- 2792 - SPACETIME - Theory of Relativity. How it messed up geometry. Our Universe isn’t made up merely of three space dimensions, but of four “spacetime” dimensions. Three of them are space and one of them is time, and that’s where we get spacetime. The shortest distance between two spacetime events isn’t a straight line any longer. Why is that?
- “What’s the shortest distance between two points?” Most of us will give the same answer that Archimedes gave more than 2,000 years ago: a straight line. If you take a flat sheet of paper and put two points down on it absolutely anywhere, you can connect those two points with any line, curve, or geometrical path you can imagine. So long as the paper remains flat, uncurved, and unbent in any way, the straight line connecting those two points will be the shortest way to connect them.
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- This is precisely how the three dimensions of space work in our Universe: in flat space, the shortest distance between any two points is a straight line. This is true regardless of how you rotate, orient, or otherwise position those two points.
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- The distance between two points depends on the path taken; displacement does not.
Normally, we measure the distance between two points by the distance traveled.
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- A straight line being the shortest distance between two points comes from the Pythagorean theorem. The Pythagorean theorem as a rule about right triangles, that if you square each of the short sides and add them together, that equals the square of the long side.
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- If the short sides are a and b while the long side is c, then the equation relating them is:
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---------------------------------- a² + b² = c².
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- Think about what this means, however, not from the perspective of pure mathematics alone, but in terms of distances. It means that if you move through one of your spatial dimensions by a certain amount (a, for example) and then move through a perpendicular dimension by another amount (b, for instance), then the distance between where you began and where you wound up is equal to c, as defined by the Pythagorean theorem.
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- The distance between any two points on a plane, where those points are separated by a in one dimension and b in another dimension, is c, where:
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--------------------------------- c = √(a² + b²).
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- In our Universe, we’re not restricted to living on a flat plain. We have not just length and width (or the x and y directions,) dimensions to our Universe, but depth (or the z direction) as well.
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- If you want to figure out what the distance is between any two points in space, it’s the exact same method as it was in two dimensions, except with one extra dimension thrown in. Whatever amount your two points are separated by in the x direction, the y direction, and the z direction, you can figure out the total distance between them in just the same way.
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- Because of the extra dimension, the distance between them is going to be given by:
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------------------------ d = √(x² + y² + z²).
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- This equation says that the distance between any two points is defined by the straight line connecting them: the line that accounts for the separation between your two points in all three dimensions: the x-direction, the y-direction, and the z-direction combined.
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- One of the interesting and important realizations about this relationship — the distance between two points being a straight line — is that it absolutely does not matter how you orient your visualization of the x, y, and z dimensions. You can either:
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---------------------------- change your coordinates so that the x, y, and z dimensions are in any (mutually perpendicular) directions you like, or
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--------------------------- rotate these two points by any amount in any direction,
and the distance between them will not change at all.
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- The individual components will change if you either rotate your perspective or rotate the line connecting those two points, as your definitions of length, width, and depth will change relative to one another for that line as the rotation occurs. But the overall distance between those two points doesn’t change at all; that quantity of the distance between those points remains what we call “invariant,” or unchanging, regardless of how you rotate them.
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- The distance between two objects doesn't depend on how your coordinates are oriented. If time is just another dimension, then the distance between any two points in spacetime will work the same way.
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- If we represent the time dimension as “t‘, you think the distance would be the straight line connecting two points through the three spatial dimensions as well as the time dimension. In mathematical terms, you might think that the equation for the separation between any two points would look something like:
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------------------------------------ d = √(x² + y² + z² + t²).
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- This is much the same change we made when we went from two dimensions to three dimensions, except this time we’re going from three dimensions to four dimensions. It’s a reasonable step to attempt, and describes exactly what reality would look like if we had four dimensions of space, rather than three.
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- But we don’t have four dimensions of space; we have three dimensions of space and one dimension of time. And despite what your intuition may have told you, time isn’t “just another dimension.”
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- There are two ways that time, as a dimension, is different from space. The first way is a small one: you can’t put space (which is a measurement of distance) and time (which is a measurement of time) on the same footing without some way to convert one to the other.
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- Fortunately, one of the great revelations of Einstein’s theory of relativity was that there is an important, fundamental connection between distance and time: the speed of light, or equivalently, of any particle that travels through the Universe without a rest mass.
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- The speed of light in a vacuum, 299,792,458 meters per second, tells us precisely how to relate our motion through space with our motion through time.
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- When we use terms like “one light-year” or “one light-second,” we’re talking about distances in terms of time: the amount of distance that light travels in one year (or one second).
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- If we want to convert “time” into a distance, we need to multiply it by the speed of light in a vacuum.
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- The key idea is that we’re all moving through the Universe, through both space and time, simultaneously. If we’re simply sitting here, stationary, and not moving through space at all, then we move through time at a very specific rate at which we’re all familiar: one second per second.
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- However, the faster you move through space, the slower you move through time. The other dimensions are not like this at all: your motion through the x dimension in space is completely independent of your motion through the y and z dimensions. But your total motion through space, relative to any other observer, determines your motion through time. The more you move through one (space or time), the less you move through the other.
