BENDING SPACE AND GRAVITY by the homeless science writer, Chongo The last article described the straight lines of "special" relativity. It explained the 'relativity' of space and time, distances and angles, in terms of tilting, two-dimensional surfaces and the two-dimensional creature inhabiting this three-dimensional universe of rising services. Surfaces were "stacked" upon each other, across time, as a series of consecutive moments, creating stacks of moments in - or rather 'across' - time. This simpler universe was used to make the concept of relativity easier to grasp. In our universe, instead of stacks of two-dimensional surfaces tilting with respect to each other, stacks of three-dimensional 'spaces' tilt with respect to each other, in a universe with 'four' dimensions, not three. The important point that the last article should have made upon the reader is that the concept of now - that corresponding to an all-inclusive, universal present-moment 'now' - is, and this is to use the same term that Einstein himself chose to use, a mere "stipulation" that we place upon physical existence: nothing actually physically existent. In other words, there is no such thing existent outside of our mind, regardless of how convinced our minds might be that a universal, present moment corresponds to the entire universe. Put bluntly, spaces tilting into the future in one direction for a surface considering itself not moving, and tilting into the past in the opposite direction with respect to such a surface means there is no such thing nor can there ever be such a thing as a universal present-moment. Instead, there are many, as many as there are unique motions in the universe. We remember that this explanation was discovered by a two-dimensional Einstein, who explained how the speed of light, and correspondingly the laws of physics, the speed of light being an indispensable element of those laws, never changed with motion. This physicist explained the most simple case example, that for 'uniform' motion: motion that does not change speed or direction. For this article, we return now to our two dimensional Einstein at the moment (albeit a stipulation that we impose upon physical existence, not existing beyond the motions of our thoughts) he explained the relativity of space and time, distances and angle for uniform motion, which is called special relativity: the relativity of space and time, distances and angles for uniform motion. Special relativity's tilting of surfaces (as well as that of spaces and lines) would 'seem' to explain a great deal about two-dimensional reality, though it would not immediately explain one rather significant aspect of it, namely gravity; though it would, quite naturally, most unavoidably, and rather obviously, lead straightto its explanation; and thus distinguish 'special' relativity as a unique and singularly 'special' case of a more 'generalized' form of relativity's geometry, namely "general relativity", which, unlike special relativity's straight-line geometry, is a geometry with curves. Surprisingly, explaining how the speed of light never changes with uniform motion in order to keep the laws of nature (of physics) from ever changing (i.e. special relativity), explains also gravity! In other words, even though we might hardly imagine how, and perhaps even further hardly imagine that it even 'should' explain it, the speed of light being constant and physical laws correspondingly never changing (a corollary) ultimately leads to a very, very, very accurate explanation, in the form of a description, for gravity. And, this explanation (this description) is absolutely the best testable explanation that has ever existed. In actual fact, every last test that science has ever made confirms relativity's validity for explaining space, time, 'big' motion, and, most significantly, gravity, making relativity's truth a cornerstone and fundamental foundation of physical theory. Just as relativity's explaining gravity by virtue of the speed of light being constant might seem a surprising relationship, so is the ease with which this relationship can be understood surprising also, because applying special relativity's straight-line geometry of unchanging, uniform motion to changing, non-uniform motion, is, surprisingly, perfectly appropriate for doing the same thing for the phenomena of gravity. Gravity's effects are completely indistinguishable from non-uniform motion. And this is how. Giving it any thought, the two-dimensional physicist who discovered the special relativity of space and time (their inseparability), with respect to 'uniform' motion, motion that does not change speed or direction, could not help but notice that not all motion occurred at the same 'uniform' speed and unchanging direction, but that a great deal of the motion in their universe, very often, occurred in a 'non-uniform' way. That is to say, that motion in their universe of surfaces would very often, just as it so very often does also in our universe of spaces, change in a 'non-uniform' way, either speed, direction, or both. Furthermore, because so much motion in their universe would be non-uniform, changing from one speed or direction to another in a (seemingly) continuous