Why does special relativity exclude gravity?

[luinthoron]

Hello,

I was unable to find a similar thread, so I would like to ask about this myself. I have several textbooks on SR and GR at my disposal but none of them gave me the answer to my question. I remeber from undergraduate course that SR brakes down if we want to include gravitation. I feel that I need to properly understand why before I continue my study of relativity. I think it might somehow go against the relativity principle but I'd really like to see some conrete reasoning behind it. Could you shed some light on it or refer me to a book that treats this topic thoroughly?

Thank you for your (space)time.

 

[WannabeNewton]

www.physicsforums.com/showthread.php?t=742809&page=2

 

[Dale]

In special relativity two objects which are moving inertially (meaning on-board accelerometers read 0) and are initially at rest with respect to one another will always maintain a constant separation. In the presence of gravity two objects which are moving inertially and are initially at rest with respect to one another may approach and even collide. The structure of SR simply cannot predict that.

 

[luinthoron]

www.physicsforums.com/showthread.php?t=742809&page=2

Although I find the discussion in the posted thread interesting, I fail to see the immediate connection with my question. I don't find answers including tensors very helpful at this point. I understand the mathematics and basic principles but the larger physical pictures eludes me.

 

[WannabeNewton]

Although I find the discussion in the posted thread interesting, I fail to see the immediate connection with my question. I don't find answers including tensors very helpful at this point. I understand the mathematics and basic principles but the larger physical pictures eludes me.

Well the issue is mostly mathematical-there is nothing deep going on here so if you understand the mathematics then you can just write down the Lagrangian density for a classical field theory of gravity on a flat background (e.g. a scalar or vector theory) and attempt to match the theory to known predictions of gravity. The point is you can have gravitational field theories defined on a flat background, nothing in SR prevents that. They just fail to predict all of the observed properties of the gravitational field. This is all discussed extensively in chapter 3 of "Gravitation: Foundations and Frontiers"-Padmanabhan.

 

[Geometry_dude]

In principle you could take Minkowski spacetime $(\mathbb{R}^4, \eta)$ along with the canonical global chart and just introduce a (covariant) force via $F= \dot p$. Then you could find the flow(=general solution) on the tangent bundle (=velocity phase space) and be happy with your solution. This is how you treat forces in special relativity.

However, what Einstein realized is that treating gravity like this is not physically meaningful - you would feel the acceleration. Imagine you were somewhere in outer space and gravitationally attracted to some massive body. Even if it is very massive, you should not feel the acceleration (neglecting tidal "forces"): You're just falling! However, if an actual force acts on you, you definitely feel its acceleration. For this reason, we needed something else.

 

[pervect]

Although I find the discussion in the posted thread interesting, I fail to see the immediate connection with my question. I don't find answers including tensors very helpful at this point. I understand the mathematics and basic principles but the larger physical pictures eludes me.

In special relativity, one assumes that the Lorentz interval can be put into the form:

ds^2 = -dt^2 + dx^2 + dy^2 + dz^2

This implies that space-time is flat. In GR, the expression for the Lorentz interval is different, and it is not assumed that space-time is flat.

The popular notions of "flat" and "curved" aren't very precise, but the precise defintions (such as flatness being the vanishing of the Riemann tensor) are rather technical. It's probalby sufficient to think of a plane as being flat, and the surface of a sphere as being non-flat, for this level of explanation. To be slightly less technical than the Riemann definition (but more precise than saying curved), we can say that we are talking about "intrinsic" curvature, something that can be defined given a local neighborhood of points, and a way of measuring the distances between them, without reference to any particular embedding. See the Wiki on intrinsic curvature, for more.

 

[haruna]

In unison with Mach's principle any coordinate change of flat spacetime resulting in space-time interval other than:

ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 must produce tidal forces ( see Newton and rotating bucket ).

Tidal forces are an immediate outcome of curved space-times, hence GR.

 

[WannabeNewton]

In unison with Mach's principle any coordinate change of flat spacetime resulting in space-time interval other than:

ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 must produce tidal forces ( see Newton and rotating bucket ).

