A Measure Of The 3d Space Matter Occupies Is: Fill & Download for Free

To be able to explain gravity according to GR, you first have to get a feel for “spacetime” as opposed to our falsely intuited separate “space” and “time”, otherwise you’ll forever be stuck at incomplete analogies, like the rubber sheet one…Just like we already feel our three dimensions of space to be clearly unified, so is 4D spacetime. So what does it mean that we feel that 3D space is “unified”?For instance, it doesn’t matter how we position a pen in space, it will remain to have the same length. This invariance is at the center of this unification. But in order for us to add time to this already very familiar unification, we have to be able to “step out” of some of our very early conceptual understandings of space and look at it afresh, from a clear physics point of view.What do we actually mean when we say “the pen keeps the same length”, when we turn the pen? If we’d take the spatial position pen [math]u[/math] had at one particular moment in time, and then place this within a coordinate system and note the three distances on the x, y and z axis, like so:…then this invariant length of the pen is found with the Pythagorean theorem [math]\sqrt{dx^2+dy^2+dz^2}[/math]. These three values can take all sorts of apparently haphazard values when we turn the pen, but this calculation keeps finding the invariant length that’s at the center of this unification of space. We take our intuition for 3D space very much for granted, but something in the background of our mind’s CPU is constantly doing these type of “calculations”, so we don’t look surprised with the pen looks like a dot when it’s pointing towards us.So why does time need to be added to this familiar 3D continuum? That has to do with the fact that light is always measured at the same speed, no matter how fast we, as observers, move towards or away from the emitter of the light and no matter how fast this emitter moves towards or away from us.Classically this would be impossible, but welcome to relativity. If we’d observe someone in motion at 25% light speed, heading towards some approaching photons at 100% light speed, we’d intuit this observer should measure that light at 125% light speed.But the thing is that all fundamental quantum mechanical interactions within matter, that at the smallest level are responsible for both keeping matter from decaying and allowing for all possible change through time, are directed by this same speed of light. If the Sun had exploded a minute ago, we won’t notice anything for another 7 minutes or so, still peacefully enjoying its historic light. This same delay is occurring inside the fundamental particles of our cells and measuring tools as well.Our brains, measuring rods and clocks appear to behave differently as observed by someone that sees us traveling. But we never see our own measuring rods and clocks change at all, because everything works perfectly in sync within the same “frame of reference”. I mean, Earth might be traveling very near to the speed of light for some distant observer.So we see a time dilated, length contracted traveler measuring the speed of things with time dilated clocks, length contracted measuring rods and with different opinions about which events occurred simultaneously. More details on this here.So, to get back to pen [math]u[/math], where we calculated the apparent invariant length of the pen at a certain moment in time. This idea of “the length of the pen at a certain moment” should now be further drilled down into “the distance between point A (the back) of the pen at moment x and point B (the tip) of the pen at moment x”. To specify this ‘moment x’ even further, let’s imagine the pen has two LED lights at point A and B of the pen that blipped simultaneously as observed by the comoving observer. But;Since simultaneity is also relative, another observer that’s in relative motion to the pen will disagree on those moments being simultaneous. Those lights will actually blip at two different moments in time (after accounting for light delay) for this other observer that’s in relative motion to this pen.The distance in space between these two events (blip A and blip B) will be different for observers in relative motion to this pen.In other words: both the time and the distance between the two blips are measured differently by all observers that are in relative motion to each other. This makes it appear as though there is no invariant interval to be found of the length of the pen at all, but thankfully, this is not true.There is an invariance at the center of the unification of 4D spacetime: if any observer subtracted (!) their observed and squared time interval in seconds between blip A and B from their observed and squared distance interval expressed in light-seconds, between blip A and B, they would all come to the exact same number.So every observer notices different time and space intervals between the same two events in spacetime, but subtracting their squared values results in the invariant spacetime interval.So time is actually also part of this Pythagorean constant, but with a very unintuitive minus sign (at light speed there’s no such thing as space and time left). The true invariant constants are [math]\sqrt{dx^2+dy^2+dz^2-dt^2}[/math] for distance and [math]\sqrt{dt^2-dx^2-dy^2-dz^2}[/math] for time, as long as we keep using compatible units for height, width, depth and time (for instance light-seconds and seconds: Natural units - Wikipedia). We don't intuitively feel time and space to be unified, simply because our brains didn't evolve together with observations at relativistic high speeds. These calculations therefore don’t take place in the background of our mind’s CPU, like it does for 3D space.So, unified 4D spacetime is this truly inseparable continuum that’s like the stage in which all events take place.For instance, in the following diagram we see what is momentarily seen as 100% time for the accelerating