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- Einstein’s relativity gives us concepts like time dilation and length contraction. If you move at very low speeds compared to the speed of light, you won’t notice these effects: time appears to move at one second per second for everyone, and lengths appear to be the same distance for everyone at speeds normally achievable on Earth.
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- But as you approach the speed of light, or rather, as you perceive an object where the relative speed between you and it is close to the speed of light, you’ll observe that it’s contracted along its direction of relative motion, and that clocks appear to run at a slower (dilated) rate relative to your own clocks.
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- The reason underlying this is because the speed of light is the same for all observers. If you imagine that a clock is defined by light bouncing back and forth between two mirrors, then watching someone else’s clock as they move close to the speed of light will inevitably result in their clock running slower than your own because it takes time for the light to reach you.
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- There is an even deeper insight here, which initially eluded even Einstein himself. If you treat time as a dimension, multiply it by the speed of light, and treat it as though it were imaginary, rather than real, then we can define a “spacetime interval” the same way we defined distance earlier. Only, since the imaginary number “i” is just √(-1), this means that the spacetime interval is actually;
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--------------------------- d = √(x² + y² + z² - c²t²).
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- Note the minus sign attached to the time coordinate!
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- In other words, the transformation from “motion through or separation in space” to “motion through or separation in time” is also a rotation, but it’s a rotation not in the cartesian coordinates of space (where x, y, and z are all real numbers), but through the “hyperbolic coordinates of spacetime“, where if the space coordinates are real, then the time coordinate must be imaginary. Read that again!
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- What’s remarkable about all of this is that Einstein, despite lacking the mathematical insight to understand exactly how the dimension of time was related to the three conventional dimensions of space, was still able to piece together this key physical insight.
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- Increasing your motion through space decreased your motion through time, and increasing your motion through time decreased your motion through space. All measurements of space and time are only meaningful relative to the observer in question, and depend on the relative motion of the observer to the observed.
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- Yet, the spacetime interval remains invariant. No matter who is doing the observing or how quickly they’re moving, the combined motion of any object through “spacetime” is something all observers can agree on.
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- Fortunately, in physics, the Universe itself, not anyone’s theory, is the ultimate arbiter of scientific truth.
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------------------------------- Other Reviews about spacetime:
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- 2722 - SPACETIME - at the macro and micro levels.? How Can Space and Time be Related?. Space and time seem to be absolute quantities to us. It is hard to see their interrelationship until you take their ratio as velocity and extend that ratio to its limits. Velocity is space / time!
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- 2707 - SPACETIME - still a conundrum in science? Our universe that we know today is extremely unnatural. It is a weird permutation among countless other possibilities, observed for no other reason than that its special conditions allowed life to arise. The properties of the universe are inevitable, predictable, ‘natural,’ locking together into a sensible pattern. It is our challenge to make sense of it.
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- 2699 - SPACETIME - Einstein and the expanding Universe ? At what speed is the universe expanding? Until now, at least two independent calculation methods have arrived at two values that are different by about 10% with a deviation that is statistically irreconcilable.
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- 2567 - SPACETIME - creating new physics? Einstein invented spacetime to explain gravity. Gravity is bent spacetime. All objects, (ie:mass), move through gravity seeking the shortest distance and shortest time through the bent spacetime created by the mass. Before Einstein almost everybody thought Isaac Newton already had figured it all out.
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- 2523 - SPACETIME - Fast Speed and Short Time. You should be fascinated by time and space. Both seem so fundamental, yet, Einstein’s Theory of relativity proves that time and space are not absolutes. They are relative. They are one thing, not two things, spacetime is one thing. Space and Time bend and curve when they interact with mass. Mass tells spacetime how to bend and spacetime tells mass how to move.
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- 2522 - Space - curved or flat? Also, lists 14 more reviews about space.
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- 2271 - Hw can space and time be related? Also lists 13 more reviews about spacetime. Space and time seem to be absolute quantities to us. It is hard to see their interrelationship until you take their ratio as velocity and extend that ratio to its limits. In order for the speed of light to be a constant physical law in the Universe, regardless of the relative motion of all observers everywhere, then space and time must change in a compensating way to keep the velocity constant.
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- 2213 - Spacetime from atoms to blackholes.
- 2180 - Velocity is space divide by time.
- 2074 - Much todo about nothing, ie space.
- 1790 - Space bends and time slows.
- 1242 - How does spacetime change at the micro level?
- 1189 - The beginning of time?
- 1006 - Is time slowing down?
- 910 - Time to think?
- 854 - Time , GPS, and Entropy?
- 842 - Pressed for time?
- 830 - Why a 24 hour day?
- 814 - Fast speed and short time?
- 783, 784 - Time is what God crested to keep everything from happening all at once.
- 747 - Why 60 minutes?
- 590 - So you want to go into space?
- 392 - Time dilation derived using the Pythagorean Theorem
- 354 - The big picture using a universal calendar.
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- August 18, 2020 2792
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