Alice, Bob, Cooper, Gravity And Time Dilation

Discussion in 'Education & Personal Growth' started by Iravati, Apr 23, 2017.

  1. Iravati

    Iravati Platinum IL'ite

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    I came across this thread today.

    There are loads of articles on the net to explain the gravitational time dilation but none explains well the science and mathematics behind gravity to grasp this notion of time. ‘Time slows down due to gravitational time dilation’ is the standard institutionalised print. When Interstellar was realised, I found no material on the Internet that explained the science behind gravity in lay terms: how it happens apart from why it happens. We have read in some form how (time dilation) but the why was fuzzy or mired in symbol grounding. Thus, I had to teach myself "gravity" ground up from the basic principles of science and maths to understand the plot of the movie. Here is my attempt to explain gravity and time dilation in simple wording by highlighting one or two underlying principles in science and mathematics.

    You might have heard in Newtonian Classical physics that gravity is a force that pulls us down. Similarly, heard in Einstein's General Relativity that gravity is a curvature of spacetime wherein time is not a fixed observation [invariant]. Again, none in a straightforward manner explains why time slows down in gravity, represented by a force or a curvature. How is a force or a curved shape manipulating something intangible like time?

    Let's step back awhile and understand how the interpretation of Newton and Einstein differed?

    We understand space well, i.e, Euclidean and Pythagorean spatial geometry on flat surfaces. Next, we identified Newtonian forces and motion, inertial frames, and acceleration in 3-dimensional space. Then something happened. We were unable to explain the erratic motion of objects travelling at a great speed (near the speed of light). We could not account the inconsistent behaviour like precession or orbital spin of objects under the influence of more gravity, for instance, Mercury, the closest planet to the Sun, experiences far more gravity than Earth does and exhibits discrepancy from its predicted orbital path. Newton's equations that described gravity as a force were failing in gravitational field closer to the Sun.

    Look at it this way. Newton developed science from a mathematical framework. He hauled maths to overlay on his scientific principles. Why is Einstein a genius, or superior, or, atleast, different from the others? Concepts used in his relativity theories were not originally developed by him. He is great because he was able to synthesise a cohesive theory from cumulative works of others. Don't undermine that effort, it took an Einstein of scientific intelligence to develop the bird's-eye view of the reality by fusing worm's-eye view of standalone work done by his peers. Einstein developed science for a mathematical framework.

    (1) Theory of Special Relativity (holds good at great speeds): Einstein described the science.
    (2) Theory of General Relativity (hold together at more gravity): Einstein described the science.

    He wrote behind-the-scenes working but then unlike Newton who pulled maths into science, Einstein pushed science onto maths. You should be able to differentiate both these approaches even with my sloppy writing. Newton knocked up mathematics (like calculus etc) to project his scientific theories. Einstein instead honoured the existing axioms propounded by the likes of Carl Friedrich Gauss Gauss, Hermann Minkowsi and Georg Friedrich Riemann and retrofitted science around their vaunted works.

    When Newton was faltering to devise new mathematics to explain the anomalies in Newton's Gravitational Force theory, his friend Marcel Grossmann suggested , “Why write new maths? There is maths already around you, why don't you write new science built upon this maths we already have”

    Grossmann's insight changed the history of scientific pursuit as we know today. Though Einstein was initially baffled with his approach, he quickly realised the significance of this revolutionising cue.

    Now let's revisit those theories again.

    Theory of Special Relativity (great speeds):

    1. Minkoswi had the maths.
    2. Einstein wrote the science around it to describe everyday observations like speed of light constant for all frames of reference (why? we will see later)

    Theory of General Relativity (more gravity):

    1. Reimann had the maths.
    2. Einstein wrote the science — why planets move, why objects falls, and why gravity affects time.

    Stay with me. The transcendence of Newton's Force (science-driven) as Einstein's Curvature (maths-driven) was a paradigm shift in scientific thinking. Newton married science to maths. Einstein married maths to science. They are fundamentally very unique approaches although they converge in solving the mysteries of the universe.

    Newton's gravitational force theory didn't even predict that time varies with gravitational intensity, whereas Einstein's gravitational dilation field theory not only correctly predicted but also quantified time distortion in field equations in that how time bends in gravity. That in turn explains why Cooper didn't age near a black hole.

