I bought a book titled ” Black Holes and Time Warps: Einstein’s outrageous legacy” by Dr Kip S. Throne about an year and a half ago and I had never imagined that a movie will be made on the book and that it will be directed by the legend himself. When I learnt that Nolan has teamed up with Kip to fire up the script, I was happy that I will finally be able to swim across Gargantua.
I bought the book for two reasons: one is that I am a space enthusiast and two, well, I stumbled upon a YouTube video with Kip and Preskill in it (both being Richard Feynman Professors of Physics at the Caltech). The video ends with Stephen Hawking making a guest appearance out of him:
The reason for me to quit my first job and work in research was due to Dr. Richard Feynman. There’s a long story behind it and I will share it with you some other time. So Kip caught my attention and I youtubed more of his videos and that was when I started getting interest in theoretical Physics and Astrophysics in particular. I bough his book from Flipkart for Rs 800 and I recommend this book for you guys; it’s worth a read.
I was surprised by another coincidence: I recently published my blog post titled “Kardashev-Shankaracharya model for exo-planet colonization of Humanoids” which is exactly the “Plan B” of the movie. If you read the post you will realize is that India could play a major role in enhancing Artificial Intelligence in times to come. The minds of Indians and the might of the west can take the world to a new paradigm. The Chinese too can play a major role because of their deep rooted history and experience in nurturing human civilization.
Now, coming to the movie, I was happy with the movie because I had read the book. I am quite sure you would have had the same face before leaving the theater hall as in the case of the “Inception” movie: that’s Chris’s style of playing with our minds. I felt that Chris wanted the audience to be able to digest science and not puke up the whole theater with confusion and popcorn. But most audience did puke.
I will attach two plots made by Nolan’s fan base. The first one is simple to understand, I hope [click on the timeline image to open it in its native resolution]:
[Stick with me and you will understand and appreciate the movie]
I will hope to keep it short and sweet before I put myself into cryo-sleep mode:
Earth is bitten by Blight that has been courteous to only corn plants: this would not be an issue for people in northern part of Karnataka because of their popular Jawar roti (corn totilla) diet. Dr. Brand is a theoretical Physicist who has a plan of building an Ark, just like Noah’s Ark, that can accommodate the living population and transport them away from the Earth which is pretty much inhabitable for corn haters.
Dr Brand also happens to be the mission director of NASAs secret space mission. He first sends 12 crew members through “Lazarus mission” through a black hole that appeared from no where: something, that Dr Brand claims, to have been set up by “them” (Future). the plan is to send the crew on a suicide mission: to warp into the black hole. The crew is supposed to split and search for habitable planets on the other side of the worm hole. Three of the crew members manage to find planets being Miller, Mann and Edmund (named after the respective crew members). The rest of the crew are plain dead. The three manage to send signals back to earth. Dr. Brand now has a strong reason to puncture NASAs money stash to pump his objective into the next mission being “Mission Endurance”.
Dr Brand comes up with two plans for Endurance mission to execute followed by partially successful Lazarus mission:
“Plan A: Go through the wormhole to the three potentially habitable planets, find the one suitable for relocation and also receive some data from the black hole and relay it to the Earth so that Professor Brand could solve the gravity problem that would enable him to carry out mass evacuation from the Earth using the scant resources available in that apocalyptic scenario.”
“Plan B: Forget about Earth if Professor Brand is unable to solve the gravity problem and instead start off a new mankind colony on whichever planet they find suitable for habitation using the frozen embryos they were carrying.”
This is my opinion: Everything in this movie is gravitated towards Dr Brand solving “Gravity Equation” so it is important for us to know a bit about it:
If Dr. Brand manages to solve the Gravity Equation, then he will be able to:
1) Creating Artificial Gravity.
2) Create a Wormhole that allows inter-galaxy travel (wormhole is not a naturally occurring phenomenon)
3) Lift the Ark (which will eventually be named Cooper-Station) into space defying the laws of gravity.
4) Travel through the worm hole, and to the habitable planet, and land his Ark there.
That’s it. Simple ain’t it?
Now a little bit into the physics of the equation [Those not interested in Physics can skip this section and head to the later part of the post]:
Firstly, why is Earth revolving around the sun and why is moon revolving around Earth? the two beautiful science enthusiasts in the video below could help us answer the question:
What you noticed is that they placed a heavy mass at the center of the spandex trampoline and the spandex warps downward, and then they roll metal balls around the warp and the metal balls revolve around it and will eventually settle into the dent and reach to a steady state condition.
Now imagine space as the spandex and Sun as the heavy mass at the center of the spandex and the planets revolving around the sun as metal balls: the concept is one and the same. In fact all the planets will collide into Sun some day but it won’t happen any soon so don’t worry about it.
Now I will ask you to tweak your imagination about the Spandex a bit: make the spandex go transparent to an extent that it is not visible to the naked eyes and that you cannot touch or feel the spandex but the warp created by the spandex is apparent: the heavy mass will now appear as if it is suspended in empty space. In this case, the small metal balls shall have no surface to trail and they have an infinite degree of freedom of movement around the heavy mass. That’s Solar system for you.
Einstein predicted that this invisible spandex called spacetime gets warped around heavy mass like earth and sun and other celestial objects. Einstein also predicted that even time trails the curvature and gets curved due to gravitation. This is the reason why time slows down as we move away from earth and speeds up as we move close to earths core. This phenomenon is called “Time Dilation”.
