Let's reproduce the calculations from Interstellar

warning this video contains spoilers here's what Interstellar could have looked like [Music] [Music] welcome back to science click today let's try to remake the calculations from Interstellar released in 2014 Interstellar directed by Christopher Nolan left its mark on audiences by presenting realistic images of a black hole but is the movie accurate what is the speed of the ship what are its Dimensions how does time behave around a black hole what does the inside of a worm whole look like in this video we will go back over the Journey of the film's characters and attempt to mathematically verify the accuracy of the different concepts we will also try to rebuild some visuals from the movie this may be a bit technical but just keep in mind you don't need to understand all the calculations we will at times rely on information from the book the science of interstellar written by Kip Thorn the main scientist behind the movie this video was originally made in collaboration with the French Channel gab which talks about science in popular movies he usually doesn't make videos in English but kindly agreed to let us translate his voice for this one so don't be surprised if the visual style of the video changes at times when we present the storyline at the start of interstellar the situation on Earth is critical humans are facing food shortages air is close to unbreathable dust is omnipresent the planet no longer seems habitable on this apocalyptic Earth we follow Cooper a former NASA pilot who lives with his two children in the bedroom of his daughter Murphy strange events occur events that Cooper ends up identifying as gravitational anomalies one day one of these anomalies mysteriously reveals coordinates which once followed lead to a secret NASA base as if the library had wanted to guide them there in this NASA Center we learn that a wormhole has opened near Saturn forming a passage to planets and that NASA is about to launch a critical Mission a space crew will cross the Wormhole and try to reach three potentially habitable exoplanets of course Cooper arrives just in time for the adventure as a former NASA pilot he is recruited and thus embarks on an Interstellar Journey aboard the endurance after takeoff the crew goes into hibernation the ship will take two years to arrive at the Wormhole near Saturn is this travel time realistic we can roughly estimate the speed of the ship by taking the shortest straight line between Earth and Saturn and factoring in the radius of their orbits we can assume that the ship travels approximately 1. 3 billion km divided by 2 years this gives us a speed of around 70,000 kmph as a basis for comparison the fastest ever manned flight was Apollo 10 with a speed of 40,000 km hour assuming that interstellar's Technologies are better than ours this 2-year duration is therefore realistic the characters need artificial gravity to feel comfortable the inurance forms a wheel which turns on itself and pushes the characters against its outer Walls by centrifugal effect the centrifugal acceleration is given by 4K ^ 2 R / t^ 2 where R is the radius of the wheel and T its period of rotation in this scene we can time a half period of 5. 7 seconds which gives us a total period of 11.

4 seconds this must equal 9. 8 m/s squared the value of gravity on Earth we deduced that the radius of the wheel must be roughly 30 m or 60 m in diameter a fairly realistic value given the International Space Station is 70 m cross the endurance's crew wakes up near Saturn we observe For the First Time The Wormhole that they will have to cross a wormhole is a distortion of space a sort of tube which connects two slices of the universe it is difficult to draw because the surface of the tube is actually three-dimensional a volume within which the ship can move light can also pass through it and we can receive an image of what is on the other side but is its appearance in the movie realistic to find out let's try to simulate it we start with a cylinder connecting two sheets of space let's smooth out this cylinder to have a better transition for the movie Nolan chose a very short cylinder and a relatively quick transition to get the appearance he liked let's place a camera and choose two images to represent the sky the celestial sphere of each of these two Universe slices the camera receives light rays some of which arrive directly from the same slice it is on and others after crossing the Wormhole all these light rays arrive on the pixels of the camera without going into details we can perform a method called Ray tracing which which involves sending a light Ray through each pixel of the camera and rewinding its trajectory to determine where it came from in the sky and what color is received by the camera on that pixel in practice this amounts to using the geodesic equation a general relativity equation which allows us to trace trajectories through curved SpaceTime by repeating the operation for each pixel we obtain an image a result which when put in motion is quite similar to that of the movie we can now play around with varying the geometry of the Wormhole to see what it might have looked like if Nolan had made different choices with a very long Wormhole light can orbit several times before reaching us and we can observe an infinite number of repeating images of the [Music] universe in interstellar the endurance enters the Wormhole to cross it we discover the inside of the Wormhole is this scene realistic to find out let's move our virtual camera along the surface of the tube this time around the images are quite different from what the movie shows us the visual effects team probably wanted to help the viewer better understand that the spaceship is moving along the surface of a cylinder and Crossing into a fourth dimension if this scene had been simulated we could have witnessed hypnotizing images such as [Music] these exiting the Wormhole the Explorers arrived near a super massive black hole called Gargantua this black hole has planets orbiting around it and our crew's first destination is one of the Millis Planet this planet is very special it is completely covered in water and its orbit is very close to Gargantua so close in fact that extreme phenomena arise near a black hole tidal forces