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Thread: Non-Euclidean Geometry

  1. #1
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    Non-Euclidean Geometry

    Someone please explain to me how there can be non-Euclidean geometry without Euclidean geometry. I understand all the analogies quite well - The 2D surface of the Earth, the Surface of a balloon, etc. - but all of these analogies simply indicate a non-Euclidean, N-dimensional surface exists as part of a Euclidean, (N+1)-dimensional object. This is more than just a difficulty of visualizing abstract concepts. Infact, I am getting fairly good at visualizing a curved space-time (a classic hard one). It just seems to me to be mathematically impossible to have such geometry without having it based on more fundamental geometry of right-angles.

    The concept is simple. Non-Euclidean geometry is based on the effects of a curve. You cannot have a curve without a dimension to curve into. I cannot draw a curved line with anything less than two dimensions.

    The notion that spacetime "does not curve 'into' anything" seems false from the start. This would mean that you can't even draw a simple graph to represent the curvature of spacetime.

    Surely the curvature of spacetime can be drawn on a graph!

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    Cthulhu could answer your question.

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    Quote Originally Posted by TheNick View Post
    You cannot have a curve without a dimension to curve into.
    Why do you think that? I want you to think about that deeply. What it is about your notion of geometry, the mental picture in your head, that demands the above be so? Think about it.

    Riemannian geometry (and even non-Riemannian geometries that go as far beyond Riemann as Riemann goes beyond Euclid!) is all about setting up the machinery to help you get rid of those notions.

    Think of a curve, or a curved surface, or a curved volume as an entity unto itself. You are bound to "live" on it. You have only those dimensions available to you, and they are curved.

    Embedding spaces, which are a higher dimensional flat space are very useful at letting you see how this machinery is developed. And it gets involved, some high powered 'rithmetic indeed. The next step is to realize that machinery does not depend on the embedding space. You've developed a (invariant) description of the curved space that stands on its own.

    A word about space-time. Even flat space-time, Minkowski, is non-Euclidean. It's flat, but the norm, the "distance", the line element is not positive definite. It's basically time^2 - space^2 (or the reverse if you like). The time-like dimension(s) is/are different from the space-like dimensions.

    So technically, space-times are "psuedo-Riemannian", because you allow that non-positive definite signature. And that has some consequences. For example, you have the notion of "null vectors", light-like paths, where a vector is zero length but with all non-zero components.

    Space-time embedding spaces can get even weirder. For example, consider a helix, and coil, a screw thread curve. That is a one dimensional curve, but it takes 3 dimensions to embed it in. Same way with surfaces. You can have a surface that requires 4 dimensions to embed (or more).

    Go to space-times, valid solutions of the EFE, and you can have manifolds that require two or more higher time-like dimensions to embed! Rather than having to add more space-like dimensions, you have to add "more time". That gets a bit weird.

    So, while you can use embedding space to help picture curved manifolds, they are not required. Since you, living in the lower space, are confined to it, and have only the lower dimensions available to you and are unaware of the higher dimensions, why worry with them?


    -Richard

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    Three Illusions

    Buckminster Fuller recalled being in geometry class while in school. His teacher explained how we use the basic elements to form geometry.

    The first is the point. A point has no length, width or depth. It does not exist but we use its concept. The second is the line. It is infinite and has only length but no width or depth..It does not exist either. Third we can imagine lining up a series of lines like logs into a plane. A plane has width and length but has no depth. Just as with the point and the line, Bucky could not see one in front of his eyes anywhere in nature. But if you make a cube out of points, lines and planes you get something that is real.

    Now that really floored Bucky and many other mathematicians. How can you raise non-existence to the 4th power and get existence?

    Now what is a better way to describe the above 3 illusions with something that represents something real?

    Points do not exist. Now imagine a point that is there for only a moment and then disappears. Magic? No. Think again. The word "moment" is the clue. What occurs at a specific location for a moment or duration of time? An EVENT does. There might have been an accident at some intersection near you just moments ago. An event. Or hydrogen gases may have just collided enough to start fusion at some great distance. Another event at another location but an event nonetheless. Therefore, instead of points we have events. Events are real.

