, L3, L4 and L5 and also known as L-points or libration points.

The Trojan Asteroids

The L4 and L5 libration points lie on the corners of two equilateral triangles. The points are balanced because the gravitational forces between the Earth and Sun keep any object in orbital equilibrium with the rest of the system. These points are sometimes referred to as “triangular Lagrange points” or “Trojan points” and come from the Trojan asteroids at the Sun-Jupiter L4 and L5 points. These asteroids were named after characters from Homer’s Iliad. Asteroids at the L4 point leads Jupiter and is referred to as the “Greek camp” while those are the L5 are referred to as the “Trojan camp”.

The Earth-Sun system, like the Jupiter-Sun system, also uses the same terms of reference. There is one known Greek asteroid in the Earth-Sun system, 2010 TK7, first detected in October 2010 by Dr. Martin Connors using the Wide-field Infrared Survey Explorer (WISE) . The asteroid’s designation come from the provisional naming system for minor planets. This asteroid which has a diameter of about 300 meters oscillates about the L4 Lagrangian point. The region around these points is known to contain interplanetary dust. As people tend to search for asteroids at much greater elongations, very few searches have been done at these locations. Currently, there are no confirmed or suspected L5 Trojans of Earth which makes both Raj’s and Alex’s work interesting and something to talk about.

The Big Bang Theory of the Doppler Effect

Season 01, Episode 06: “The Middle-Earth Paradigm”

Sheldon Cooper dresses as the Doppler Effect on the Season 1, Episode 6 “The Middle-Earth Paradigm” of the Big Bang Theory

Yes. It’s the apparent change in the frequency of a wave caused by relative motion between the source of the wave and the observer.

-Sheldon Cooper

In the “Middle-Earth Paradigm” episode, Sheldon Cooper dresses as the “Doppler Effect” for Penny’s Halloween party. The Doppler Effect (or Doppler Shift) describes the change in pitch or frequency that results as a source of sound moves relative to an observer; moving relative can mean either the source is moving while the observer is stationary or vice versa. It is commonly heard when a siren approaches and recedes from an observer. As the siren approaches, the pitch sounds higher and lowers as it moves away. This effect was first proposed by Austrian physicist Christian Doppler in 1842 to explain the color of binary stars.

In 1845, Christophorus Henricus Diedericus (C. H. D.) Buys-Ballot, a Dutch chemist and meteorologist conducted the famous experiment to prove this effect. He assembled a group of horn players on an open cart attached to a locomotive. Ballot then instructed the engineer to rush past him as fast as he could while the musicians played and held a constant note. As the train approached and receded, Ballot noted that the pitch changed as he stood and listened on the stationary platform.

Physics of the Doppler Effect

A stationary sound source has sound waves radiating outward and can be viewed as concentric circles.

As a stationary sound source produces sound waves, its wave-fronts propagate away from the source at a constant speed, the speed of sound. This can be seen as concentric circles moving away from the center. All observers will hear the same frequency, the frequency of the source of the sound.

When either the source or the observer moves relative to each other, the frequency of the sound that the source emits does not change but rather the observer hears a change in pitch. We can think of the following way. If a pitcher throws balls to someone across a field at a constant rate of one ball a second, the person will catch those balls at the same rate (one ball a second). Now if the pitcher runs towards the catcher, the catcher will catch the balls faster than one ball per second. This happens because as the catcher moves forward, he closes in the distance between himself and the catcher. When the pitcher tosses the next ball it has to travel a shorter distance and thus travels a shorter time. The opposite is true if the pitcher was to move away from the catcher.

If instead of the pitcher moving towards the catcher, the pitcher stayed stationary and the catcher ran forward. As the catcher runs forward, he closes in the distance between him and the pitcher so the time it takes from the ball to leave the pitcher’s hand to the catcher’s mitt is decreased. In this case, it also means that the catcher will catch the balls at a faster rate than the pitcher tosses them.

Sub Sonic Speeds

The source radiates sound waves outward. As it moves, the center of each new wavefront is slightly displaced to the right and the wavefronts bunch up on the right side (front) and spread out further apart on the left side (behind) the source.

We can apply the same idea of the pitcher and catcher to a moving source of sound and an observer. As the source moves, it emits sounds waves which spread out radially around the source. As it moves forward, the wave-fronts in front of the source bunch up and the observer hears an increase in pitch. Behind the source, the wave-fronts spread apart and the observer standing behind hears a decrease in pitch.

The Doppler Equation

When the speeds of source and the receiver relative to the medium (air) are lower than the velocity of sound in the medium, i.e. moves at sub-sonic speeds, we can define a relationship between the observed frequency, $$f$$, and the frequency emitted by the source, $$f_0$$.
$f = f_{0}\left(\frac{v + v_{o}}{v + v_{s}}\right)$
where $$v$$ is the speed of sound, $$v_{o}$$ is the velocity of the observer (this is positive if the observer is moving towards the source of sound) and $$v_{s}$$ is the velocity of the source (this is positive if the source is moving away from the observer).

Source Moving, Observer Stationary

We can now use the above equation to determine how the pitch changes as the source of sound moves towards the observer. i.e. $$v_{o} = 0$$.
$f = f_{0}\left(\frac{v}{v – v_{s}}\right)$
$$v_{s}$$ is negative because it is moving towards the observer and $$v – v_{s} < v$$. This makes $$v/(v - v_{s})$$ larger than 1 which means the pitch increases.