    If anyone is interested or not, I might write up ordered steps to solve Cooper's ageing conundrum.
    1. What is 3-D space? What is 4-D spacetime? How time is integrated as 4th coordinate/dimension in a 4-D representation of reality. Not only integrate but also convert into spatial units. Why do we need to do that? What does it even mean to interchange space and time?
    2. How objects move in 4-D flat spacetime and what are geodesics?
    3. How objects move in 4-D curved spacetime — also known as "gravity"?
    4. How did ideas of Newton and Einstein converge if gravity is a force according to Newton and gravity is also a curvature according to Einstein?
    5. Why time is not invariant in this 4-D curved spacetime?
    6. We will demonstrate from points (3) and (5) why Cooper didn't age.

    I will try to keep the maths to the absolutely necessary bits which means all advanced field equations will be simplified for easy visualisation. All rubber sheets will be replaced by intuitive modelling and diagrams.

    I will write this up as if I am talking to my imaginary 5-year-old niece who has inquired, “Gravity means an apple falling down but you are telling me gravity also means time recorded by different people to track that falling apple as an event in spacetime. I wish apples were more simpler to study”. I wish too ...

    I don't want to drift too much while talking about gravity so will focus only on time dilation aspect of gravity to explain why Cooper aged slow in a seven-step write-up. Einstein took ten years to formulate this phenomenon and another 1200 pages for Charles Misner and Kip Thorne to publicise Einstein's work, I will have to settle for seven posts for now.

    Welcome to the mind-boggling world of Gravity where things are not pulled down but gravity is pushed up in spacetime.
     
    Last edited: Apr 23, 2017
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  2. Iravati

    Iravati Platinum IL'ite

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    Remember we will just focus on the impact of gravity on time? We will touch upon but won't elaborate on other aspects of mechanics and spacetime.

    Step#1: What is 3D space? What is 4D spacetime?

    Recap: In Einstein's gravitational model, mathematics preceded science. We need to understand this 4D maths before solving Cooper's aging enigma.

    Go on ...

    What is 3D space? 3D space is what we see around us. The proverbial length, breadth and height of objects. In 3D terminology, we have points and then distance between the points. To ease the calculation of distance between two points in space, we shall flatten our 3D space to 2D plane and use Cartesian frame of reference. Let's say, there are two points A (ax, ay) and B (bx, by) in space. The distance between two points in Cartesian coordinate system is simple to calculate.

    Δx = bx-ax
    Δy = by-ay

    Knowing Δx and Δy, we calculate the distance (d) using Pythagorean theorem. Let's invite three surveyors to measure the distance between A and B. They measure and provide Δx, Δy and Δd. You notice that Δd is same, but Δx and Δy are calculated differently. You inquire how they calculated the distance. First surveyor, from ground, second surveyor, from a ramp, the last one hanging mid-air. Each surveyor had a different location in space.They have different frames of reference in that their X and Y axes did not coincide. They won't universally agree on Δx and Δy because they are using different referential planes but everyone agrees on Δd, the distance. The distance between two points in flat space is fixed regardless of the frame of reference.

    Next, some more mathematical concepts and we are done. Here's the real scoop! Mathematics can model higher dimensional space. 1D has a line , 2D has a plane, 3D has a cube, and 4D has a tesseract. When Einstein published his special relativity, 3D space + time were separate though entwined. Minkowski integrated and scaled Einstein's theory from a (3D space) + (time) to a unified (4D spacetime) mathematical model.

    If 3D space is what we see around us, 4D spacetime is what we experience around us. For simplicity, let's represent passage of time as travel along the time axis with a constant speed. And to project time units on a spatial fourth dimension, instead of saying one sec has elapsed, I can say that I've travelled a distance Δd' in temporal plane equal to constant speed of time in metres/sec multiplied by one sec. If Δd is my change of location in space, then I can say Δt has elapsed or Δd' (distance prime) is my displacement on time axis. Watch, I have converted my time in spatial units by measuring how I have moved in time. This is done so that we have all axes in same spatial units.

    Let's see how the features of 3D correspond in 4D.

    In 3D we have space, in 4D we have spacetime.

    In 3D space, we have location (x,y,z), similarly, in 4D spacetime we have a time-sensitive location called event (x,y,z,t).

    In 3D space, distance between two locations is fixed and is called distance. Likewise in 4D, some measurement should be fixed and should be called something. What is that and what is it called? Not space and time because space and time correspond to shifting Δx and Δy in 4D. This fixed measurement which is 4D's equivalent of distance is spacetime interval.

    Let's bring back those surveyors again. If those surveyors are measuring in spacetime, they will all agree on spacetime interval but not on space and time. What does it mean?