This is entirely incorrect. First of all Mach's principle is not an established principle of physics. Secondly, GR violates Mach's principle in more ways than one. Lastly, and most importantly, coordinate transformations do not introduce tidal forces. I can transform the Minkowski metric into a coordinate system rigidly rotating with uniform angular velocity about a preferred axis and there will still be no tidal forces. Newton's rotating bucket was a demonstration of the fact that a circulating fluid will feel centrifugal forces that pull it up at the edges hence rotation must be absolute-this has nothing to do with tidal forces. Coordinate transformations are gauge transformations of the gravitational field which tidal forces (i.e. space-time curvature) are gauge invariant of.

Again, you can write down the Lagrangian for a field theory of gravity on a flat static background. Nothing in SR prevents you from doing this. A weak gravitational wave in vacuum is in fact a symmetric tensor field propagating on a flat static background with dynamics governed by a simple wave-equation. The problem with gravitational field theories on a flat static background is they don't fit all the observed effects of gravity; one can find the motivation for the description of gravity as a dynamical curved background in chapter 2 of "General Relativity"-Straumann.

 

[Wes Tausend]

Hello,

I was unable to find a similar thread, so I would like to ask about this myself. I have several textbooks on SR and GR at my disposal but none of them gave me the answer to my question. I remeber from undergraduate course that SR brakes down if we want to include gravitation. I feel that I need to properly understand why before I continue my study of relativity. I think it might somehow go against the relativity principle but I'd really like to see some conrete reasoning behind it. Could you shed some light on it or refer me to a book that treats this topic thoroughly?

Thank you for your (space)time.

I think DaleSpam gave you the most concise and simple answer in post #3, and it is a classic.

Fundamental principles of all theories, SR and GR included, try to boil down the essence of comparing physical properties, and their initial definition must be considered very carefully. GR does not, "somehow go against the relativity principle" of SR, if that is what you are thinking. They are combined to work together.

The other post answers are good treatises on the associated math to form a precise blueprint, but detailed math is not imperative to recognise some important observations and deductions. Some characteristics can be roughly "drawn out on a napkin" (decribed) to get the basic idea. So, in simple terms, what are the differences, besides what DaleSpam said?

First of all, SR deals in velocities in a straight line, or linear direction. Starting with a straight line, what happens as speed increases? Einstein pondered this and came up with a simple, but non-intuitive, logical theory based on other ideas and measurements. This thinking helped simplify an otherwise complicated decription and get that down pat before anything more than straight lines were tackled. I think SR is harder to describe and accept, than the changes that came later by merely curving the proven straight SR relativity lines.

GR tries to explain what happens when bodies, or light, travel in a curve, relative to one another, such as the all important orbits of bodies, or gravitational bending of light. It just so happens Einstein noticed that gravity is an "attraction" acceleration, just like the momentum acceleration of highway curves in an automobile. SR, designed for simple straight lines, cannot adequately describe curves, attractions, or accelerations, if at all. Einstein basically deduced that if orbits follow a curved road, so should light. By experimentation humans found light does follow curved roads. Voila! Einstein's GR theory worked; mass causes space-roads to curve. The converted SR-math of straight lines could now be successfully applied to gravity-curves of space at various velocities.

I had the best luck reading science books by Isaac Asimov to gain basic understanding. Good luck.

Wes
...

 

[WannabeNewton]

SR, designed for simple straight lines, cannot adequately describe curves, attractions, or accelerations, if at all.

SR can more than adequately describe accelerations; it most certainly is not restricted to uniform motion.

 

[Nugatory]

SR, designed for simple straight lines, cannot adequately describe curves, attractions, or accelerations

SR handles these just fine. Many people don't realize this because introductory SR texts usually don't cover these problems; the math is appreciably more challenging without contributing any additional understanding of the basic principles so the texts don't go there.

You might want to google for "Rindler coordinates" to find a complete SR treatment of linear acceleration, and "Ehrenfest paradox" will get you started on circular motion and radial acceleration.

The essential difference between SR and GR is that SR limits itself to situations in which spacetime is flat (described by the Minkowski metric) and GR does not. Either way, we can handle any path through spacetime, whether curved or straight, accelerated or not.

 

[bhobba]

A weak gravitational wave in vacuum is in fact a symmetric tensor field propagating on a flat static background with dynamics governed by a simple wave-equation.