From the local frame of reference (the blue clock), a relatively accelerated (e.g. red) clock moves more slowly. Time dilation explains why two working clocks will report different times after different accelerations. For example, International Space Station (ISS) astronauts return from missions having aged slightly less than they would have been if they had remained on Earth, and GPS satellites work because they adjust for similar bending of spacetime to coordinate with systems on Earth. The atomic clock of the GPS must offset the latency to stay in sync with the clocks on ground stations. This explains the relative velocity time dilation.
But my focus is towards Gravitational time dilation: Gravitational time dilation is at play for ISS astronauts too, and it has the opposite effect of the relative velocity time dilation. To simplify, velocity and gravity each slow down time as they increase. Velocity has increased for the astronauts, slowing down their time, whereas gravity has decreased, speeding up time (the astronauts are experiencing less gravity than on Earth). Nevertheless, the ISS astronaut crew ultimately end up with “slower” time because the two opposing effects are not equally strong. The velocity time dilation (explained above) is making a bigger difference, and slowing down time. The (time-speeding up) effects of low-gravity would not cancel out these (time-slowing down) effects of velocity unless the ISS orbited much farther from Earth.
It is important to understand time dilation without which the movie would be boring and meaningless. Let me remind you: Interstellar is not a Science fantasy movie like Harry Potter: it’s a science fiction movie with plausible evidences. The deeper you dive into the Physics of it, the more convinced you will be of it.
General Relativity says:
a) Geometry of spacetime is described by metric tensor g.
b) Matter tells spacetime how to curve and curvature of spacetime tells the matter how to move.
c) Field equations of General Relativity are called Einstein equations:
It has been of interest for Quantum Physicists to be able to understand gravity at sub-atomic scale for which they need to quantize gravity: What they realized is that quantizing Einsteins General relativity to fit their curiosity would lead to non-renormalizable theory which cannot be considered as a proper theory.
In 1995, a theory called M-Theory was proposed which basically unifies all known interactions, including gravity. it is defined in 11 dimensions and predicts that fundamental constituents of the universe are strings, membranes and higher dimensional objects. This theory has a weakness though: it makes sense only in the case when supersymmetry is realised in nature, and an even more serious weakness: compactification problem (In physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be periodic, just like how FIR is to IIR in DSP).
However, M-theory offers many topology of additional dimensions of which one of the most simplest and interesting were:
This topology is called the 5D equation. In Astrophysics there is a term called “brane”: brane is nothing but the same invisible spandex, and can span depending on the size of interest i.e., the whole universe is assumed to be a brane and there are other universes in parallel and that the bulk between the branes is what is the 5th dimension:
The black disc or brane in the picture above is the universe which has been expanding post Big-Bang. The 5th dimension is orthogonal to the brane and I think the equation above is the one Dr. Brand is working on because based on what I understood form other sources, the possible size of the 5th dimension is:
“There are two possibilities:
a) Additional dimension is so small that we cannot observe it in our experiments (compactification). maybe possible in future
b) We assume very special properties of the model (space in the bulk, cosmological constant…). Then we can have even infinite size of the fifth dimension without violating known formula of gravitational force.“
So the only way to to complete the equation is by calculating S ( holding times as the random time elapsed between two subsequent events) and Z (I am still figuring out what Z is, I will update this post as I get info on Z, thanks for excusing me)
Following is the visualization of 5th Dimension (4th D being time is not shown but assumed)
in the above visualization, notice the small white circles: they are gravitons which are not confined to either of the branes and are moving freely between the branes.
This equation holds the key to quantize and hence control gravity and controlling of gravity implies altering the space-time spandex and thus creating “man-made worm hole”. I got this piece of info from Quora (thanks to Casey Handmer, PhD student in Mathematical Physics, Caltech):
“This is left up to speculation. However you can infer from the blackboard equations etc that he and Murph were trying to unify formalism used for gravity and quantum physics. A lot of the physics is highly speculative, but if you lived in a universe where (stable) wormholes were known to exist then you’d know that the average null energy condition can be violated, which would really encourage you to work out the details.
If you can get around various null energy conditions you can build worm holes, time machines (to a point) and warp drives. I surmised from the film that they used warps to get the large concrete spaceships into orbit. Though why, I’m not sure – living in a concrete box on Earth would be easier in space. And if you can manipulate gravity, there is no need to spin it up.
Kip Thorne suggests in his book that when all the colonies were loaded Murph locally reset Newton’s gravity constant G, effectively turning off the Earth’s gravity so that small rockets could boost the space stations to Saturn. And also destroying the Earth in the process…”
So Murph would need information regarding quantum gravity to balance the equation which would be furthered by hours of simulations. So, how will Murph solve the equation? I will answer to this question in a while.
Back to Plan A and Plan B of the mission: The first planet that Cooper and his crew visit is Miller’s. The most impressive things about this planet are the extreme slowing of time there, gigantic water waves, and huge tidal gravity. All three are related, and arise from the planet’s closeness to Gargantua.