    Lines do not exist. But what does move forward or backward? How about pushes and pulls? One word that describes pushes and pulls better is the word FORCES. And there is direction and magnitude in those forces. Therefore, VECTORS describe those forces and represent something that is real.

    Planes do not exist. But surfaces do exist and they have all kinds of shapes and now we have a realistic description of the world around us. Points, lines and planes do not suffice. They are incomplete descriptions. Events, forces and surfaces and stronger and realistic fundamentals to base geometry upon.

    This does not imply that Euclidian geometry should be thrown out the window. It is still great for building porches and cars. But is has a weakness when trying to describe all the energy events in the universe. It is incomplete.

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    Quote Originally Posted by TheNick View Post
    Someone please explain to me how there can be non-Euclidean geometry without Euclidean geometry.
    You are under a severe misconception here. Whether a geometry is Euclidean or not is dependent one only one thing: the status of Euclid's fifth postulate. Euclid found that to get his geometry (which is plane goemetry) to work, he had to include that fifth postulate, which was really rather a pain. His first four are short (under 20 words) and quite obvious. His fifth is
    onger, rather convoluted , and not at all obvious. Euclid himself tried to prove the fifth (along with many after him) but it was eventually shown that it can't be done.
    The status of parallel lines determines what kind of geometry you have. In Eulcidean, they never meet, but maintain the same distance. If they meet, you have Elliptic geometry (first described by Riemannian). If they never meet, but continue to increase the distance between each other, then you have hyperbolic, first described by JŠnos Bolyai and Nikolai Ivanovich Lobachevsky. You can ignore the fifth altogether and have what's called Neutral, or Absolute Geometry. There is Affine Geometry, where Euclid's third and fourth postulates are basically meaningless. Spherical Geometry, used in measuring distances on a sphere (such as distances on the Earth) is also a non-Euclidean Geometry.

    Quote Originally Posted by TheNick View Post
    I understand all the analogies quite well - The 2D surface of the Earth, the Surface of a balloon, etc. - but all of these analogies simply indicate a non-Euclidean, N-dimensional surface exists as part of a Euclidean, (N+1)-dimensional object.
    You have to ignore the N+1 part of it, or the analogy breaks down.

    Quote Originally Posted by TheNick View Post
    This is more than just a difficulty of visualizing abstract concepts. Infact, I am getting fairly good at visualizing a curved space-time (a classic hard one). It just seems to me to be mathematically impossible to have such geometry without having it based on more fundamental geometry of right-angles.
    Well, see, Euclidean isn't based on right angles, it's all based on that silly fifth postulate. The "seems" in your sentence indicates you really don't have a concrete complaint, it's more of a something outside your experience kind of thing. And, as such, you don't have a point of reference (or enough math) to really get a good handle on what the math is describing (not knocking you here).

    Quote Originally Posted by TheNick View Post
    The concept is simple. Non-Euclidean geometry is based on the effects of a curve. You cannot have a curve without a dimension to curve into. I cannot draw a curved line with anything less than two dimensions.
    Sorry, this is a huge misstatement. I would suggest looking into the different kinds of non-Euclidean goemetry and being a bit more specific before issuing a blanket statement like this.

    Quote Originally Posted by TheNick View Post
    The notion that spacetime "does not curve 'into' anything" seems false from the start.
    Does it seem false because you can't imagine how or is there something more concrete you can show?


    Quote Originally Posted by TheNick View Post
    This would mean that you can't even draw a simple graph to represent the curvature of spacetime.

    Surely the curvature of spacetime can be drawn on a graph!
    Exactly how do you propose to draw a four-dimensional graph? I'd really be interested in your explanation of how you place the t dimension 90 degrees from the other three? For that matter, when you get into higher dimensional spaces, how do those go out 90 degrees each time you add a dimension. While it's quite easy to manipulate these things, mathematically, it's quite another to say they can be visualized.