I know Wannabe knows this, but not only is it inconsistent but actually tells you how to fix it up. You will find this approach in Ohanian:
https://www.amazon.com/dp/1107012945/?tag=pfamazon01-20

Thanks
Bill

 

[HomogenousCow]

Keep in mind, that QFT is done in an SR compatible framework.
In that light, it kind of makes no sense that accelerations cannot be handled in SR.
The reason fully dynamical motion is not often taught in those algebra-only SR books is because they are best treated in a lagrangian formulation.
The equations of motion in SR (for a particle in a scalar field) are more complicated than those in classical mechanics and contain (four) velocity dependant terms.

 

[Wes Tausend]

SR handles these just fine. Many people don't realize this because introductory SR texts usually don't cover these problems; the math is appreciably more challenging without contributing any additional understanding of the basic principles so the texts don't go there.

You might want to google for "Rindler coordinates" to find a complete SR treatment of linear acceleration, and "Ehrenfest paradox" will get you started on circular motion and radial acceleration.

The essential difference between SR and GR is that SR limits itself to situations in which spacetime is flat (described by the Minkowski metric) and GR does not. Either way, we can handle any path through spacetime, whether curved or straight, accelerated or not.

Hi Nugatory,

I've taken my time to think it over.

Maybe we can make lemonade out of a lemon here. The OP asks a question and some basic answers seem unnecessarily complicated and one does not. Another naive student (myself) has raised his hand and volunteered a slightly expanded basic answer that is not quite right. Confusion reigns amongst students with widely differing backgrounds. The class can still benefit.

A green student might naively ask, if SR is so comprehensive, why do we even need GR?

Both Rindler (born 1924) coordinates and Ehrenfest paradox(1909) came along after SR but the Ehrenfest paradox before GR. There is little doubt that Einstein was aware of Rindler-like Minkowski space since Minkowski was his former instructor, and that influenced SR. But, keeping in mind the fallibilty of my sources, it seems the following is true.

'Einstein himself at first viewed Minkowski's treatment as a mere mathematical trick, before eventually realizing that a geometrical view of space-time would be necessary in order to complete his own later work in general relativity (1915).'

This indicates to me that Einstein had a very simple mental picture of SR. This Occams Razor view is what a naive student may best start with. What was it?

The beauty of DaleSpams simple, classic answer to the OP in #3 was in that it was axiomatic. It is easy for all to see as the difference of SR vs GR is self-evident as long as, in part, "In the presence of gravity two objects which are moving inertially and are initially at rest with respect to one another may approach and even collide." In keeping it simple, DaleSpam doesn't even expand enough to define gravity as an attraction, so he still allows that any collision caused by acceleration could be gravity, something dear to my heart and the Equivalence Principle.

As Asimov has taught us in The Relativity of Wrong and other wisdoms, it is the weakness of axioms that fool us and the very weakness of axioms is the weakness of human intuition. Intuition never fails us entirely, but may be later supplanted by better intuition. It is the root of SR vs GR that the OP seeks, that foundation we all here continue to seek. What is the simple essence of our world in the fewest words?

Linear is not quite the same as flat and I understand SR does allow for at least acceleration in a straight line, or perhaps flat spacetime acceleration. How can we reword the error(s) in my two short paragraphs up above to remain simple, yet be truthful? In other words, in the spirit of learning, yet keeping it simple, can you explain in pure logical concise prose, why we need GR if SR is so adequate?

Thanks,
Wes
...

 

[Nugatory]

Linear is not quite the same as flat and I understand SR does allow for at least acceleration in a straight line, or perhaps flat spacetime acceleration. How can we reword the error(s) in my two short paragraphs up above to remain simple, yet be truthful? In other words, in the spirit of learning, yet keeping it simple, can you explain in pure logical concise prose, why we need GR if SR is so adequate?
...

SR covers all the situations in which space-time is flat, but only those situations. Thus, we need GR to cover situations in which space-time is not flat. That includes all situations in which gravitational influences are significantly different from one point to another.