Keeping the same spandex analogy in mind, we see that Endurance is parked (yellow band) at a distance quite safe from the abysmal black hole Gargantua, and Miller is much closer to the event horizon. In layman’s term, Event horizon is defined as the “point of no return”: not even light can escape from it if it comes in contact with the event horizon. Miller is positioned in a very uncompromising orbit: Time slows near Gargantua, and the slowing becomes more extreme as we get closer and closer to Gargantua’s event horizon. Therefore, according to Einstein’s law of time warps, gravity becomes ultra-strong as we near the horizon. So an hour in Miller’s planet would be equivalent to 7 years on Earth.
The huge tides: Being so close to Gargantua, Miller’s planet is subjected to enormous tidal gravity, so enormous that Gargantua’s tidal forces almost tear the planet apart. Almost, but not quite. Instead, they simply deform the planet. Deform it greatly. It bulges strongly toward and away from Gargantua. Earth bulges towards and away from Moon’s gravity which is the reason for surfers enjoying in beaches playing with tides. Unlike Earth, Miller is harassed by Gargantua’s enormous gravitational pull and hence the 1.2 Kilometers high waves:
The centrifugal force that the planet feels has a more gradually changing slope. There the planet can travel around Gargantua with the outward centrifugal force balancing the inward gravitational force. If Gargantua was a smaller black hole, Miller would not exist at all: this is similar to a blue whale swallowing a small fish: the fish might not die at all because of the whales enormous size whereas the chances of the same fish surviving a shark would be very small.
Following is depiction of how Garagntua’s enormous accretion disk would appear if viewed from the ranger craft. Notice how the disk is eclipsed partially by the planet:
Clearly Miller’s planet is hopeless for human colonization. When Cooper and Amelie head back to Endurance having failed to get the black-box back, they bestow their hopes on the next planet being Mann’s planet. When the crew head back to Endurance, enough number of years would have passed on earth and in Endurance. Romilly (the black Physicist-cum- Astronaut) will wake up from his cryogenic sleep mode and welcome the crew back to Endurance. Cooper has 20 years worth of video transmitted from earth and he will see Murph and his son all grown up while he has aged by few years. Romilly tells Amelia, “I learned what I could from studying the black hole, but I couldn’t send anything to your father. We’ve been receiving but nothing gets out.” What did Romilly observe? Gargantua’s vibrations:
. . His data will include Quasinormal Mode freqencies of Gangantua’s resonant vibration. While Cooper and Amelia Brand are on Miller’s planet, Romilly stays behind in the Endurance, observing Gargantua. From exquisitely accurate observations, he hopes to learn more about gravitational anomalies and he hopes that quantum data from Gargantua’s singularity will leak out through the event horizon, bringing information about how to control the gravitational anomalies. Or, in Romilly’s pithy language, bringing information for “solving gravity” back on Earth.
Mann’s planet’s orbit is quite tricky:
If you notice in the Interstellar timeline chart, you see that Mann’s planet is farther away from Gargantua but by the time Cooper and crew make it to Mann’s planet it would have orbited towards Gargantua.
Endurance parks in orbit around Mann’s planet, and then Cooper and his crew descend to the planet in the Ranger. The planet is covered with ice since it spends most of its life far from the warmth of Gargantua’s accretion disk. As the Ranger nears the planet, we see it maneuver among what appear to be clouds, but then it scrapes along one and we discover the cloud is actually made from some sort of ice.
these clouds are largely frozen carbon dioxide, “dry ice,” and they are starting to be warmed as the planet is on its inward excursion toward the accretion disk, When warmed, dry ice sublimates and vaporizes, and so what appears to be clouds may be a mixture of dry ice and sublimating vapor; perhaps mostly vapor. At lower altitudes, where the Ranger lands, temperatures are higher and the ice on which they land is presumably all frozen water.
In the meanwhile, Dr. Brand will be on his death bed and will kick the bucket leaving behind an incomplete equation that Murph has to now solve to save humanity.
Mann happens to be a cruel man and we know what happens there. Mann is a disgrace to humanity. Mann will falsely claims with fudged geological data that the planet is habitable and his intention will be to steal the Ranger, head to Endurance and escape to Earth. We lose Romilly here in a blast that results into a greenish yellow flame, this is due to the presence of Ammonia in air.
So Cooper will be saved by Amelia and TARS after his fight with Mann and they will chase Mann who is heading towards Endurance in Ranger craft. Mann will manhandle Endurance while trying to manually override the docking of ranger into it. Now why Nolan must be appreciated as a director is because the explosion of Endurance that we see in the movie is in fact not a computer generated special effect: it is real and done in movie set: This is Christopher Nolan’s commitment to Science accuracy.
One of the ring modules explodes when Mann’s Ranger vehicle collides with it. The result is that the explosive force happens perfectly in plane with the vehicle and perfectly orthogonal to the ring, inducing a perfect roll at 68 rpm.
This means that Cooper “just” has to match the 68 rpm roll and close range with the center of the vehicle.
That just wouldn’t happen. The collision and explosion would be off-axis and would impart roll, pitch, and yaw on the vehicle – it would tumble. Any pitch or yaw and it would be impossible for Cooper to dock his ranger.
But, let’s just assume that the one in one hundred-trillion odds were in their favor and the vehicle somehow only began to roll, what would be the next obstacle?
Cooper has to match the 68 rpm roll. If both vehicles are rolling at exactly the same rate, it is as if there is no rotation. The spinning Ranger vehicle will have a centrifugal force pushing outwards from the center of the spin axis. The two seats were separated and off the axis. I’d guess Cooper’s seat was between 1 and 2 meters from the center-line of the vehicle.