are gigantic and Gargantua attracts the planet more on one side than the other distorting it and forming immense 1 km high waves on its surface because of its proximity to the black hole time also passes more slowly so much so that for each hour that passes on this planet 7 years pass on Earth in short Miller's planet is mysterious and deserves our attention let's start with the waves Kip Thorn indicates that they stretch over 1 km in height how close to the black hole would the planet need to be for such waves to form we can attempt a rough estimate we can assume that these waves are in fact Tides generated not by the attraction of the moon but by that of the black hole the waves in the movie look thinner than simple Tides they might be closer to what we call tidal BS but for a rough estimate our approach should still work the waves thus result from the combination of gravity from the black hole and the centrifugal effect produced by the orbit of the planet we can calculate the resulting net force with Newtonian physics this is the tidal Force this Force displaces water from low tide left and right to high tide up and down the force supplies each water droplet a certain energy which we can calculate as the product of the vertical displacement which is the radius of the planet and the average value of the force along this path it is this energy brought by the tidal Force which allows the water to rise it must therefore equal the gravitational energy that the water acquires from this equality we can deduce the distance at which the planet should orbit for such tides to form if the planet's dimensions are similar to those of Earth knowing that the black hole weighs 100 million solar masses then the planet should orbit at a distance of 4. 5 billion km to compare the spart Shield radius of the black hole which we can deduce from its mass is 300 million km across if we bring this diagram to scale and compare it to this image from the movie showing Gargantua a scene from Millis Planet the results seems coherent but is this close enough to also explain the strong time dilation each hour that passes on the planet corresponds to seven years on Earth time passes 7times 365 time 24 times slower a factor of 60,000 for a static black hole this is the time dilation experienced by a planet in circular orbit if we insert the distance obtained with the calculation of the tides we get a time dilation of only 5% far from what the movie shows us there is an incoherence to obtain a factor of 60,000 the planet would have to orbit just above the photon sphere where light itself can remain in orbit an absurd result because any orbit so close is inevitably unstable but this calculation is only valid for a static black hole general relativity allows black holes to rotate the faster a black hole spins the more likely stable orbits exist near its Horizon the calculations are more complex but we can determine that it is possible to reach a factor of 60,000 provided that the planet orbits 6,000 km from The Horizon and that the black hole spins at a trillion of a percent slower than the speed of light extremely fast but not impossible in theory of course this situation remains incompatible with what the movie shows us the visual appearance of the black hole is surely what brought attention to Interstellar in France jeanpier Lum had done a first simulation in 1979 but in 2014 it is the first time that we see such images in theaters could we reproduce these simulations at our scale let's try the ray tracing method we place a camera in the center of a celestial sphere in front of a black hole rotating almost at maximum speed general relativity allows us to rewind the trajectory of light rays to find their origin in the sky some rays are blocked by the black hole and the corresponding pixels do not receive any light in this way we can stru an image the rays are deflected by gravitational lensing causing Optical distortions we observe the shadow of the black hole the area from which we received no light for a static black hole this Shadow would be a disc but for a spinning black hole it seems squashed on one side because SpaceTime spins and drags light with it in the movie this phenomenon was reduced to render the visual effect less strange a round Gargantua orbits an extremely luminous plasma disc let's place a disc in our simulation we see that its image is distorted the back of the disc seems folded above and below because the black hole bends the Rays coming from it in reality such an accretion disc would be extremely hot it would emit intense radiation which would instantly destroy the ship the disc temperature has therefore been drastically brought down for the movie implying that it is also very thin given its lower internal pressure this plasma rotates very quickly around the black hole such that the light we received from it is propelled on one side and slowed down on the other this is the Doppler effect the disc appears bright and slightly blue where it moves towards us and dark and red where it moves away from us this effect was considered too strange and was not included in the film we can play around with comparing our simulation to images from [Music] Interstellar here's what it could have looked like if they had included the asymmetry the Doppler effect and a higher temperature dis [Music] [Music] [Music] while our Interstellar mission is going on back on Earth the professor refines his mathematical model trying to account for the discovery of the gravitational anomalies this Quest had already led him towards a very complex equation to solve and his goal in solving this equation is to have answers about the nature of gravity and then use this new knowledge to launch a gigantic spaceship into space and save Humanity but we are not there yet at this stage of the research the professor's conclusion is that the gravitational anomalies must be caused by something outside our universe according to him our space could be one slice inside a larger universe with one more Dimension the anomalies would be caused by Fields present in this bigger space outside our world we see the details of this model on Professor Bran's blackboards in the movie The Universe would contain three brains our brain a brain above and a brain below forming a sort of Sandwich between