way, then he or she could only conclude that where surfaces tilt as a result of uniformmotion, surfaces must also be 'bending' with any motion that was non-uniform. Such a conclusion would be inescapable, since bending is really nothing more than just a progressivelymore and more, or less and less tilted frame (though in actual fact it is much more complex), corresponding to progressively changing speeds or directions (non-uniformly). Realizing that this bending created the 'equivalent' effect as gravity on his or her two-dimensional surface, this physicist who discovered the first half of the geometry of their universe would now have discovered the other half of the geometry of his or her universe. This physicist would have discovered the relativity of space and time measures (distances and directions [angles]), subject to the effects of non-uniform motion and hence, equivalently, to the effects of gravity.Like Einstein, such a physicist would have discovered the 'general' relativity of surface-time, describing how surface-time 'bends', relative to two-dimensional gravity and to non-uniform motion. Finally, there would exist a description of two-dimensional gravity that worked accurately. Just as the great two-dimensional physicists discovering his or her two-dimensional version of 'special' relativity could extend that theory and its straight-line geometry, by using this singularly simple geometry as a 'special' case "curve" for a broader, more comprehensive geometry, that of two-dimensional 'general' relativity, so can we three-dimensional creatures do the same thing; which is exactly what the great three-dimensional physicist, Albert Einstein, did, by extending the description of special relativity's effects upon uniform motion (motion that does not change speed or direction) upon three-dimensional space (as opposed to a two-dimensional surface) and time, to non-uniform motion (again, motion that does change speed, direction, or both), and 'equivalently', to the effects of gravity, since the effects of gravity are absolutely indistinguishable from the effects of inertia. By imagining special relativity's straight lines as curved lines, we can create a conceptual description of space and time together (space-time), and a way to imagine the effects of mass, as the force of either inertia or gravity upon it.In this way, we can logically derive the Theory of General Relativity in terms of a concept (namely, bending), just as Albert Einstein did (in a formal and more precise way using mathematics, instead of using words as we are doing here). The Theory of General Relativity extends Special Relativity's principles for 'uniform' motion, applying them to the non-uniform motion of acceleration (motion that continuously changes speed, direction, or both, corresponding to a 'bent' set of space and time measures [distances and angles]) and "turning" (changing direction, corresponding to a "skewed" set of space and time measures) to those of gravity, in order to describe these effects upon space and time measurements. Gravity, like motion, affects both the pace of time, and the length of spatial distances. Its effects are the same as those of acceleration and/or turning, compressing space, and dilating time, in a way that is non-uniform (and so curved or skewed, instead of straight). General Relativity uses the 'equivalence' of the 'inertial' effects of non-uniform motion, in a way that is non-uniform (and so, "bent" [curved], instead of straight). General relativity describes how space and time bend (bending being a complexity of tilting), as a consequence of the effects of mass's gravity, and likewise, as a consequence of the non-uniform motion of acceleration (the former, gravity, being indistinguishable from the latter, acceleration). Needless to say, every experiment ever conducted by science repeatedly confirms relativity's remarkably accurate description of space, time, 'big' motion, and gravity Because it works so flawlessly, relativity's accuracy is unquestionably true - it is as true as space, time, mass, and energy's gravity are 'true'; as true as the speed of light being constant and the laws of nature (physics) not changing as a consequence of motion or of anything else are true; as true as the conservation of energy and momentum are, as true as the passage of time is, while you read these words and think the thoughts that they yield as you do so. General relativity is a brilliantly stunning example of natural truth's expression and of the marvels that such an expression can accurately reveal about nature's phenomena. (c) 2007, C. Tucker (Chongo). All rights reserved. (From the first ten minutes of RELATIVITY IN HALF AN HOUR, by Chongo in collaboration with Jose. To see all the books that Chongo has written on nature (physical theory), go to www.chongonation.com, which is a web site dedicated to educating those who have least opportunity for learning the scientific foundations that describe nature (accurately). Chongonation provides books that allow such opportunity, in lay terms, without any math. Simply go to the www.chongonation.com home page and click on either 'Nature' or 'Products & Prices' to see just how many books are available.)