There is a formal mathematical definition of "flat", but you asked for an intuitive one . Here it is (but be aware that from a mathematical point of view it is somewhat bogus): Pick three points in space-time. Draw straight lines ("geodesics", in the lingo) between them to form a triangle. Do the three interior angles of this triangle add up to 180 degrees? If so, you're in a flat space-time (or at least a good approximation of flat).

 

[WannabeNewton]

In other words, in the spirit of learning, yet keeping it simple, can you explain in pure logical concise prose, why we need GR if SR is so adequate?

Unfortunately your post is filled with a lot of unnecessary rhetoric so I'm just going to reply to the above. The simple answer is SR can deal with any and all forms of acceleration be it linear or rotation or combinations thereof. The difference between SR and GR is simply that SR is a theory of a flat static background whereas GR is a theory of a dynamical curved background codifying the notion that nature doesn't provide us with a fixed space-time geometry a priori but rather all the energy-momentum sources in a given system cause the space-time geometry to propagate and in return makes space-time geometry dependent the notions of rotation and acceleration.

GR is obviously much more than a theory of gravity. It is what Wheeler called a theory of "geometrodynamics", one that just also happens to accurately predict observational gravitational effects. To put it conversely, there is nothing in the underlying theoretical framework of SR that prevents one from establishing a relativistic theory of gravity on its static flat background. What weeds them out is experiment. However historically speaking Einstein wasn't led by experiment to GR but rather by purely theoretical considerations partly on the basis of aesthetics (and who can blame him? He gave us what many physicists consider to be the most beautiful theory of physics). In other words we don't need GR, it is just the theory that manages to consistently and accurate agree with experiment through an extremely elegant framework so we adopt it as the proper classical theory of gravity.

 

[Nugatory]

Both Rindler (born 1924) coordinates and Ehrenfest paradox(1909) came along after SR but the Ehrenfest paradox before GR. There is little doubt that Einstein was aware of Rindler-like Minkowski space since Minkowski was his former instructor, and that influenced SR.

There is little doubt that Einstein was NOT aware of the Minkowski geometric formulation of SR when he developed and published SR in 1905. It's not in the 1905 paper, and it is all over the GR publications a decade later - because it was discovered in between.

Just about all of the mathematical apparatus of modern special relativity was developed after the fact, after other physicists recognized the value of Einstein's insight and started building on his original formulation - this is why you'll occasionally hear Einstein's formulation of SR described as "old-fashioned" or even "obsolete". Much of the modern treatment of SR didn't appear until after the discovery of GR, when people went back to reformulate SR in a way that made it clearer that it was the zero-curvature special case of a more general theory that worked for all values of curvatures including zero.

 

[Wes Tausend]

SR covers all the situations in which space-time is flat, but only those situations. Thus, we need GR to cover situations in which space-time is not flat. That includes all situations in which gravitational influences are significantly different from one point to another.

Can light curve in SR? I would think not, but flat mathematical graphs may/must display a curve. According to WannabeNewton, I must be wrong about light curving within SR, if SR can encompass gravity.

There is a formal mathematical definition of "flat", but you asked for an intuitive one . Here it is (but be aware that from a mathematical point of view it is somewhat bogus): Pick three points in space-time. Draw straight lines ("geodesics", in the lingo) between them to form a triangle. Do the three interior angles of this triangle add up to 180 degrees? If so, you're in a flat space-time (or at least a good approximation of flat).

Ok, that makes sense, the triangular enclosed angles on a globe (sphere), nor any other curved surface, do not add up to 180. So... the curve of which we speak is merely the surface of a graph to calculate coordinates, not necessarily the path light must take (and is observed to), which is/forms the real dimension(s) of space?

There is little doubt that Einstein was NOT aware of the Minkowski geometric formulation of SR when he developed and published SR in 1905. It's not in the 1905 paper, and it is all over the GR publications a decade later - because it was discovered in between.

I first worded it NOT, then decided that it is quite possible some discussion took place in class that might say, for instance, "...and we could add more dimensions if we wish". Even though Einstein made no use of it in SR, he might have heard it before and tucked it in a dusty bin. Pretty speculative I know, but after you so graciously introduced it, I felt I could not positively claim he couldn't have previously heard it in an earlier Minkowski timeline. I used to love little trivia tidbits offered by instructors. It offered additional usefulness for bothering to learn the subject material.