these two brains the geometry of the universe would be curved into an anti- deit geometry a geometry allowing distances to be greatly stretched above and below our universe such that gravity cannot Escape in fact in our universe gravity propagates in all directions and its intensity is therefore distributed over spheres which is why the force of gravity is inversely proportional to the square of the distance just like the surface of a sphere but if we were to add an extra dimension of space gravity wouldn't form spheres but hyperspheres extending beyond our slice of universe and gravity would decrease like the cube of the distance which we obviously do not observe with the anti- deito warping gravity remains confined near our universe and still behaves as one/ r s the other two brains help in delimiting this warping leaving enough volume outside for possible Adventures within the fourth dimension this is a necessary Precision to justify the end of the movie back on Earth Murphy has now taken over from Professor brand in their quest for Gravity it's now her task to find a solution to the equation and describe the nature of the fields to explain the anomalies observed on Earth although the professor's model seemed promising at first no results follow and Murphy ends up realizing what he had known all along without telling her to complete the model they need information about the quantum nature of gravity Murphy must understand what happens at a point where gravity meets Quantum for example by diving near the singularity of a black hole coincidentally this is exactly what her Father Joseph Cooper is doing in another [Music] galaxy following tars Cooper detaches himself from the endurance and Falls towards the horizon of the black hole can he survive without being spaghettified up until the 1960s singular ities were considered as pointlike in the 1970s however physicists understood that these singularities undergo chaotic distortions they are known as bkl singularities in the 1990s researchers discovered that when we fall into a black hole all the matter falling behind us seems to pile up because of the slowing down of time we Face a mass inflation Singularity finally we discovered around 2010 that older black holes have a third Singularity stemming from the accumulation of matter that fell prior this Singularity is gentle it generates relatively weak tidal forces while this is still highly speculative it might be possible to survive there but we would have to dive almost at the speed of light to prevent the other Singularity from catching up with us this is precisely the advice that romaly offers Cooper in the film Gargantua is an old black hole containing a gentle Singularity he could survive it by diving at high speed it is this possibility although very speculative that Nolan chose for his movie in a mysterious way Cooper manages to accelerate enough to hit this Singularity first and survive no one knows what lies at the center of a black hole we would probably need a theory of quantum gravity nonetheless in the movie Nolan imagines that when Cooper hits the singularity he is transported by a four-dimensional object called the Tesseract an object believed to have been placed there by the same beings who opened the Wormhole this Tesseract is a sort of four-dimensional cube while a three-dimensional Cube's faces are two-dimensional squares the Tesseract faces are not squares but three-dimensional cubes and this structure allows for Koopa to stay inside one of the faces of the Tesseract while the tesseract transports him back to Earth all the way to Murphy's room by the way if this journey feels quite fast despite the huge distance it is because the Tesseract has been lifted above our brain into the anti- DEA warping we mentioned previously where distances are greatly contracted the Tesseract acts as an elevator towards the fourth dimension of space once he arrives Cooper can see Murphy's room in the past which is possible because light can travel within the faces of the Tesseract up to his eyes however in the model imagined by Nolan light cannot go back in time from Cooper to Murphy he therefore has no way to communicate with her at least this is until Cooper discovers that gravity can cross this barrier and return to the past an idea which is probably inspired by the fact that in theories describing universes with additional dimensions in the context of brain cosmology all fundamental interactions are generally can find within the brains except gravity which propagates through all Dimensions Cooper can therefore send the quantum data to Murphy using gravity giving his daughter all the elements necessary to solve the equation we finally understand that he was behind the gravitational anomalies from the beginning recall once again though that the things we have just described are simply the physical rules set by Nolan for this very speculative sequence of the movie thanks to the data Murphy understands that it is possible to control gravity we can reduce the intensity of gravity on the surface of the Earth to lift off a huge space station and save Humanity the kooper station a large centrifuge thus heads towards Saturn to cross the Wormhole for our last calculation let's try to determine the dimensions of this space station we could imagine several methods but let me propose an unexpected Ed approach in this scene of the movie we see a baseball player hitting a ball which travels through the entire station and ends up breaking a window on the opposite side above the field the ball experiences no force and therefore moves in a straight line at constant speed but from the inside the trajectory of the ball seems curved because we are spinning with the wheel at this stage we don't know the param of this trajectory the speed of the ball the angle of the trajectory the radius of the wheel and its rotation speed but we see in the movie that the gravity in the station is similar to Earth's this gives us a restriction on the rotation of the wheel it must spin at a precise speed to generate this gravity we also know that the ball ends up breaking a window above the field we can therefore restrict the angle of the trajectory such that it reaches is a point above the field watching the scene we can time the trajectory of the ball and measure 4.