BENDING SPACE AND GRAVITYby the homeless science writer, ChongoThe last article described the straight lines of "special" relativity. It explained the 'relativity' of space and time, distances and angles, in terms of tilting, two-dimensional surfaces and the two-dimensional creature inhabiting this three-dimensional universe of rising services. Surfaces were "stacked" upon each other, across time, as a series of consecutive moments, creating stacks of moments in - or rather 'across' - time. This simpler universe was used to make the concept of relativity easier to grasp. In our universe, instead of stacks of two-dimensional surfaces tilting with respect to each other, stacks of three-dimensional 'spaces' tilt with respect to each other, in a universe with 'four' dimensions, not three.

The important point that the last article should have made upon the reader is that the concept of now - that corresponding to an all-inclusive, universal present-moment 'now' - is, and this is to use the same term that Einstein himself chose to use, a mere "stipulation" that we place upon physical existence: nothing actually physically existent. In other words, there is no such thing existent outside of our mind, regardless of how convinced our minds might be that a universal, present moment corresponds to the entire universe. Put bluntly, spaces tilting into the future in one direction for a surface considering itself not moving, and tilting into the past in the opposite direction with respect to such a surface means there is no such thing nor can there ever be such a thing as a universal present-moment. Instead, there are many, as many as there are unique motions in the universe. We remember that this explanation was discovered by a two-dimensional Einstein, who explained how the speed of light, and correspondingly the laws of physics, the speed of light being an indispensable element of those laws, never changed with motion. This physicist explained the most simple case example, that for 'uniform' motion: motion that does not change speed or direction.

For this article, we return now to our two dimensional Einstein at the moment (albeit a stipulation that we impose upon physical existence, not existing beyond the motions of our thoughts) he explained the relativity of space and time, distances and angle for uniform motion, which is called special relativity: the relativity of space and time, distances and angles for uniform motion.

Special relativity's tilting of surfaces (as well as that of spaces and lines) would 'seem' to explain a great deal about two-dimensional reality, though it would not immediately explain one rather significant aspect of it, namely gravity; though it would, quite naturally, most unavoidably, and rather obviously, lead straight to its explanation; and thus distinguish 'special' relativity as a unique and singularly 'special' case of a more 'generalized' form of relativity's geometry, namely "general relativity", which, unlike special relativity's straight-line geometry, is a geometry with curves. Surprisingly, explaining how the speed of light never changes with uniform motion in order to keep the laws of nature (of physics) from ever changing (i.e. special relativity), explains also gravity! In other words, even though we might hardly imagine how, and perhaps even further hardly imagine that it even 'should' explain it, the speed of light being constant and physical laws correspondingly never changing (a corollary) ultimately leads to a very, very, very accurate explanation, in the form of a description, for gravity. And, this explanation (this description) is absolutely the best testable explanation that has ever existed. In actual fact, every last test that science has ever made confirms relativity's validity for explaining space, time, 'big' motion, and, most significantly, gravity, making relativity's truth a cornerstone and fundamental foundation of physical theory.

Just as relativity's explaining gravity by virtue of the speed of light being constant might seem a surprising relationship, so is the ease with which this relationship can be understood surprising also, because applying special relativity's straight-line geometry of unchanging, uniform motion to changing, non-uniform motion, is, surprisingly, perfectly appropriate for doing the same thing for the phenomena of gravity. Gravity's effects are completely indistinguishable from non-uniform motion. And this is how.