I want to spend time in this department because the crux of the movie lies here. Of the 5 dimensions, we are familiar with three: Up-Down, East-West and North-South. The Forth dimension is Time, at least that’s what physicists claim.
Look at the diagram below:
This is our universe in one brane, one spandex. There is a small brane which is revolving around the large brane. The small brane could be Gargantua Black hole and the large one could be a super-massive black hole at the center of the universe. The bulk or the dent has additional dimensions which cannot be visualized on a 3D diagram. Time dilation is the only evidence that it is knitted into space: The only way to visualize time is by living it 🙂
Well, there’s another dimension that will make it to 5 plausible dimensions out of 11 dimensions proposed in Super-String theory. What is of interest is this one extra dimension after space and time: the rest of the six are too complex for me to cover in this post, I wish to take them up some other time. Let’s stick to 5 dimensions for now.
The fifth dimension, “Out-back”, extends perpendicular to our brane:
Atomic limitation: Atoms have three space dimensions. They can only exist in three space dimensions, not four. And this is true of subatomic particles as well. And it is true also of electric fields and magnetic fields and the forces that hold atomic nuclei together.
So when we talk about 4th and 5th dimensions, our imagination is the only way to visualize them. Some of the world’s most brilliant physicists have struggled to understand how matter and fields and forces behave if our universe really is a brane in a higher-dimensional bulk. Those struggles have pointed rather firmly to the conclusion that all the particles and all the forces and all the fields known to humans are confined to our brane, with one exception: gravity, and the warping of spacetime associated with gravity. There might be other kinds of matter and fields and forces that have four space dimensions and reside in the bulk. But if there are, we are ignorant of their nature. We can speculate. Physicists do speculate. But we have no observational or experimental evidence to guide our speculations. In Interstellar, on Professor Brand’s blackboard, we see him speculating.
It’s a reasonable, half-educated guess that, if bulk forces and fields and particles do exist, we will never be able to feel them or see them. When a bulk being passes through our brane, we will not see the stuff of which the being is made. The being’s cross sections will be transparent. On the other hand, we will feel and see the being’s gravity and its warping of space and time. For example, if a hyper-spherical bulk being appears in my stomach and has a strong enough gravitational pull, my stomach may begin to cramp as my muscles tighten, trying to resist getting sucked to the center of the being’s spherical cross section.
All the characters in Interstellar are convinced that bulk beings exist, though they use that name only rarely. Usually, the characters call the bulk beings “They.” A reverential They. Early in the movie, Amelia Brand says to Cooper, “And whoever They are, They appear to be looking out for us. That wormhole lets us travel to other stars. It came along right as we needed it. ”One of Christopher Nolan’s clever and intriguing ideas is to imagine that They are actually our descendants: humans who, in the far future, evolve to acquire an additional space dimension and live in the bulk.
Late in the movie, Cooper says to TARS, “Don’t you get it yet, TARS? They aren’t beings. They’re us, trying to help, just like I tried to help Murph.” TARS responds, “People didn’t build this tesseract, Not yet,” Cooper says, “but one day. Not you and me but people, people who’ve evolved beyond the four dimensions we know.”
Cooper, Brand, and the crew of the Endurance never actually feel or see our bulk descendants’ gravity or their space warps and whirls. That, if it ever occurs, is left for a sequel to Interstellar. But older Cooper himself, riding through the bulk in the closing tesseract, reaches out to the Endurance’s crew and his younger self, reaches out through the bulk, reaches out gravitationally. Amelia feels and sees his presence, and thinks he is They.
In 1999, Lisa Randall at Princeton University and MIT and Raman Sundrum at Boston University conceived another way to stop gravitational force lines from spreading into the bulk: the bulk could suffer what is called “Anti-deSitter warping.” This warping might be produced by what are called “quantum fluctuations of bulk fields”. By contrast, the Anti-deSitter (AdS) warping itself does not look natural at all. It looks downright weird:
You notice two purple vectors orthogonal to the brane. The one in the left is, say, you and the onl in the right is, say, your pal. You and your pal are,say, a Kilometer away on Earth. Now if both of you manage to enter into the 5th dimension i.e., orthogonal to the brane then a displacement of 0.1mm perpendicular to brane would translate into a distance of 100 meters between you and your pal and further 0.1mm would shorten it to 10m and 1m and so on: It is as if you and your pal are so close that the tip of both your nose is touching each other while both of you have perpendicular displaced yourself of just 0.3mm in the 5th dimension while you are, in reality or on brane, a Kilometer away from your pal. Wow!
Now, let’s apply the same AdS warping on a mega scale:
Here, Gargantua is in the far reaches of the observable universe: roughly 10 billion light-years from Earth. Cooper, in the tesseract, rises through the AdS layer, from Gargantua’s core into the bulk. There the distance to Earth is 10 billion light-years divided by a thousand trillion, which is about the same as the distance between the Sun and the Earth, one “astronomical unit” (1 AU). Cooper then travels that 1 AU distance through the bulk, parallel to our brane, to reach the Earth and visit Murph.
Professor Brand, working with Einstein’s relativity equations, rediscovers the AdS sandwich.