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    I certainly don't have the strongest base in mathematics. I only just completed calculus 3 in my last semester and am beginning differential equations now (so now you know what you're dealing with ). Among the things I was introduced to in calc 3 was the curvature value. Surely such a simple mathematical process can be applied to the numbers of spacetime geometry.

    If we choose a point in space, can we not theoretically calculate the degree to which spacetime is curved at that point?

    This is the first half of a larger question. I'll ask the second half depending on the answer to this.

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    Quote Originally Posted by TheNick View Post
    If we choose a point in space, can we not theoretically calculate the degree to which spacetime is curved at that point?
    Yes, this is what gravity is-- and what general relativity gives us.

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    Quote Originally Posted by TheNick View Post
    I certainly don't have the strongest base in mathematics. I only just completed calculus 3 in my last semester and am beginning differential equations now....
    It sounds like you're constructing a pretty strong base there to me. Keep paying attention. Congratulations on what you've completed.

    Quote Originally Posted by TheNick View Post
    If we choose a point in space, can we not theoretically calculate the degree to which spacetime is curved at that point?
    That's pretty much what Einstein's Field Equations do, if my vague understanding is correct.

    As it happens, I barely survived Calculus and D.E. myself - went surfing a lot - but then I really enjoyed abstract algebra and modern geometry. Yeah, me and Garrett Lisi go way back.

    In my imagination only.
    Everyone is entitled to his own opinion, but not his own facts.

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    Quote Originally Posted by Ken G View Post
    Yes, this is what gravity is-- and what general relativity gives us.
    And that's why I personally have trouble picturing euclidean space, for every time I graph something with right angles, I know the graph is subtley warped by the mass of the pencil lead I'm laying down, yea, even by my own mass as I might move about trying to view the graph. Those right angle just don't remain right angles. I'm amazed by anyone who can tolerate the notion that they do.
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    Quote Originally Posted by TheNick View Post
    I certainly don't have the strongest base in mathematics. I only just completed calculus 3 in my last semester and am beginning differential equations now (so now you know what you're dealing with ).
    Actually, your probably ahead of about 95% of most of the people here. And your complaint about Euclidean geometry now really surprises me. I would have thought you would have been quite aware of the specifics of the fifth postulate/non-fifth postulate definition for a geometry. Although, I really think that your questions have nothing to do with the Euclidean Geometry, per se, but are more related to the normal experience(and expectation) of having a requirement that something of n dimensions have to curve into a space of n+1 dimensions. Just my opinion.

    Quote Originally Posted by TheNick View Post
    Among the things I was introduced to in calc 3 was the curvature value. Surely such a simple mathematical process can be applied to the numbers of spacetime geometry.

    If we choose a point in space, can we not theoretically calculate the degree to which spacetime is curved at that point?
    Of course. It's dependent on the amount of energy plus the pressure of each of the three orthogonal directions in a given volume of spacetime. That's the basic Einstein Field Equation. The curvature at each point in that volume is determine be the solution of the equation. You end up with what's called a metric for that volume. That metric, tells you how distance is measure in each of the four dimension at each point within that volume. Note however, that each dimension can have a different warpage. Just FYI, I prefer warpage over curvature, simply becuase if a point has each dimension curving differently, it's easier for my mind to visulize it as space at that point being warped, not curved.

    Quote Originally Posted by TheNick View Post
    This is the first half of a larger question. I'll ask the second half depending on the answer to this.
    Ok, fire away. I will point out that if you ever get to Differential Geometry (The study of differentialble manifolds), the answers to your questions may become obvious. I will warn you that I am mostly self taught at this level and there are holes (that keep getting smaller) and that your question may sail through one of those holes.

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    Quote Originally Posted by TheNick View Post
    Among the things I was introduced to in calc 3 was the curvature value. Surely such a simple mathematical process can be applied to the numbers of spacetime geometry.
    You ain't seen nothin' yet, my boy.

    I'm certain the curvature you were introduced to there is the plane curvature, which can be applied to curves in a plane. That is, 1D objects that curve around in a plane, a flat 2D space.

    In the plane, things are very simple, and the curvature (at each point on the curve) can be described by a simple scalar. That's the number you were introduced to.