    For simplicity, let's assume the surveyors are trying to calculate the distance between A on Jan 01, 2017 and B on April 01, 2017. Say, the space time interval between A (as, at) and B (bs, bt) = 1 spacetime interval. Though all surveyors in their respective standing, running, and hanging positions will compute one unit of space time, they will differ in their input Δs and Δt.

    This is fundamentally and substantially what is meant by your 1 metre is not my 1 metre, and your 1 sec is not my 1 sec, but your 1 spacetime interval is my 1 spacetime interval in 4D.

    Note, this is the most reductive and simplistic representation of mathematical spacetime that can be intuitively reconciled if not visualised. So far we have only demonstrated that time is not the same for everyone. In the next instalments, we will learn how gravity affects Δs and Δt in spacetime and in the process define gravity.

    Mind you, I am talking to my five-year-old niece.

    Further reading: Reference frames, inertial and non-inertial reference frames, equivalence principle, Galilean transformation, Lorentz transformation, worldlines, four-dimensional manifold to understand spacetime, why is there a negative sign in Minkowski's spacetime equation, why is speed of light constant, why is that constant for all observers, why spacetime interval is zero for light, what happens to causal effect if spacetime interval is not zero for light, what happens to spacetime if we travel faster than light, why is there a speed limit in the universe, what does any of this have to do with Maxwell's electromagnetic field theory. If spacetime interval is zero for light, how is space and time experienced by light. For photon's sake, what does any of these even mean. Go and read!

    Spacetime.png
     
    Last edited: Apr 23, 2017
  3. Iravati

    Iravati Platinum IL'ite

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    What are we doing here? Still on track? Yes, very much. Our aim is to demonstrate how Cooper aged slow near a black hole which can be generalised as why time distorts/dilates in gravity.

    Step#2: How objects move in 4D flat spacetime and what are geodesics?


    Recap: In the previous step, we stated that in 4D spacetime, observers will agree only on spacetime interval but not on space and time independently. Because we wish to tackle only time and gravity here, I am emphasising, your one sec is not my one sec. When you hear this statement, recall Minkowski's spacetime geometry. Ok, time is different for different observers, the question still remains how gravity affects time.

    We will briefly touch upon geodesics and move on to Step (3).

    What is a geodesic? A straight line on a curved surface.
    That's it, what does it mean. It means there are no straight lines on a curved surface.

    A straight line is a shortest path between two points on a plane surface. A geodesic is the shortest path between two points on a curved surface. You don't call the shortest path on a curved surface a line.

    That's it, again I don't get it.
    Let's say you walk straight on a curved surface, you traverse a circle. But a circle is not a line.

    Just remember that. Why do I need a geodesic? Because we will be moving to the fun part soon. Toss away all your intuition and known modelling of Newtonian gravitation force of orbits and falling apple trajectories. Have you ever thought, what-if, an irrational what-if, the objects are not moving in a curved path because of a force acting upon them but they are moving straight with no external forces yet they appear curved because the surface along which they are moving is curved. Planets orbit the sun in elliptical paths, tossed balls in parabolic paths, apples fall straight, how can all these shapes be explained by one force or phenomenon. From here on, there is no more force, we will talk only of shapes and trajectories which give the appearance of a force.

    OK, I get it. No more forces but only geometrical shapes. How does travelling on a curved surface create so many shaped trajectories? Watch below video.



    A car goes in a straight line (no curved surface), forms a parabola (edge of the curve surface), completes a circle (around the curve surface). This is why we need to understand geodesics. Let's move on to actual gravity and time distortion/dilation.

    Further Reading: Principle of equivalence and accelerating frames of reference. None of what I say will make sense if you have not thoroughly understood equivalence principle. Everything I talk henceforth is counter-intuitive. But first let's get rid of the curved rubbersheet diagram.
     
    Last edited: Apr 24, 2017
  4. Iravati

    Iravati Platinum IL'ite

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    Pre-step#3:Let's get rid of the rubber sheet

    They say not many people understand Einstein's relativity theories. Two things could have happened.

    (1) My understanding of General Relativity is completely wrong.
    (2) The rubber sheet diagram is so misleading that it obscures everyone from imagining the constructs behind his theories. You have to be imaginative with just high-school science and maths and Einstein is your mate.

    Shall carry on next week. But remember this diagram. If you are still fond of the rubber , then the only feature you want to preserve from rubber is its elasticity, in that, time stretches, warps, bends and distorts in spacetime inducing all those earlier seen configurable time coordinates.

    We shall examine in the next step what John Wheeler meant by “Spacetime tells matter how to move; matter tells spacetime how to curve.”

    CurvedSpacetime.png
     
    Last edited: Apr 24, 2017

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