Giving it any thought, the two-dimensional physicist who discovered the special relativity of space and time (their inseparability), with respect to 'uniform' motion, motion that does not change speed or direction, could not help but notice that not all motion occurred at the same 'uniform' speed and unchanging direction, but that a great deal of the motion in their universe, very often, occurred in a 'non-uniform' way. That is to say, that motion in their universe of surfaces would very often, just as it so very often does also in our universe of spaces, change in a 'non-uniform' way, either speed, direction, or both. Furthermore, because so much motion in their universe would be non-uniform, changing from one speed or direction to another in a (seemingly) continuous way, then he or she could only conclude that where surfaces tilt as a result of uniform motion, surfaces must also be 'bending' with any motion that was non-uniform. Such a conclusion would be inescapable, since bending is really nothing more than just a progressively more and more, or less and less tilted frame (though in actual fact it is much more complex), corresponding to progressively changing speeds or directions (non-uniformly). Realizing that this bending created the 'equivalent' effect as gravity on his or her two-dimensional surface, this physicist who discovered the first half of the geometry of their universe would now have discovered the other half of the geometry of his or her universe. This physicist would have discovered the relativity of space and time measures (distances and directions [angles]), subject to the effects of non-uniform motion and hence, equivalently, to the effects of gravity. Like Einstein, such a physicist would have discovered the 'general' relativity of surface-time, describing how surface-time 'bends', relative to two-dimensional gravity and to non-uniform motion. Finally, there would exist a description of two-dimensional gravity that worked accurately.

Just as the great two-dimensional physicists discovering his or her two-dimensional version of 'special' relativity could extend that theory and its straight-line geometry, by using this singularly simple geometry as a 'special' case "curve" for a broader, more comprehensive geometry, that of two-dimensional 'general' relativity, so can we three-dimensional creatures do the same thing; which is exactly what the great three-dimensional physicist, Albert Einstein, did, by extending the description of special relativity's effects upon uniform motion (motion that does not change speed or direction) upon three-dimensional space (as opposed to a two-dimensional surface) and time, to non-uniform motion (again, motion that does change speed, direction, or both), and 'equivalently', to the effects of gravity, since the effects of gravity are absolutely indistinguishable from the effects of inertia. By imagining special relativity's straight lines as curved lines, we can create a conceptual description of space and time together (space-time), and a way to imagine the effects of mass, as the force of either inertia or gravity upon it. In this way, we can logically derive the Theory of General Relativity in terms of a concept (namely, bending), just as Albert Einstein did (in a formal and more precise way using mathematics, instead of using words as we are doing here).

The Theory of General Relativity extends Special Relativity's principles for 'uniform' motion, applying them to the non-uniform motion of acceleration (motion that continuously changes speed, direction, or both, corresponding to a 'bent' set of space and time measures [distances and angles]) and "turning" (changing direction, corresponding to a "skewed" set of space and time measures) to those of gravity, in order to describe these effects upon space and time measurements. Gravity, like motion, affects both the pace of time, and the length of spatial distances. Its effects are the same as those of acceleration and/or turning, compressing space, and dilating time, in a way that is non-uniform (and so curved or skewed, instead of straight). General Relativity uses the 'equivalence' of the 'inertial' effects of non-uniform motion, in a way that is non-uniform (and so, "bent" [curved], instead of straight).

General relativity describes how space and time bend (bending being a complexity of tilting), as a consequence of the effects of mass's gravity, and likewise, as a consequence of the non-uniform motion of acceleration (the former, gravity, being indistinguishable from the latter, acceleration). Needless to say, every experiment ever conducted by science repeatedly confirms relativity's remarkably accurate description of space, time, 'big' motion, and gravity

Because it works so flawlessly, relativity's accuracy is unquestionably true - it is as true as space, time, mass, and energy's gravity are 'true'; as true as the speed of light being constant and the laws of nature (physics) not changing as a consequence of motion or of anything else are true; as true as the conservation of energy and momentum are, as true as the passage of time is, while you read these words and think the thoughts that they yield as you do so. General relativity is a brilliantly stunning example of natural truth's expression and of the marvels that such an expression can accurately reveal about nature's phenomena.

(c) 2007, C. Tucker (Chongo). All rights reserved.

(From the first ten minutes of RELATIVITY IN HALF AN HOUR, by Chongo in collaboration with Jose. To see all the books that Chongo has written on nature (physical theory), go to www.chongonation.com, which is a web site dedicated to educating those who have least opportunity for learning the scientific foundations that describe nature (accurately). Chongonation provides books that allow such opportunity, in lay terms, without any math. Simply go to the www.chongonation.com home page and click on either 'Nature' or 'Products & Prices' to see just how many books are available.)