    In more precise terms, this is dT/ds, the change in tangent vector (slope vector) of the curve per arc length along the curve. This is a measure of how the curve is deviating from a straight line at that point. The tangent vector of a straight line is constant, and so you're comparing how the slope is changing. This is a second derivative level operation, involving how the slop changes. The defintion in terms of arc length along the curve itself makes that number be *invariant*. It doesn't depend on the coordinate system used to map the plane and the curve in it. That gets important.

    For example consider a parabola, y = x^2. Well, that's just one coordinate system. We can rotate that parabola around 90 degrees, and now we have y = +/- sqrt(x). The slope at x = 0 in the first case is 0, but the slope in the second case is infinite, a vertical line. However, it is the same point on the curve.

    But that parabola is geometrical entity of itself that doesn't on the coordinate system. What is desirable is to find a way to deal with things that doesn't depend on the coordinates. The curvature is the way to do this. The parabola is specified by its curvature. All we need is one parameter, s the length along it (starting somewhere), and the curvature as a function of that s. We then have a invariant description of the curve.

    In the plane, a circle turns out to have constant curvature, and the curvature, dT/ds = 1/R, where R is the radius of the circle. Thus, you can think of the inverse of the curvature as the radius of circle tangent to the curve at that point, and that's known as the radius of curvature. If the curvature remained constant, it would trace out that circle.

    So circles are special. And that holds as you increase dimensions as well. N-spheres are constant curvature.

    But as you increase dimensions, it turns out simple scalars are no longer enough to completely specify things. Consider the helix I mentioned before. That is a curve that is not confined to a plane. dT/ds is no longer enough to desribe a curve in a 3 dimensions.

    What else do we need, there? Well, if the plane curves are restricted to a plane, then it stands to reason we want some measure of the tendency of the curve to come out of a plane at a point. What is the plane at a point? The tangent plane of a curve can defined by the tangent vector, and another vector representing dT, the change in the tangent vector, which is given by the curvature.

    A curve comes out of the plane if the change in the change in the tangent vector has a component normal to that plane. The change in the change is a third derivative operation. This number is sometimes called the "torsion", and a helix has constant torsion in this sense. This notion of torsion is not to be confused with a much more complex notion that is beyond our scope here.

    Those two numbers, curvature and torsion, can then specify a curve in 3 dimensions. So, one number describes a curve in the plane, but two are required for a volume.

    In general, it turns out the curvature must be a *tensor*, which can be thought of as an array of numbers, and that array can be more than just 2dimensions.

    The curvature of 4D space-time, which is the sort of the main big thing in General Relativity is the Riemann curvature tensor. That is *4 dimensional* square array of 4 components in each dimension. That is a 4x4x4x4 array of numbers. Fortunately symmetries and other constraints reduce the number of independent components down greatly.

    Anyway, that's why I said you ain't seen nothin' yet!

    -Richard

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    Quote Originally Posted by TheNick View Post
    If we choose a point in space, can we not theoretically calculate the degree to which spacetime is curved at that point?
    Yes, and we can approximate it pretty well without a lot of high-powered machinery. The stuff in publius's post is on target, but just like the radius of a black hold calculation, we can come up with a decent swag at curvature too. The idea is borrowed from early in the book Gravitation, by Misner, Thorne, and Wheeler.

    We naturally use 1/r as a measure of curvature, and, as publius remarks, that's pretty much a valid mathematical definition as well. So, how do we apply it to the surface of the earth, or gravity in general? Let's just say we're looking at something that rises to a height h in distance d, then falls back after another distance d. We know it would be close to a parabola, but let's just approximate it with a circle. What would be the radius of that circle? One chord would cross with length 2d, and it would be bisected by a diameter, one side length h, the other (2r - h)

    A geometry theorem says that d2 = h(2r-h) = 2rh-h2, but since h2 is so small comparred to the other terms, we're going to ignore it (remember, we're just after an approximation! ) What we have left is:

    2rh = d2
    r = d2/2h
    1/r = 2h/d2

    And there we have our formula for curvature, in terms of h and d.

