Let's reproduce the calculations from Interstellar
Summary
TLDRThis video explores the scientific accuracy of 'Interstellar', analyzing the ship's speed, artificial gravity, and black hole depictions. It delves into wormhole geometry, time dilation near a black hole, and the film's portrayal of gravitational anomalies. The video also discusses the theoretical physics behind the movie's climax, including the Tesseract and the potential for quantum gravity to save humanity.
Takeaways
- 🚀 *Interstellar* explores the realism of its scientific concepts, such as black holes and wormholes, with calculations to verify their accuracy.
- 🛰️ The ship in *Interstellar* travels at approximately 70,000 km/h, a speed feasible with advanced technology, taking two years to reach Saturn.
- 🌍 Artificial gravity is created on the spaceship *Endurance* through centrifugal force, with realistic dimensions of about 60 meters in diameter.
- 🌀 The movie's portrayal of a wormhole as a three-dimensional tunnel connecting two slices of space is consistent with theoretical physics.
- 🌊 On Miller’s planet, 1 km high waves result from tidal forces due to its proximity to the supermassive black hole Gargantua, though the time dilation portrayed is exaggerated.
- ⏳ Time dilation near Gargantua leads to one hour on Miller’s planet equating to seven years on Earth, achievable with a rapidly rotating black hole.
- 🌌 Gargantua’s visual effects include a plasma accretion disk with gravitational lensing and Doppler effects, though some aspects were altered for cinematic purposes.
- 🧑🔬 Professor Brand's equation suggests that gravitational anomalies may be caused by fields in a larger universe with extra dimensions.
- 🕳️ Cooper survives falling into the black hole due to a theoretical gentle singularity, and enters a four-dimensional tesseract created by advanced beings.
- 🏟️ The *Cooper Station* space habitat likely has a diameter between 50 to 100 meters, based on a baseball trajectory used to calculate its rotation and gravity.
Q & A
What is the main theme of the video 'Interstellar'?
-The video discusses the scientific accuracy and concepts presented in the movie 'Interstellar', such as black holes, wormholes, and the behavior of time and space.
What is the estimated speed of the ship in 'Interstellar'?
-The estimated speed of the ship in 'Interstellar' is around 70,000 km/h, calculated by dividing the distance of approximately 1.3 billion km by 2 years of travel time.
How does the artificial gravity system work in the ship Endurance?
-The artificial gravity system in the ship Endurance works by using a rotating wheel, which creates a centrifugal force that pushes the characters against its outer walls, simulating gravity.
What is a wormhole and how does it appear in the movie 'Interstellar'?
-A wormhole is a theoretical 'shortcut' through spacetime, connecting two distant points in the universe. In the movie, it appears as a three-dimensional tube connecting two celestial spheres.
What is the concept of time dilation as depicted in 'Interstellar'?
-Time dilation in 'Interstellar' refers to the difference in the passage of time between two observers due to differences in their relative motion or gravitational fields. On Miller's planet, one hour corresponds to seven years on Earth.
How are the massive waves on Miller's planet explained?
-The massive waves on Miller's planet are explained as tidal forces caused by the close proximity to the black hole Gargantua, which distorts the planet and creates immense waves.
What is the significance of the black hole Gargantua in 'Interstellar'?
-Gargantua is a supermassive black hole with a strong gravitational pull that affects the time dilation experienced by the characters and the extreme tidal forces on nearby planets.
How does the Tesseract, a four-dimensional object, play a role in the movie?
-The Tesseract in 'Interstellar' is a four-dimensional space that allows Cooper to interact with his past and communicate with his daughter Murphy, providing a means to send crucial data.
What is the role of the character Cooper in the movie 'Interstellar'?
-Cooper, a former NASA pilot, plays the protagonist who embarks on an interstellar journey to find a new home for humanity and eventually interacts with a higher dimension to influence past events.
What scientific theories are explored in the movie's depiction of the universe?
-The movie explores theories such as general relativity, quantum mechanics, and the possibility of extra dimensions, as well as the effects of black holes on time and space.
How does the movie 'Interstellar' visualize the scientific concept of gravitational anomalies?
-Gravitational anomalies in 'Interstellar' are visualized as strange occurrences that guide characters to secret locations and are later explained as being influenced by higher-dimensional fields.
Outlines
🌌 Introduction to 'Interstellar' and Scientific Accuracy
The video script introduces a deep dive into the scientific accuracy of the movie 'Interstellar' directed by Christopher Nolan. It discusses the movie's realistic portrayal of a black hole and the various scientific concepts it explores, such as the speed of the ship, its dimensions, the behavior of time around a black hole, and the appearance of a wormhole. The script mentions a collaboration with Kip Thorne, the main scientist behind the movie, and the use of his book 'The Science of Interstellar' to verify the movie's scientific claims. The video aims to recreate some visuals from the movie and understand the technical aspects of the film, such as the travel time to the wormhole near Saturn and the artificial gravity on the spaceship Endurance.
🔭 Simulating the Wormhole and Black Hole in 'Interstellar'
This section of the script discusses the attempt to simulate the wormhole and black hole as depicted in 'Interstellar'. It explains the process of creating a realistic wormhole image using a method called ray tracing, which involves tracing the path of light rays through curved spacetime. The script also explores the appearance of the inside of a wormhole and how it was visualized in the movie. Additionally, it delves into the characteristics of the black hole named Gargantua, including its massive size, the extreme tidal forces it generates, and the significant time dilation near it. The script compares the simulated images with those from the movie to assess their realism.