How the confining branes are stabilized then gets intertwined with the Professor’s struggle to understand and control gravitational anomalies. In the movie, that struggle is spelled out mathematically on the sixteen blackboards in Professor Brand’s office.
The gravitational anomalies: the falling of books that Murph attributes to as Ghost. Or was it just Ghost? try recollecting from the movie:
a) Romilly tells Cooper so in the movie: “We started detecting gravitational anomalies on Earth almost fifty years ago,” and also, around that same time, the most significant anomaly of all: the sudden appearance of a wormhole near Saturn, where before there was none.
b) In the movie’s opening scene, Cooper experiences an anomaly himself, while trying to land a Ranger spacecraft. “Over the Straights something tripped my fly-by-wire,” he tells Romilly.
c) The GPS system that Cooper has adapted to control harvesting machines, as they roam through corn fields, has also gone haywire, and a bunch of harvesters have converged on his farmhouse. He attributes this to gravitational anomalies that screwed up the gravity corrections that any GPS system relies on.
d) Early in the movie, we see Murph watch, as dust falls unnaturally fast to the floor
of her bedroom, collecting in a bar-code-like pattern of thick lines. And then we see Cooper stare at the lines and toss a coin across one. The coin sticks to the floor:
These sudden changes must have a source right?. If the source is not in our universe, on our brane, then there is only one other place it can be, the Professor reasons: in the bulk.
The Professor can think of just three ways that something in the bulk could produce these anomalies, and the first two he quickly rejects:
1. Some object in the bulk—perhaps even a living object, a bulk being—might come near our brane but not pass through it. The object’s gravity reaches out through all the bulk’s dimensions and so could reach into our brane. However, the AdS layer surrounding our brane would drive the object’s tidal lines parallel to our brane, allowing only a minuscule portion to reach our brane. So the Professor rejects this.
2. A bulk object, passing through our brane, could produce tidal gravity that changes as the bulk object moves. The lines tend to be more diffuse than those from a localized object. Some tidal anomalies might be from localized objects, but most must be something else.
3. Bulk fields passing through our brane could produce the changing tidal gravity. This, the Professor concludes, is the most likely explanation for most of the anomalies.
Bulk fields are not just the key to gravitational anomalies on Earth, Professor Brand believes. Bulk fields also plays the very crucial role of holding the wormhole open: since Professor Brand thinks the wormhole has been constructed and placed near Saturn by bulk beings, bulk fields holding it open seem natural to him:
The left wormhole is pinched off at the middle while the right one is held open by bulk field.
The bulk field also is the reason for our universe to be alive today:
without the bulk field, the confining brane sandwich would destroy our universe. Bulk field is the reason behind the confining branes not buckling towards our brane.
Finally, Dr. Brands Equation we all have been waiting to see:
If you scroll up, I have mentioned four points that Dr. Brand would have been able to achieve had he succeeded in solving the above equation.
This equation is called an “Action.” There is a well-known mathematical procedure to begin with such an Action, and from it deduce all the non-quantum physical laws. The Professor’s equation, in effect, is the mother of all non-quantum laws. But for it to give birth to the right laws—the laws that predict correctly how the anomalies are produced, how the wormhole is held open, how G is controlled, and how our universe is protected—the equation must have precisely the correct mathematical form. The Professor doesn’t know the correct form. He is guessing. His is an educated guess, but a guess nevertheless.
His equation contains lots of guessing: guesses for the things called “U(Q), H ij (Q 2 ), W ij ,and M (standard model fields)” on his blackboard. In effect, these are guesses for the nature of the bulk fields’ force lines, and how they influence our brane, and how fields in our brane influence them.
When the Professor and his team speak of “solving his equation,” they mean two things.
a) First, figure out the right forms for all these things they are guessing: “U(Q), H ij (Q 2 ), W ij, and M(standard model fields).”
b) Second, deduce, from his equation, everything he wants to know about our universe, about the anomalies,and most important, about how to control the anomalies so as to lift colonies off the Earth.
In the movie, when the Professor is very old, we see him and grown-up Murph trying to solve his equation by iterations. On a blackboard, they make a list of guesses for the unknown things. Then Murph inserts each guess into a huge computer program that they’ve written.
The program computes the physical laws for that guess, and those laws’ predictions for how the gravitational anomalies behave.
the Professor and Murph keep trying. They keep iterating: making a guess, computing the consequences, abandoning the guess, and going on to the next guess, one guess after another after another after another, until exhaustion sets in. Then they begin again the next day.
A bit later in the movie, when the Professor is on his death-bed, he confesses to Murph: “I lied, Murph. I lied to you.”. Murph infers that he knew something was wrong with his equation, and Dr. Mann tells the Professor’s daughter the very same thing.
But, in fact—Murph realizes, soon after the Professor’s death—“His solution was correct. He’d had it for years. It’s half the answer.” The other half can be found inside a black hole. In a black hole’s singularity.
All right, glad that we have made it this far. I am feeling dizzy now, I haven’t had proper food in the afternoon. But, ironically, my hunger for understanding the science of Interstellar is keeping my mind stable somehow.