    For the ideal case of projecting something at a velocity v at an angle Θ in gravity g, the time it takes to go a height h is v sinΘ/g, and the average vertical velocity is v sinΘ/2, so h equals the product of those two: (v sinΘ/g)(v sinΘ/2). The distance d is just the time multiplied by the horizontal velocity, or (v sinΘ/g)(v cosΘ). Plugging them into our formula for curvature, we get:

    2h/d2 = 2(v sinΘ/g)(v sinΘ/2) / ((v sinΘ/g)(v cosΘ))2 = g/(v2cos2Θ)

    So the curvature depends upon g, v, and cosΘ. The greater g, the greater the curvature (duh). The faster we throw (or shoot, like a bullet) the flatter the arc. And the smaller Θ means the cosine is greater, and the arc is flatter. Actually, the denominator is just the horizontal velocity squared. All that makes sense, but it seems to mean that the curvature is different for bullets than baseball, and yes the arc of a bullet is fairly flat.

    In relativity though, we have to use spacetime. Now, our distance d includes a time component. The height h, being the instanstaneous height at the middle of the arc, does not have a time component. And the time has to be converted to distance somehow so that we can use it for d. The way to do that, in relativity, is multiply the time by the speed of light c. But that gives us a distance so large that it completely dwarfs all other contributions to d. So, in our approximation, we're left with d = cv sinΘ/g

    Plugging that into our formula, with h the same as before, we get

    2h/d2 = 2(v sinΘ/g)(v sinΘ/2) / (cv sinΘ/g)2 = g/c2

    So, now, the curvature does not depend upon anything but g and the speed of light! And so it's the same for bullets and baseballs!

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    hh,

    That was very nice.

    I'll point that relation, curvature = g/c^2 holds exactly for Rindler observers. The constant path curvature is exactly that for a constant proper acceleration of 'g' in flat space-time. Hyperbolas are the equivalent of circles in Minkowski, and the Rindler familiy of hyperbolic world lines are curves of constant curvature, just like circles are in Euclid.

    The radius of curvature there, c^2/g is the distance to the Rindler horizon.

    And that brings up another point, the value g/c^2 is actually a path curvature, seen by stationary, accelerated observers in the earth's gravitational field. It is the same view a Rindler observer would see of something he dropped in his accelerating frame.

    So g/c^2 is actually telling us more about the path curvature of the *stationary observer*, accelerated resisting gravity, than it is the free-falling object itself! That's the Equivalence Principle having fun with us here. The stationary observer is the one who is doing most of the curving, not the free faller.

    To get the true curvature of space-time, we need to look at the *tidal g*, the difference in acceleration seen by a free faller of another free faller. And we'll find that is much, much smaller than g/c^2 at the surface of the earth. The earth is a very weak field indeed.


    -Richard

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    Quote Originally Posted by publius View Post
    To get the true curvature of space-time, we need to look at the *tidal g*, the difference in acceleration seen by a free faller of another free faller. And we'll find that is much, much smaller than g/c^2 at the surface of the earth.
    Excellent point-- if I keep reading your posts, they will eventually sink in. Keep at it.

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    Ken,

    The symmetry of Schwarzschild is high enough that a single scalar can be pulled out that contains all the information. That is the Kretschmann scalar/invariant, and that sucker is

    12R^2/r^6

    where R is the usual 2GM/c^2, and r is the radial coordinate.

    Note this is fine and dandy at r = R and only blows up at r = 0, the true singularity. However, that r^6 seems a bit too high a power.

    The Kretcschmann scalar (IIRC, and with my -1 caveats) is some sort of quadratic contraction of the Riemann tensor, sort of adding up squares of the components, much like the norm of a vector is the sum of the squares. But this is doing that with a rank-4 structure.

    So this thing actually takes the form of the square of Riemann components, so the square root would go as ~1/r^3, which is the expected tidal behavior.