observer in the middle on the y-axis, and we see what is momentarily seen for him as 100% space shown on the x-axis. The big X represents the speed of light (seconds and light-seconds are of equal distances) and all those dots are events in spacetime, like many blips of LED lights:That’s how length contraction, time dilation and relative simultaneity behave in unison. It’s all very symmetrical, isn’t it? Note that this is really just an unchanging picture of spacetime, continually being adjusted for the observer’s relative view.Now, what we call gravity according to general relativity is not some direct effect that the mass-energy of a massive body like Earth has on some object like an apple (and vice versa), the way that for instance a rope could be directly pulling it. Earth does nothing at all to the apple, only to the stage in which the apple happens to reside. And the apple in its turn, only responds to this “stage of all events”.That’s about the only thing the rubber sheet analogy has going for it, really...Therefore one cannot use shields of special material to block this apparent “pull" of gravity as this shielding would merely appear on the very same stage.So what would happen if this stage became curved for some reason? I’ve already mentioned that the idea of constant motion is completely relative to an observer. We could be moving at near light speed for some distant observer. The property of an object to remain at rest or to remain in motion with constant velocity is called inertia.Here we are following an inertial object in flat spacetime (meaning, no gravity) that’s not moving relative to us as observers. It’s just floating there in front of us. You can see that this object describes a perfectly straight line within flat 4D spacetime:Now what if some planet, let’s say without an atmosphere, is observed to approach this floating inertial object? Would it experience the force of a sudden pull, you think?It really wouldn’t, as there are no differences between the experience of either a free-fall inside a vacuum or floating still inside the vacuum of a flat spacetime. It’s both the same experience of weightlessness.This “free-fall” actually follows a perfectly straight line on the stage of curved spacetime, just like it did in flat spacetime:But you can see that in respect to that curved “space” stage, it is observed to be constantly falling. This is why feathers and rocks will be observed to follow the exact same paths.If there was a hole throughout this planet, objects would start floating according to the following pattern (credits for the following images go out to John D. Norton taken from his beautiful Einstein for Everyone lectures):If we’d try to draw these straight 4D worldlines in curved spacetime on a flat 2D paper, it would look curved like this:Now, imagine the following four flat-Earth believers meticulously following straight lines on this curved surface of Earth, initially starting out perfectly parallel at different points on the equator and ending up at the other side of the equator:If we’d try to draw these straight paths on Earth's curved surface on a flat 2D paper, it will look curved like this:Looks familiar, doesn’t it?Note that the 2D surface of Earth is obviously simply curving into an actual existent third dimension of space, but 4D spacetime isn’t curving into some real 5th dimension at all, although that is how it is often represented, visually. For instance, rings of equal distances within two dimensions of curved space is often shown curving into a completely imagined 3rd dimension “in which it curves”:Although that does reveal them having equal distances, these following rings of equal distances of 2D space are closer to our reality:Another thing to note is the reason that the path of a ball thrown in the air appears so extremely curved to us (how could this possibly follow a straight line in spacetime, right?), is only because we subjectively feel that a second isn’t that long, while at the same time we feel a light-second must take up a much larger interval in spacetime (a whopping 299,792,458 m), but both take up the exact same interval within 4D spacetime.In other words, we need to stretch that second of “falling time” in our spacetime diagram, so that it occupies the same distance a light-second does. This way Earth’s curvature of spacetime is shown to actually be quite small.Our conscious experience of time is highly subjective and depends very much on the animal in question. Every animal has a different “operating system” running inside their brains, making moments appear strongly different within their subjective conscious experience. To dig a bit deeper into understanding this highly subjective experience of the moment, definitely check out What is the difference between time and the flow of time?.So, to sum up, gravity according to general relativity basically says that a nearby mass-energy source (mostly nuclear: 99% of the energy within mass is the binding energy between the constituent quarks inside the protons and the neutrons) directly curves the spacetime around it. Objects (worldlines) inside this curved 4D spacetime therefore automatically start following a path through space, just by the simple passage of time.The “downward force” that we feel on the surface of Earth is actually a Fictitious force - Wikipedia, because this surface limits us from following straight lines in spacetime, which would be the perpetual back-and-forth fluctuation of a free-fall.In other words, we are not being “pulled down” towards the “curved travel path” of a free-fall by this “force of gravity”, but it’s really the other way around: the surface of Earth is constantly pushing us upwards into this curved path through curved spacetime. But since we are thereby following the same curvature as spacetime itself, we don’t think of ourselves constantly following curved paths at all.Here’s a well-explained short clip that helps intuit this a bit further:Hope this helped.