So, Cooper and Amelia manage to miraculously dock the Ranger into Endurance. If you remember moment before Endurance is parked closer to Mann’s planet Romilly tells Cooper about singularities: “I have a suggestion for your return journey from Mann’s planet. Have one last crack at the black hole. Gargantua’s an older, spinning black hole. It has what we call a gentle singularity.” “Gentle?” Cooper asks. “They’re hardly gentle, but their tidal gravity is quick enough that something crossing the horizon fast might survive.” Cooper, lured by this conversation and the quest for quantum data, later plunges into Gargantua. It’s a brave plunge. He can’t know in advance whether he’ll survive. Only the laws of quantum gravity know for sure. Or the bulk beings…It’s a suicide mission over an already suicide mission.
Cooper makes a quick decision: “The navigation mainframe’s destroyed and we don’t have enough life support to make it back to Earth. But we might scrape to Edmunds’ planet.” “What about fuel?” Amelia Brand asks. “Not enough,” Cooper responds. “Let Gargantua suck us right to near the horizon, then a powered slingshot around to launch us at Edmunds’ planet.” “Manually?” “That’s what I’m here for. I’ll take us just inside the critical orbit.” Within minutes they are at the critical orbit and all hell breaks loose.
To understand the climax scenes before Cooper and TARS eject their Ranger out of Endurance , we need to understand their positions with respect to Gargantua Black Hole:
First, when Mann blows up Endurance module, following diagram shows the position of Mann’s planet with respect to Gargantua:
Following is the position of Mann’s planet while Cooper maneuvers Endurance towards the critical path of Gargantua to achieve the sling shot propulsion:
This is exactly what ISRO scientists did with Mangalyaan to throw it out of Earth’s sphere of influence, remember?
Another view of the sling-shot effect:
So, Cooper will blast ignition when Endurance has accumulated enough momentum to offset shortage of fuel to reach the third and final plant “Edmund’s planet” (See the interstellar timeline) .
While Endurance is on it’s trajectory towards Edmunds’ planet, which is their final hope, Cooper will plunge into the black hole jettisoning his Ranger with TARS in it so as to lower the mass of Endurance and to save fuel for Amelia’s safe journey to Edmund’s planet).
As Brand watches the Ranger approach the horizon, she must see time on the Ranger slow and then freeze relative to her time, Einstein’s laws say. This has several consequences: She sees the Ranger slow its downward motion and then freeze just above the horizon. She sees light from the Ranger shift to longer and longer wavelengths (lower and lower frequencies, becoming redder and redder), until the Ranger turns completely black and unobservable. And bits of information that Cooper transmits to Brand one second apart as measured by his time on the Ranger arrive with larger and larger time separations as measured by Brand. After a few hours Brand receives the last bit that she will ever receive from Cooper, the last bit that Cooper emitted before piercing the horizon. Cooper, by contrast, continues receiving signals from Brand even after he crosses the horizon. Brand’s signals have no trouble entering Gargantua and reaching Cooper. Cooper’s signals can’t get out to Brand.
As the Ranger carries Cooper deeper and deeper into the bowels of Gargantua, he continues to see the universe above himself. Chasing the light that brings him that image is an infalling singularity. The singularity is weak at first, but it grows stronger rapidly, as more and more stuff falls into Gargantua and piles up in a thin sheet Einstein’s laws dictate this. Below the Ranger is an outflying singularity, created by stuff that fell into the black hole long ago and was backscattered upward toward the Ranger. The Ranger is sandwiched between the two singularities. Inevitably, it will be hit by one or the other:
The infalling singularity is produced by stuff that falls into Gargantua long after Cooper falls in (long after, as measured by the external universe’s time; Earth’s time). If Cooper is hit by that singularity and survives, the universe’s far future will be in his past. He will be so far in our future that, even with the help of bulk beings, he won’t be able to return to the solar system until billions of years after he left, if ever. That would prevent him from ever reuniting with his daughter, Murph. So Nolan chose Cooper to be hit by the outflying singularity, not the infalling one: hit by the singularity arising from stuff that fell into Gargantua before the Ranger, not after it.
Looking downward, Cooper should see light from objects that fell into Gargantua before him and are still falling inward. Those objects need not emit light themselves. He can see them in reflected light from the accretion disk above, just as we see the Moon in reflected sunlight. I expect those objects to be mostly interstellar dust, and this could explain the fog he encounters in the movie as he falls.
Now for the most interesting part: as the Ranger nears the outflying singularity, it encounters mounting tidal forces. Cooper ejects just in the nick of time. Tidal forces tear the Ranger apart. Visually, it splits in two. At the singularity’s edge the tesseract awaits Cooper placed there, presumably, by bulk beings.
This is the most controversial than the already controversial plot:
The entrance to the tesseract is a checkerboard pattern. Each square is the end of a beam. Cooper, entering the tesseract, falls down a channel between beams, lashing out at what appear to be bricks along the channel wall, but turn out to be books. The channel leads to a large chamber, where he floats and struggles, gradually getting oriented.
The chamber is Christopher Nolan’s unique take on one three-dimensional face of the four-dimensional tesseract. The chamber is complex and needs some bit of understanding of the science behind it. So, let’s get started, lets build a Tesseract:
Take a Dot (zero-dimension) and move it perpendicularly and you get a line (one dimensional). Take the line and move it perpendicularly and you get a square (two dimensional). Take the square and move it perpendicularly and you get a cube(three dimensional). Now, take the cube and move it perpendicularly and you get a hyper cube or a Tesseract (four dimensional):
It looks like two cubes, inside each other. The inner cube has expanded outward, in the picture, to sweep out the four-dimensional volume of the tesseract, just like we did for dot, line, square and cube.