    When your curvature is this big tensor thing, with all sorts of components telling you which way this part "curves" with respect to that part which can all be different and coordinate dependent, pulling some single invariant number or even numbers out can be sort of ambiguous. When the symmetry is high enough, all the components sort of depend on the same factor, then you're good to go.

    -Richard

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    Above, we spoke of path curvature vs. curvature of the space-time itself, and it might be helpful to expand on the difference between the two.

    In flat space, as we went through above, the curvature is a notion of the deviation from a straight line (and even added a notion of deviation out of plane). So in flat spaces, a curvature is the deviation of a path from straight. Now, when the space itself is curved, the notion of "straight line" becomes that of a *geodesic*.

    So, we if we have a curve in curved space, the notion of path curvature becomes deviation from the geodesic. In flat spaces, the geodesics are trivially straight lines, so the path curvature is comparing the curve to a straight line.

    However, in curved spaces, the geodesic itself curves, so path curvature is sort of the difference between the "coordinate curvature" of the path vs the geodesic curvature. And that path curvature is simply the proper acceleration of that path.

    The covariant form of Newton works in exactly those terms. The force is inertial mass times the path curvature, the deviation from the geodesic, not the deviation from what you think is "straight" in your own coordinates.

    Sitting here on earth, our geodesic wants to fall down. We are the ones who have path curvature, and require an acceleration of 'g' upwards to maintain what we think is a "straight" path in our coordinates.

    -Richard

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    And finally, as we saw above, the actual curvature of space-time at the surface of the earth is incredibly small, going as roughly the tidal g over c^2, and not g itself. Most of the apparent curvature of a free fall path, small as it itself is, g/c^2, is still due to our own proper accleration against space-time that what space-time itself is doing.

    But that just doesn't seem right. How can it be that small? Heck, if we go to the other side of the earth, just a few thousand miles away, the direction of g is opposite. Surely that means there's more curvature there than that tiny tidal g value seems to say.

    It's all in 'c', the speed of light. To get curvature, we have to put space and time on equal footing, measuring them with the same units. A small of amount of time is a huge distance by x = ct. So just as hh introduced by converting times to distances, you "stretch" things out greatly.

    Go back to the simple case of measuring the path curvature in the plane. It would be as if we used a very small unit of distance normal to the curve, but a very long distance unit along the curve, and expressed the curvature in units of mm per km per km. That number is factor of a million squared larger. We want mm per mm per mm, or km per km per km.

    And that's what our minds our doing. A second is a very long distance indeed. Eight of them get you to the sun. So, roughly, when we speak of meters per second and meters per second per second, we're comparing meters to AU. We get lots of meters per AU, but not too many meters per meter.

    The human time scale is in AU, but our spatial scale is meters. If our time scale was in microseconds, then things would take *forever* to fall, just floating around, barely moving at all.

    -Richard

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    Quote Originally Posted by publius View Post
    A second is a very long distance indeed. Eight of them get you to the sun.
    I am quite sure you meant that a minute is a very long distance here...

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    Quote Originally Posted by montebianco View Post
    I am quite sure you meant that a minute is a very long distance here...


    Sheesh. Yes, scale everything down by a factor of 60. Our time scale is in AU/100 compared to meters.

    -Richard

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    Quote Originally Posted by publius View Post
    Yes, scale everything down by a factor of 60. Our time scale is in AU/100 compared to meters.
    Time dilation!
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    All good stuff. Of course, I prefer to think of time as the fundamental quantity, not distance, because we actually experience time but we conceptualize distances based on time delays (a three-minute walk, etc.). So we should convert distances into times, and say that g is 3.3 x 10-8 light-seconds per second per second.

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    And if we want to feel even smaller/weaker, think of things in the Planck scale, or God's units, or the Universe's units. One Planck acceleration is one Planck velocity per one Planck time. Which is just c per one Planck time. That is pretty darn steep. One earth g is absolutely nothing compared to that.

    But this g business is just coordinates -- the "real gravity" is in the tides, which is acceleration per length (or time as Ken likes) -- per area, per square length or time is the dimension of the tide, and the dimension of the curvature. And you'll note Lambda, the Cosmological Constant, has those units as well. That's how that works.