In order to prove 4d space exists one must first prove 4d matter exists. Luckily, it may be possible to prove 4d matter exists.The proton may be four dimensional, and here's why. From experiments using muonic hydrogen, researchers have determined the radius of the proton to be about .841 fm (.841 x 10^-15 m). Multiplying that by 2(pi) you find the proton's circumference to be about 5.284 fm. If one calculates the wavelength of a photon having energy equal to one quarter the mass of a proton's mass one gets a wavelength of 5.285 fm, which is almost exactly the same as the circumference of a proton calculated from its radius measurement. The difference could be due to the experimental error associated with the radius determination.So, within the error limits of the experimentally determined values, the circumference of a proton as determined from its radius, and, the wavelength of a photon having energy equal to a quarter of the proton's mass, are the same. Could it be that the proton is composed of four photons (or photon like things) traveling in a tight orbit in 3d space? Well, they can't be in 3d space because the proton's 3d volume is too small. The volume of a photon's enclosing cylinder equals its wavelength (2(pi)r) times pi times its amplitude squared (r^2) = 2(pi^2)(r^3). I'm using 'r' for amplitude. This just so happens to also be the formula for the volume of the surface of a hypersphere. What that says is that the volume of the cylinder that just encloses any photon (a classical Maxwellian sine curve shaped photon) is equal to the surface volume of a hypersphere with radius equal to the amplitude of the photon.So, one of these quarter proton mass photons has a volume that takes up all the 3d space in a proton's hypersphere's surface. What about the other three quarter proton mass photons that are needed to account for all the proton's mass? What about them? Because photons are bosons (not fermions) they can all occupy the same space, at the same time.And remember, the surface of a hypersphere is 3 dimensional, so a photon from our 3D space would feel right at home there. But what central force would keep it traveling in such a tight circle? Remember, it is turning in 4D space. Mass is determined by the Higgs field, which is a 3D phenomenon, so maybe there is no inertial mass when making 4D turns. Maybe the photon thinks it is still in our 3D space and is traveling in a straight line. The topology of the two situations is the same. (the proton's hypersphere's surface vs the universe's hypersphere's surface) Both are 3D spaces with 4D space on either side. The only difference being the greater curvature of the proton's hypersphere's surface.These facts strongly suggest the proton is a four dimensional object, and since 4 dimensional objects can only exist in 4 dimensional space, four dimensional space exists. (Actually, any dimension of space exists if matter of that dimension exists to bring it into coexistence.) We can see the proton because it is not entirely in 4d space. It is intersected by our 3D space so we can see some of it -- the part that intersects our 3D space. The rest of it, about 85% of it, I conjecture, resides in 4d space. How can that be, you ask? We live in 3D space. Where is the 4D space?Think of a marble embedded in the surface of a beach ball so that half of the marble protrudes into the interior space of the beach ball and the other half of the marble sticks out into the exterior space surrounding the beach ball. For 2D creatures living in the 2D surface of the beach ball (Flatlanders), the 3D marble (3D sphere) will look like a circle (2D sphere). They cannot see the part of the marble sticking out into 3D space. They think the marble is 2D because that's all they can 'see'. (Their light is 2D and only moves around in the 2D surface of the beach ball.) The marble can move anywhere on the 2D surface of the 3D sphere, and wherever it is, it will be mostly in 3D space.Likewise with beings (us) living in the 3D surface of a 4D sphere (hypersphere). Wherever a proton is in our 3D world, it is bisected by the surface of the hypersphere, and, most of it is sticking out into 4D space. How can that be if our space is 3D? Just like the 2D space in which the Flatlanders live has ZERO THICKNESS in the third dimensional direction (which they cannot point to), our 3D space has ZERO THICKNESS in the fourth dimensional direction (which we cannot point to). So just like in the case of the Flatlanders, where every point in their 2D world is adjacent to 3D space (in two opposite 3D directions, by the way), every point in our 3D world is adjacent to 4D space (in two opposite 4D directions). All this follows directly from the fact that for a sphere of any dimension, all points in the surface of the sphere are the same distance from the center of the sphere, which implies that the thickness of a sphere's surface is ZERO in the n-dimensional direction.