Because Cooper is made of atoms held together by electric and nuclear forces, all of which can exist only in three space dimensions and one time, he is confined to reside in one of the tesseract’s three-space-dimensional faces (cubes). He can’t experience the Tesseract’s fourth spatial dimension.
The tesseract ascends from the singularity into the bulk. Being an object with the same number of space dimensions as the bulk (four), it happily inhabits the bulk. And it transports three-dimensional Cooper, confined in its three-dimensional face, through the bulk.
I had explained about the distance from Gargantua to Earth which is about 10 billion light-years as measured in our brane (our universe, with its three space dimensions). However, as measured in the bulk, that distance is only about 1 AU (the distance from the Sun to the Earth). So, traveling with whatever propulsion system the bulk beings provided, the tesseract can quickly carry Cooper across our universe, via the bulk, to Earth.
This trip is very quick, just a few minutes, while Cooper is still dazed and falling:
As he comes to rest, floating in the large chamber, the tesseract docks beside Murph’s bedroom. How does this docking work? arriving in the bulk near Earth the tesseract must penetrate the 3-centimeter-thick AdS layer that encases our brane in order to reach Murph’s bedroom. Presumably the bulk beings who built the tesseract equipped it with technology to push the AdS layer to the side, clearing the way for its descent.
Following figure shows the tesseract, after the clearing, docked alongside Murph’s bedroom in Cooper’s farmhouse:
Now the distance between Gargantua and Earth is squeezed down to Cooper being able to see his daughter Murph. This is similar to the nose-touching analogy I had given before.
In the figure above, The back face of the tesseract is engineered to coincides with Murph’s bedroom (may be my great great granddaughter will have that Engineering course pertaining to controlling AdS warping). The back face is a three-dimensional cross section of the tesseract that resides in Murph’s bedroom. So everything in Murph’s bedroom, including Murph herself, is also inside the tesseract’s back face. When a light ray traveling out from Murph reaches the common edge of Murph’s bedroom and the tesseract, it has two places to go: The ray can stay in our brane, traveling along route 1 out an open door or into a wall where it is absorbed. Or the ray can stay in the tesseract, traveling along route 2 into and through the next tesseract face, and then onward to Cooper’s eyes.
When Cooper looks through the right wall of his chamber, he sees into Murph’s bedroom through its right wall (right white light ray). Looking through the left wall of his chamber, Cooper sees into Murph’s bedroom through its left wall (left white light ray). Looking through his back wall, he sees into the bedroom through its back wall. Looking through his front wall (orange light ray), he sees into the bedroom through its front wall. Looking along the yellow ray, he sees down through her ceiling. Looking along the red ray, he sees up through her floor. To Cooper, as he changes his gaze from one direction to another to another, it seems like he is orbiting Murph’s bedroom:
Since Time is a physical dimension, Cooper has the flexibility of choosing time by moving in XY, YZ or XZ represented in Blue, Green and Brown lines:
With the same frame of reference being the rear of the book shelf of Murph’s room, Cooper can play with time like a kid plays with her dogs. In the movie, the time lapse between adjacent bedrooms is closer to a tenth of a second than a full second. By watching adjacent bedrooms carefully as the curtains in Murph’s bedroom window blow in the wind, you can estimate the time between bedrooms.
Cooper can move far faster than the flow of time in the bedroom extrusions, so he can easily travel through the tesseract complex to most any bedroom time that he wishes!
if it takes one second for cross sections to travel along each extrusion from one bedroom to the next, then all the bedrooms in the above figure are to the future of image 0 by the number of seconds shown in black. In particular, bedroom 2 is one second ahead of bedroom 0, bedroom 9 is two seconds ahead of bedroom 0, and bedroom 8 is four seconds ahead of bedroom 0 and so on.
To travel most rapidly into the future of Murph-bedroom time, Cooper should move along a diagonal of his chamber in the direction of increasing blue, green, and brown time (rightward, upward, and inward)—that is, along the diagonal dashed violet line. This is how Cooper manages to make the books fall and play with wrist-watch.
The biggest question is: Is Cooper really traveling forward and backward in time as he moves diagonally up and down through the complex? Forward and backward in the manner that Amelia speculates bulk beings can when she says: “To Them time may be just another physical dimension. To Them the past might be a canyon they can climb into and the future a mountain they can climb up. But to us it’s not. Okay?”
Nolan has mandated two rules that need to be ensured and enforced:
Rule 1: No Time travel is possible, Cooper can never travel to his own past.
Rule 2: Gravitational forces can carry messages into our branes past.
When falling into and through the tesseract, Cooper truly does travel backward relative to our brane’s time, from the era when Murph is an old woman to the era when she is ten years old. He does this in the sense that, looking at Murph in the tesseract bedrooms, he sees her ten years old. And he can move forward and backward relative to our brane’s time (the bedroom’s time) in the sense that he can look at Murph at various bedroom times by choosing which bedroom to look into. This does not violate rule 1 because Cooper has not reentered our brane. He remains outside it, in the tesseract’s three-dimensional channel, and he looks into Murph’s bedroom via light that travels forward in time from Murph to him. But just as Cooper can’t re-enter our brane in Murph’s ten-year-old era, so he can’t send light to her. That would violate rule 1. The light could bring her information from Cooper’s personal past, which is her future; information from the era when she is an old woman—backward-in-time information from one location in our brane to another. So there must be some sort of one-way spacetime barrier between ten-year-old Murph in her bedroom and Cooper in the tesseract, rather like a one-way mirror or a black-hole horizon. Light can travel from Murph to Cooper but not from Cooper to Murph.