Just about all of the mathematical apparatus of modern special relativity was developed after the fact, after other physicists recognized the value of Einstein's insight and started building on his original formulation - this is why you'll occasionally hear Einstein's formulation of SR described as "old-fashioned" or even "obsolete". Much of the modern treatment of SR didn't appear until after the discovery of GR, when people went back to reformulate SR in a way that made it clearer that it was the zero-curvature special case of a more general theory that worked for all values of curvatures including zero.

Very good insight! Thanks. I don't always trust pure math (basically a shorthand) as it can, and does let us down at times. Einsteins descriptive thought experiments are more invaluable than numbers to understanding in this way.

It is clear you gave some time and thought to your reply(s). I hope the OP and other readers have gotten as much out of this discussion as I have.

Nugatory, thank you so much for your time and the reply.

Wes
...

 

[Wes Tausend]

Unfortunately your post is filled with a lot of unnecessary rhetoric so I'm just going to reply to the above. The simple answer is SR can deal with any and all forms of acceleration be it linear or rotation or combinations thereof. The difference between SR and GR is simply that SR is a theory of a flat static background whereas GR is a theory of a dynamical curved background codifying the notion that nature doesn't provide us with a fixed space-time geometry a priori but rather all the energy-momentum sources in a given system cause the space-time geometry to propagate and in return makes space-time geometry dependent the notions of rotation and acceleration.

You have reinforced Nugatory's post with a few more and differently said words and added a cause and effect geometry loop between matter (mass) and energy. Thank you for your effort.

In respect to the rhetoric, my meaning is to emphasize the importance of employing basic pictoral intuition in understanding truth. Numbers and math are useless to reaching understanding without it. To a neophyte, the common suggestion to manipulate higher math does not paint much of a picture, and can deceive men altogether. I much appreciate your worded descriptions.

GR is obviously much more than a theory of gravity. It is what Wheeler called a theory of "geometrodynamics", one that just also happens to accurately predict observational gravitational effects. To put it conversely, there is nothing in the underlying theoretical framework of SR that prevents one from establishing a relativistic theory of gravity on its static flat background. What weeds them out is experiment. However historically speaking Einstein wasn't led by experiment to GR but rather by purely theoretical considerations partly on the basis of aesthetics (and who can blame him? He gave us what many physicists consider to be the most beautiful theory of physics). In other words we don't need GR, it is just the theory that manages to consistently and accurate agree with experiment through an extremely elegant framework so we adopt it as the proper classical theory of gravity.

Asimov once showed how only high school algebra was required to derive E=mc², and that is a beautiful equation. Asimov attempted no such thing with GR that I know of, and the sorrow is that GR is only beautiful to those learned few that can read it.

This eludes me as to how:
"...there is nothing in the underlying theoretical framework of SR that prevents one from establishing a relativistic theory of gravity on its static flat background.

Unrelated, I once thought I could see a form of gravity in SR. I believe it's a fact that whenever one sees V² in a formula, it designates an acceleration. Since E= mc², and c is not just a constant, but a velocity, one could attempt to read the formula as "energy is related to mass by an acceleration". I know this is somehow not proper, but it was appealing at the time.

Thank you for your time and a very thoughtful response.

Wes
...

 

[WannabeNewton]

In respect to the rhetoric, my meaning is to emphasize the importance of employing basic pictoral intuition in understanding truth. Numbers and math are useless to reaching understanding without it. To a neophyte, the common suggestion to manipulate higher math does not paint much of a picture, and can deceive men altogether. I much appreciate your worded descriptions.

Yes this is all very true, perhaps with the exception of the deception of men. Actually the physical intuition is almost always harder to master than the math by several orders of magnitude, especially for a theory like GR. As a quick aside, I would recommend you buy the following book: https://www.amazon.com/dp/0226288641/?tag=pfamazon01-20

The book has barely any math at all but don't let that deceive you because the book is filled with extremely important insights into SR/GR that standard GR books never (properly) delve into. It took me multiple reads to properly understand the various physical concepts that Geroch expounds upon. Geroch is one of the masters of GR after all :)

 

[Nugatory]

Can light curve in SR? I would think not, but flat mathematical graphs may/must display a curve. According to WannabeNewton, I must be wrong about light curving within SR, if SR can encompass gravity.
...
So... the curve of which we speak is merely the surface of a graph to calculate coordinates, not necessarily the path light must take (and is observed to), which is/forms the real dimension(s) of space?