    So, ponder a Planck curvature, one Planck acceleration per one Planck distance. A difference of however many gazillion g's it is per a tiny little Placnk length. Or per Planck time -- in Gods's units it's all the same anyway, of course All those factors of G, c, and h are just converting to our puny and/or silly human/earthly scale units.

    That would be a Planck curvature. And so we can imagine that mind-boggling strong tidal gravity is where quantum gravity, whatever it is, should be firmly taking over.

    -Richard

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    Quote Originally Posted by Ken G View Post
    All good stuff. Of course, I prefer to think of time as the fundamental quantity, not distance, because we actually experience time but we conceptualize distances based on time delays (a three-minute walk, etc.).
    I prefer to go the other way. A three-minute walk is a city block (a nice mosey), so I might take five blocks on my coffee break.
    So we should convert distances into times, and say that g is 3.3 x 10-8 light-seconds per second per second.
    Hey! a light-second is not a unit of time, but I know what you mean.

    The seconds cancel, so g is 3.3E-8 c per second, or, just 3.3E-8 per second

    So, I'd prefer g = 5.9E-6 per block
    Quote Originally Posted by publius View Post
    But this g business is just coordinates -- the "real gravity" is in the tides, which is acceleration per length (or time as Ken likes) -- per area, per square length or time is the dimension of the tide, and the dimension of the curvature.
    I'm glad you put the quotes around real, there, so we can distinguish "real" gravity from real gravity

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    Quote Originally Posted by hhEb09'1 View Post
    I prefer to go the other way.
    OK, I was joking.

    I was trying to get to geometrized units. In that system, used by a lot of physicists (Ken G excepted ), G and c are equal to one, period. If you go back to my formula for curvature
    Quote Originally Posted by hhEb09'1 View Post
    2h/d2 = 2(v sinΘ/g)(v sinΘ/2) / (cv sinΘ/g)2 = g/c2
    In geometrized units, since c=1,

    2h/d2 = g

    In other words, the curvature of spacetime due to gravity is g. Can it be any simpler?

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    Quote Originally Posted by hhEb09'1 View Post
    I prefer to go the other way. A three-minute walk is a city block (a nice mosey), so I might take five blocks on my coffee break.
    But... is your coffee break a certain length, or a certain time? That is, if you feel peppy and double your walk rate, which changes? That's my point, it's no coincidence that we structure our lives around time, not distance.
    Hey! a light-second is not a unit of time, but I know what you mean.
    Actually, common misuses of the word notwithstanding, a light second is very much a unit of time, in my way of looking at what distance really is. All distances are constructed in such a way that they should actually be thought of as units of time, that's my point, that's what relativity is trying to tell us. Had we known relativity in Newton's day, he should have built every distance in every one of his formulae as a time-- a light travel time. Then every process is seen as a rate, and every input to a process is seen as a time over which that rate is in effect. For example, a four-velocity is a rate of change of proper time, as seen from arbitrary reference frames. But the reason Newton didn't know about relativity is the same reason that actually doing this would be very clunky in human affairs-- the speed of light is just so darn fast in the human sphere.

    And by the way, I am well aware of geometrized units. All physical parameters must be unitless to be "real", I certainly agree there. What I'm talking about is not the way the universe looks at itself (unitlessly and scale invariant, down to scales we don't understand anyway), but rather how we perceive the universe. Our perceptions are all locked into time, and time is our fundamental unit of perception. Call it a second if you like, some unit that means something to us.

    You see, the universe doesn't give a hoot if we are talking about a gram or a solar mass, or a second or a millennium, if the same physics is at play-- the physics is scale invariant. It is only we who care about that difference-- hence the invention of units, hence the importance of time. Since the conversion of distances into times makes normal distances ultra short times, it is awkward, and we find more value in treating distance like it was a separate thing, hence the invention of distance units. But in relativity, we escape that fundamental imbalance of the scales of importance, and it makes more intuitive sense to convert distances into light travel times than to convert times into light travel distance (as is generally done, sadly). My main point is that observers are local entities, and time unfolds locally-- but distance is a nonlocal construct to make the time behavior make sense. Physics is about understanding change, and time is what allows room for change-- not distance. That's as true on the cosmological scale (the cosmological principle) as it is on the atomic scale (atomic rates), and I personally feel it is high time we informed our insight by attaching 1/c to distance instead of c to time.