We are actually still working these things out. Hence the study of Cosmology[1]It is generally understood to begin with the Big Bang, followed almost instantaneously by cosmic inflation - an expansion of space from which the universe is thought to have emerged ~13.7±0.2×10^9 (roughly 13.7–13.9 billion) years ago.[1]Where did the Big Bang "come from" is still being debated. There a number of theories and little to no proof.[6]The shape of The Universe is interesting. When you look at the sky, there is no preferential center to the coordinate system. [2] There simply doesn't seem to be a "center of the Universe".The recent Wilkinson Microwave Anisotropy Probe (WMAP) measurements have led NASA to state, "We now know that the universe is flat with only a 0.5% margin of error."[1] Within the Friedmann–Lemaître–Robertson–Walker (FLRW) model, the presently most popular shape of the Universe found to fit observational data according to cosmologists is the infinite flat model,[2] while other FLRW models that fit the data include the Poincaré dodecahedral space[3][4] and the Picard horn.[5]The local geometry is the curvature describing any arbitrary point in the observable universe (averaged on a sufficiently large scale). Many astronomical observations, such as those from supernovae and the Cosmic Microwave Background (CMB) radiation, show the observable universe to be very close to homogeneous and isotropic and infer it to be accelerating.The above statement basically mean, anywhere you look on a large scale, The Universe looks the same and is of the same "density" There isn't a huge empty portion and super dense portion when you look at the visible Universe as a "whole".The Universe is expanding fairly rapidly, a discovery made by Edwin Hubble who observed the majority of galaxies and celestial object were red shifted and therefore moving away from our galaxy. [3]Matter seem to occupy the areas were dark matter energy were/are. Dark matter comprises the bulk of the mass of the universe[4]. We know this through observational inferences and a considerable amount of the work in particle physics in trying to define what "it is".[4]3D map of the large-scale distribution of dark matter, reconstructed from measurements of weak gravitational lensing with the Hubble Space TelescopeA simulation of the filament-like distribution of Dark Matter in the Universe[5]The cosmic microwave background spectrum measured by the FIRAS instrument on the COBE satellite is the most-precisely measured black body spectrum in nature[6]don't know how the universe will end or when[3]. As it appears now, The Universe will continue to expand and stars and galaxies born, decay, and explode until The Universe is so diffuse it goes black. We estimate The Universe is expanding at:The newly refined value for the Hubble constant is 74.3 plus or minus 2.1 kilometers per second per megaparsec. [7]As the Universe expands, the density of radiation and ordinary and dark matter declines more quickly than the density of dark energy (see equation of state) and, eventually, dark energy dominates. Specifically, when the scale of the universe doubles, the density of matter is reduced by a factor of 8, but the density of dark energy is nearly unchanged (it is exactly constant if the dark energy is a cosmological constant).Current observations indicate that the dark energy density is already greater than the mass-energy density of radiation and matter (including dark matter). In models where dark energy is a cosmological constant, the universe will expand exponentially with time from now on, coming closer and closer to a de Sitter spacetime. In this scenario the time it takes for the linear size scale of the universe to expand to double its size is approximately 11.4 billion years. Eventually all galaxies beyond our own local supercluster will redshift so far that it will become hard to detect them, and the distant universe will turn dark.In other models, the density of dark energy changes with time.In quintessence models it decreases, but more slowly than the energy density in ordinary matter and radiation. In phantom energymodels it increases with time, leading to a big rip.[1]http://en.wikipedia.org/wiki/Cosmology[2]http://en.wikipedia.org/wiki/Shape_of_the_Universe[3]http://en.wikipedia.org/wiki/Accelerating_universe[4]http://en.wikipedia.org/wiki/Dark_matter[5]http://magic.mppmu.mpg.de/physics/index.html[6]http://en.wikipedia.org/wiki/Big_Bang[7]http://www.msnbc.msn.com/id/49282518/ns/technology_and_science-space/#.UG9SpE3XZOE