No, the curve is not just the surface of a graph that we're playing coordinate games with

Another of the ways that we know when we're dealing with a curved space is that in a curved space parallel lines can intersect (for example, draw parallel lines due north from the Earth's equator, they'll meet at the north pole).

Light always travels in a straight line (a "light-like geodesic"). In the flat space covered by SR, parallel light beams will never intersect. But in curved space-time they can; indeed, this is the basis for gravitational lensing, one of the standard observational confirmations of GR.

A string held at each end and stretched taut is a straight line (a "space-like geodesic"). The intersecting parallel lines from the equator to the north pole are examples of such geodesics.

The world-line of a body moving through space-time without experiencing proper acceleration (which is to say that the body is in free-fall) is a straight line (a "time-like geodesic").

In all three of these cases, the straight lines that we're talking about are defined by real observable physical phenomena (the taut string, the path of the light beam, the path of the free-fall body) and I need no graphs or coordinates to describe them. I identify the presence or absence of curvature by looking at the geometrical relationships between these physically meaningful straight lines - again no graphing or coordinates required. The lines intersect or they don't; the interior angles of the triangles add to 180 or they don't.

(At this point you may be asking why I need three different flavors of straight line (time-like, space-like, light-like). The mathematical answer is in the metric ($\Delta{s}^2$ can be positive, zero, or negative) but the intuitive answer is that I can stretch a string between two points that are spatially separated but not temporally separated: I can stretch a string between my nose and my toe, but not between noon and midnight.)

 

[A.T.]

Can light curve in SR?

Yes, in accelerated frames of reference light rays can curve in empty space, or flat space-time.

I must be wrong about light curving within SR, if SR can encompass gravity.

SR can encompass the gravity you have in linearly accelerated frames, in empty space. It cannot deal with tidal gravity around masses.

 

[WannabeNewton]

Can light curve in SR? I would think not, but flat mathematical graphs may/must display a curve. According to WannabeNewton, I must be wrong about light curving within SR, if SR can encompass gravity.

Any theory that is conformally flat will fail to predict bending of light due to gravity. However what I said was SR, on the level of framework, has no trouble allowing gravitational theories but that no such theory manages to conform to experiment, including the conformally flat theories that fail to predict light bending.

 

[Nugatory]

Yes, in accelerated frames of reference light rays can curve in empty space, or flat space-time.

However, that really is a coordinate artifact - the light is still following a geodesic. You can plot the coordinates of the light ray using the coordinates of different frames and get differently shaped lines on a piece of paper, and you can stretch and twist the lines by applying various coordinate transformations, but you cannot change the basic geometry that way. For example, the "curved" lines that you refer to cannot be made to intersect in one frame but not another.

SR can encompass the gravity you have in linearly accelerated frames, in empty space.

Calling this "gravity" is, I think, reading the equivalence principle backwards; the EP says that all gravitational effects are equivalent to some local acceleration, but not that all accelerations are gravitational. In particular, there is no gravitational field that can reproduce exactly the effects of uniform linear acceleration.

It cannot deal with tidal gravity around masses.

Yes. That's the reason why back in #16 I used the phrase "all situations in which gravitational influences are significantly different from one point to another". If the differences from point to point (which make tidal effects) are small enough, you can ignore them and use the SR solution for uniform linear acceleration - but this is always an approximation.

 

[Wes Tausend]

In respect to the rhetoric, my meaning is to emphasize the importance of employing basic pictoral intuition in understanding truth. Numbers and math are useless to reaching understanding without it. To a neophyte, the common suggestion to manipulate higher math does not paint much of a picture, and can deceive men altogether. I much appreciate your worded descriptions.