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    Quote Originally Posted by Ken G View Post
    Actually, common misuses of the word notwithstanding, a light second is very much a unit of time, in my way of looking at what distance really is.
    George Lucas, somewhere, is grinning
    And by the way, I am well aware of geometrized units. All physical parameters must be unitless to be "real", I certainly agree there. What I'm talking about is not the way the universe looks at itself (unitlessly and scale invariant, down to scales we don't understand anyway), but rather how we perceive the universe. Our perceptions are all locked into time, and time is our fundamental unit of perception.
    As that link sorta makes clear, time (and mass and energy and momentum) are measured in meters, in geometrized units. But that's a quibble, time is distance afterall.

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    Quote Originally Posted by hhEb09'1 View Post
    As that link sorta makes clear, time (and mass and energy and momentum) are measured in meters, in geometrized units.
    I'm certain you are aware there is nothing about "geometric units" that requires that association. In fact, this is precisely the issue I am decrying-- the tendency to treat distance as fundamental and time as emerging via the time of flight across a distance, whereas I am arguing that it is time that is fundamental because all physics is local (being an observational science), so distance emerges fundamentally as the distance that light moves in a given observable time. That we use any other means to measure distance is purely a function of the awkwardness of dealing with the extreme contrast between elapsed times and times of flight in our everyday experience. Were that not the case, we'd do fine with watches and laser-rangers-- rulers would be seen as a very awkward way to measure anything.

    Obviously one can always use either distance or time as a proxy for the other, via the movement of light, but in everyday life we use the two concepts entirely independently (which is why you think a light-second is a distance not a time). Relativity showed us the loss of insight we accept when we do that. But then the question is, which one should most insightfully be used as a proxy for the other? Well, do you care about where you are in the universe, or the age of the universe? Do you care about how much distance the Earth will cover in your lifetime, or how many days you'll live? Do you care about size of a neuron, or how long it takes to fire? And above all: can you measure the distance between two events at the same time, or the time between two events at the same location? See?
    But that's a quibble, time is distance afterall.
    Does an objection with no basis at all count as a "quibble"?

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    Quote Originally Posted by Ken G View Post
    Does an objection with no basis at all count as a "quibble"?
    What? that time is distance? I'm surprised that you are so adamant about one over the other, is all.

    So, yes, that's a quibble.

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    But note that your statements have been "I prefer distance" and "distance is the standard", never even "they are equivalent". I have actually presented an argument, with examples, to establish why time is the more central to physics, so if there is to be any break in the convention that they are equivalent, then it is time that should emerge as the standard. If all you are saying is "I'm fine with them being viewed as completely equivalent", that's fine-- but that's never been what you said. Indeed, even treating them as perfectly equivalent is something I would view as a step in the right direction.

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    Quote Originally Posted by Ken G View Post
    But note that your statements have been "I prefer distance" and "distance is the standard", never even "they are equivalent".
    The only post where I say I prefer one over the other is followed immediately by a post where I admit I was joking, and where I say that I was trying to get to geometrized units where c = 1. If c = 1, the units of distance cancel with the units of time, they are equal.

    Not only that, I have in other threads taken that position many times. That they are equal.
    I have actually presented an argument, with examples, to establish why time is the more central to physics, so if there is to be any break in the convention that they are equivalent, then it is time that should emerge as the standard.
    I am not arguing for a break, you are.
    If all you are saying is "I'm fine with them being viewed as completely equivalent", that's fine-- but that's never been what you said.
    It's all I've ever said, except in that one joking fashion, where I only said my preferred units were city blocks because that is what I experience (to make light of what you said: "because we actually experience time")
    Indeed, even treating them as perfectly equivalent is something I would view as a step in the right direction.
    A lot of us have taken that step a long time ago.

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