🌊 Tidal Forces and Time Dilation on Miller's Planet
The script examines the extreme tidal forces and time dilation on Miller's Planet, which is depicted as being very close to the black hole Gargantua. It discusses how these forces create enormous waves and how time passes much slower on the planet compared to Earth. The calculations provided aim to estimate the distance at which a planet would need to orbit a black hole to experience such phenomena. The script also addresses the discrepancy between the movie's portrayal of time dilation and the actual calculations based on general relativity, suggesting that for the extreme time dilation shown in the movie to be accurate, the planet would need to be in an unstable orbit extremely close to the black hole.
🌒 Visualizing the Black Hole and Gravitational Lensing
This part of the script focuses on the visual representation of the black hole Gargantua and the surrounding environment in 'Interstellar'. It describes the use of ray tracing to simulate the gravitational lensing effects caused by the black hole, which distorts the light from the accretion disk and creates the shadow of the black hole. The script also discusses the Doppler effect on the appearance of the accretion disk, where one side appears brighter and bluer while the other appears darker and redder due to the movement of the disk relative to the observer. The video script compares these simulated visuals with the images from the movie to highlight the artistic choices made in the film.
🔬 Theoretical Physics and the Quest for Quantum Gravity
The script explores the theoretical physics behind the movie's plot, particularly the professor's quest to understand gravity and the nature of the universe. It discusses the idea of our universe being one slice of a larger multiverse with additional dimensions, and how this concept is used to explain the gravitational anomalies observed in the movie. The script also touches on the concept of singularities within black holes and the speculative idea that one might survive a fall into a black hole with a gentle singularity. It describes how Cooper, the protagonist, uses this knowledge to communicate with his daughter Murphy across time and space.
🚀 The Cooper Station and the Future of Humanity
In this final section, the script discusses the construction of the Cooper Station, a massive centrifuge designed to save humanity by lifting off into space. It provides a unique approach to estimating the dimensions of the station by analyzing a scene where a baseball is hit and travels the length of the station. The script uses the principles of gravity and motion to calculate the radius and rotation speed of the station. It concludes by reflecting on the movie's attention to scientific detail and the thought-provoking concepts it presents, encouraging viewers to delve deeper into the scientific aspects of the film.
Mindmap
Keywords
💡Interstellar
💡Black Hole
💡Wormhole
💡Time Dilation
💡Gravitational Anomalies
💡Accretion Disk
💡Centrifugal Force
💡Photon Sphere
💡Tesseract
💡Singularity
Highlights
Interstellar, directed by Christopher Nolan, presents realistic images of a black hole and explores scientific concepts like time dilation and wormholes.
The video attempts to mathematically verify the accuracy of various concepts presented in the film.
The speed of the ship in the movie is estimated to be around 70,000 km/h, which is considered realistic given the context.
Artificial gravity in the ship is achieved through centrifugal force, calculated to have a radius of roughly 30 m or 60 m in diameter.
The appearance of the wormhole in the movie is simulated using a method called Ray tracing, which involves tracing light rays through each pixel of a camera.
The inside of the wormhole is explored, with simulations suggesting different visual effects based on the geometry of the wormhole.
The movie's depiction of extreme tidal forces on Miller's planet due to proximity to a black hole is analyzed.
Time dilation on Miller's planet is discussed, with the movie showing a factor of 60,000, which is higher than calculated estimates.
The visual appearance of the black hole Gargantua in Interstellar is examined, with simulations attempting to reproduce the movie's images.
The concept of a fourth dimension is introduced through the Tesseract, a four-dimensional object in the movie.
Cooper's survival near a black hole's singularity is explored, suggesting the possibility of a gentle singularity in older black holes.
The movie imagines gravity as the force that can cross dimensions and communicate across time.
Murphy uses the data sent by Cooper to understand how to control gravity, which is crucial for humanity's survival in the movie.
The dimensions of the space station in the movie are estimated using the trajectory of a baseball hit inside the station.
The movie's scientific accuracy is praised, with detailed calculations and simulations enhancing the understanding of complex concepts.
The video concludes by enjoying simulations built throughout, showcasing the intersection of science and cinema in Interstellar.
Transcripts
warning this video contains
spoilers here's what Interstellar could
have looked like
[Music]
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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
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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 4.6
seconds between the hit and the impact
and we can then estimate its initial
speed by analyzing frame by frame and
comparing to the size of the player we
find an estimate of 30
m/s with all these restrictions we
determine the radius of the wheel
approximately 50 m or 100 m in diameter
for roughly 3 00 m in circumference a
relatively small station which would
rotate at four revolutions per minute
the trajectories of baseballs would be
chaotic to say the
least with other methods by comparing
with the size of a building for instance
we find very different values sometimes
even 1 km I'll leave it up to you to
develop your own techniques to estimate
these
Dimensions born from the creative mind
of an expert in relativity Interstellar
is a unique movie with breathtaking
visuals through these few calculations
we can appreciate the attention to
detail brought to the construction of
the scenario and delve deeper into
issues that may not seem obvious at
first viewing as a conclusion let's
enjoy once again some of the simulations
we've built throughout this video
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