Source Stationary, Observer Moving

Now if the source of sound is still and the observer moves towards the sound, we get:
$f = f_{0}\left( \frac{v + v_{o}}{v} \right)$
$$v_{o}$$ is positive as it moves towards the source. The numerator is larger than the denominator which means that $$v + v_{o}/v$$ is greater than 1. The pitch increases.

Speed of Sound

As the source of sound moves at the speed of sound the wave fronts in front of the source all bunch up at the same point.

As the source of sound moves at the speed of sound, the wave fronts in front become bunched up at the same point. The observer in front won’t hear anything until the source arrives. When the source arrives, the pressure front will be very intense and won’t be heard as a change in pitch but as a large “thump”.

The observer behind will hear a lower pitch as the source passes by.
$f = f_{0}\left( \frac{v – 0}{v + v} \right) = 0.5 f_{0}$

Early jet pilots flying at the speed of sound (Mach 1) reported a noticeable “wall” or “barrier” had to be penetrated before achieving supersonic speeds. This “wall” is due to the intense pressure front, and flying within this pressure front produced a very turbulent and bouncy ride. Chuck Yeager was the first person to break the sound barrier when he flew faster than the speed of sound in the Bell X-1 rocket-powered aircraft on October 14, 1947.

Bell X-1 rocket plane of the United States Air Force (NASA photography)

As the science of super-sonic flight became better understood, engineers made a number changes to aircraft design that led the the disappearance of the “sound barrier”. Aircraft wings were swept back and engine performance increased. By the 1950s combat aircraft could routinely break the sound barrier.

Super-Sonic

As the source moves faster than the speed of sound, i.e. faster than the sound waves it creates, it leads its own advancing wavefront. The sound source will pass by the stationary observer before the observer hears the sound.

As the sound source breaks and moves past the “sound barrier”, the source now moves faster than the sound waves it creates and leads the advancing wavefront. The source will pass the observer before the observer hears the sound it creates. As the source moves forward, it creates a Mach cone. The intense preseure front on the Mach cone creates a shock wave known as a “sonic boom”.

Twice the Speed of Sound

Something interesting happens when the source moves towards the observer at twice the speed of sound — the tone becomes time reversed. If music was being played, the observer will hear the piece with the correct tone but played backwards. This was first predicted by Lord Rayleigh in 1896 .

We can see this by using the Doppler Equation.
$f = f_{0}\left(\frac{v}{v-2v}\right)$
This reduces to
$f=-f_{0}$
which is negative because the sound is time reversed or is heard backwards.

Applications

U.S. Army soldier uses a radar speed gun to catch speeding violators at Tallil Air Base, Iraq.

The Doppler Effect is used in radar guns to measure the speed of motorists. A radar beam is fired at a moving target as it approaches or recedes from the radar source. The moving target then reflects the Doppler-shifted radar wave back to the detector and the frequency shift measured and the motorist’s speed calculated.

We can combine both cases of the Doppler equation to give us the relationship between the reflected frequency ($$f_{r}$$) and the source frequency ($$f$$):
$f_{r} = f \left(\frac{c+v}{c-v}\right)$
where $$c$$ is the speed of light and $$v$$ is the speed of the moving vehicle. The difference between the reflected frequency and the source frequency is too small to be measured accurately so the radar gun uses a special trick that is familiar to musicians – interference beats.

To tune a piano, the pitch can be adjusted by changing the tension on the strings. By using a tuning instrument (such as a tuning fork) which can produce a sustained tone over time, a beat frequency can be heard when placed next to the vibrating piano wire. The beat frequency is an interference between two sounds with slightly different frequencies and can be herd as a periodic change in volume over time. This frequency tells us how far off the piano strings are compared to the reference (tuning fork).

To detect this change in a radar gun does something similar. The returning wave is “mixed” with the transmitted signal to create a beat note. This beat signal or “heterodyne” is then measured and the speed of the vehicle calculated. The change in frequency or the difference between $$f_{r}$$ and $$f$$ or $$\Delta f$$ is
$f_{r} – f = f\frac{2v}{c-v}$
as the difference between the speed of light, $$c$$, and the speed of the vehicle, $$v$$, is small, we can approximate this to
$\Delta f \approx f\frac{2v}{c}$
By measuring this frequency shift or beat frequency, the radar gun can calculate and display a vehicle’s speed.

“I am the Doppler Effect”

The Doppler Effect is an important principle in physics and is used in astronomy to measure the speeds at which galaxies and stars are approaching or receding from us. It is also used in plasma physics to estimate the temperature of plasmas. Plasmas are one of the four fundamental states of matter (the others being solid, liquid, and gas) and is made up of very hot, ionized gases. Their composition can be determined by the spectral lines they emit. As each particle jostles about, the light emitted by each particle is Doppler shifted and is seen as a broadened spectral line. This line shape is called a Doppler profile and the width of the line is proportional to the square root of the temperature of plasma gas. By measuring the width, scientists can infer the gas’ temperature.

We can now understand Sheldon’s fascination with the Doppler Effect as he aptly explains and demonstrates its effects. As an emergency vehicle approaches an observer, its siren will start out with a higher pitch and slide down as as it passes and moves away from the observer. This can be heard as the (confusing) sound he demonstrates to Penny’s confused guests.