“Light can travel toward the future from Murph to him. It can’t travel to the past from him to Murph. However, gravity can break that one-way barrier”
To explain Gravitational waves, let’s take up a depiction of an instance in time between Cooper and young Murph:
Each object in the bedroom, for example each book, contributes to the bedroom’s extrusion. In fact, the book has its own extrusion, which travels forward in time along the blue-arrow direction as part of the bedroom’s larger extrusion. A variant of this extrusion is called the book’s “world tube.” And we call the extrusion of each particle of matter in the book the particle’s “world line.” So the book’s world tube is a bundle of world lines of all the particles that make up the book, clear?. The thin lines that you see in the movie, running along the extrusions, are world lines of particles of matter in Murph’s bedroom. Cooper slams his fist on the book’s world tube over and over again, creating a gravitational force, which travels backward in time to the moment in Murph’s bedroom that he is seeing and then pushes on the book’s world tube. The book’s tube responds by moving. The tube’s motion appears to Cooper as an instantaneous response to his pushes. And the motion becomes a wave traveling leftward down the tube. When the motion gets strong enough, the book falls out of the bookcase.
By the time Cooper has received the quantum data from TARS, he has mastered this means of communication. In the movie we see him pushing with his finger on the world tube of a watch’s second hand. His pushes produce a backward-in-time gravitational force, which makes the second-hand twitch in a Morse-encoded pattern that carries the quantum data. The tesseract stores the twitching pattern in the bulk so it repeats over and over again.
When forty-year-old Murph returns to her bedroom three decades later, she finds the second hand still twitching, repeating over and over again the encoded quantum data that Cooper has struggled so hard to send her.
Recall that the tesseract is an object whose faces have three space dimensions and interior has four. The interior is part of the bulk. Everything we see in the movie’s tesseract scenes lies in the faces: Cooper, Murph, Murph’s bedroom, the bedroom’s extrusions, the world tubes of the book and watch—all lie in tesseract faces. We never see the tesseract’s bulk interior. We can’t see it, since light can’t travel through four space dimensions, only three. However, gravity can do so. When Cooper sees a book in Murph’s bedroom, he does so via a light ray that travels in faces of the tesseract. And when he pushes on a book’s world tube, or on the world tube of the watch’s second hand, he generates a gravitational signal (a gravitational wave in the bulk) that spirals into and through the tesseract’s bulk interior, along the violet curve in Figure below:
The signal travels forward in local, bulk time, but backward in bedroom time, arriving before it started out. It is this gravitational signal that pushes the book out of the bookcase and twitches the watch’s second hand.
With the quantum data safely in Murph’s hands, Cooper’s mission is finished. The tesseract, carrying him through the bulk, begins to close. As it is closing, he sees the wormhole. And within the wormhole, he sees the Endurance on its maiden voyage to Gargantua. As he sweeps past the Endurance, he reaches out and gravitationally touches Amelia across the fifth dimension. She thinks she has been touched by a bulk being. She has . . . by a being riding through the bulk in a rapidly closing tesseract: By an exhausted, older Cooper.
Murph now has the data, She kisses Topher Grace, throws her papers and shouts “Eureka!”
What she did before kissing Topher is : she discarded quantum fluctuations from those laws I have mentioned and Murph will learn the non-quantum laws that govern gravitational anomalies. And from those laws, she figures out how to control the anomalies.
Murph must have figured out how to reduce Newton’s gravitational constant G inside the Earth. We know that the Earth’s gravitational pull is given by Newton’s inverse square law: g = Gm/r^2 , where r^2 is the squared distance from the Earth’s center, m is the mass of the Earth, and G is Newton’s gravitational constant. Now, cut Newton’s G in half and you reduce the Earth’s gravity by two. Cut G by a thousand and you reduce the Earth’s gravity by a thousand. With Newton’s G reduced inside the Earth to, say, a thousandth its normal value for, say, an hour, rocket engines could lift the enormous colonies into space.
As a byproduct, the Earth’s core—no longer compressed by the enormous weight of the planet above—must have sprung outward, pushing the Earth’s surface upward. Gigantic earthquakes and tsunamis must have followed, wreaking havoc on Earth as the colonies soared into space, a terrible price for the Earth to pay on top of its blight-driven catastrophe.
When Newton’s G was restored to normal strength, the Earth must have shrunk back to its normal size, wreaking more earthquake and tsunami havoc. But humanity was saved. And Cooper and ninety-four-year-old Murph were reunited. Then Cooper set out in search of Amelia in the far reaches of the universe.
Every piece of this movie is carefully glued with science. If you didn’t understand the movie then read this blog post and watch the movie again if possible. Chris is a kind of a director who just doesn’t want a movie out for the audience: he want the ghost of the movie to haunt them until they can free themselves by understanding the science behind the movie.
Thanks Kip, I found you out of luck. May be your ghost gravitated my thoughts.