Yes this is all very true, perhaps with the exception of the deception of men. Actually the physical intuition is almost always harder to master than the math by several orders of magnitude, especially for a theory like GR. As a quick aside, I would recommend you buy the following book: https://www.amazon.com/dp/0226288641/?tag=pfamazon01-20

The book has barely any math at all but don't let that deceive you because the book is filled with extremely important insights into SR/GR that standard GR books never (properly) delve into. It took me multiple reads to properly understand the various physical concepts that Geroch expounds upon. Geroch is one of the masters of GR after all :)

Thank you very much for the book link. I see it is not in my local library, but it is reasonably priced and I will buy it. My primary interest was originally in gravity and I found the basics quite simple. The first book I ever read on any Relativity was https://www.amazon.com/dp/0486273784/?tag=pfamazon01-20, and I was intently interested in the Equivalence principle, which may be further enlightened by your book. I did start backwards with gravity first.

My narrow focus on the Equivalence principle is because of a pictoral "thought experiment" Einstein did concerning a "chest" that is drawn up in such a way that two scientists inside cannot immediately distinguish whether they reside within a gravitational field or not. It is amazing to me that Mother Nature could provide two nearly identical, yet separate inertial physics properties, and I immediately set out to solve this marvelous riddle with further simple thought experiments. It can't be solved (united) in a thought experiment without establishing light as "at rest" and that immediately violates a root assumption (axiom) of SR, and can't be openly discussed in depth here. So I am alone with my books again. It's been a burden for 30 years off and on, and it's on my bucket list to somehow pin it down before the inevitable last book.

Regarding, "the exception of the deception of men", the best example to the contrary I can think of is the story of Ptolemy. He mapped our solar system with predictive math so well, that I understand NASA still uses a version today for space shots, and we our clocks and calenders. Any new Ptolemic observations only needed the addition of new epicycles. Yet, as perfectly valid as the math was, he missed half of all early relativity by not being able to envisualize "the big picture" as well as his successor, Copernicus, a fellow mathematician.

Had not math fellow Copernicus "seen further", Galileo, and his wonderful telescope, may not have recognized the moons of Jupiter as more than just another epicycle. Who knows how long this "deception of men" could have gone on, if not for the astute visual imagination of Copernicus? And the mistake was all based on the erroneous assumption, the fragile intuition, the axiom if you will, that Earth rested while the sun did not. Beautiful mathematics is certainly not enough.

WannabeNewton, thanks again for your time and insight.

Wes
...

 

[pervect]

Ok, that makes sense, the triangular enclosed angles on a globe (sphere), nor any other curved surface, do not add up to 180. So... the curve of which we speak is merely the surface of a graph to calculate coordinates, not necessarily the path light must take (and is observed to), which is/forms the real dimension(s) of space?

The popular word curvature as popularly understood is very vague But it's NOT just a matter of coordinates as to whether a surface is intrinsically curved. The sum-of-angles test , for instance, doesn't depend on how you assign coordinates.

Distances and angles (in ordinary geometry) and the Lorentz interval and relative velocity (in space-time geometry) are geometric properties that are independent of any assignment of coordinates.

It's a bit unfortunate that "distance" in relativity does depend on the observer, that's why a geometrical discussion of special relativity will replace the observer-dependent distance with an observer-indpendent Lorentz interval.

Curvature is being used in multiple ways by you - the way in which we talk about the surfaces being curved, using the sum-of-angles test, is different than the way in which we talk about a line being curved, as there isn't any sum-of-angles test for a line.

The mathematics by which we can most directly determine if a line is curved depends on a concept that you may not be familiar with, called "parallel transport". Straight lines proceed in some direction starting from a point, and parallel transport that starting direction so as to keep moving in "the same direction". (Parallel transport can be used to good effect to define the intrinsic curvature of a surface as well - flat surfaces don't change a direction (mathematically a vector) when it's parallel transported around a closed loop.)

Issues like this make non-mathematical discussion difficult, as many of the needed distinctions are not easily expressed in every-day language. When one actually want to make physical predictions, one needs an understanding that reaches through to the underlying mathematics . Unfortuanttely if all one his is pictures and popularizations based on non-mathematical language, one may (and usually does not) not have enough understanding of a theory to actually get good predictions out of it - not even qualitative ones :(.

 

[Wes Tausend]

...Issues like this make non-mathematical discussion difficult, as many of the needed distinctions are not easily expressed in every-day language. When one actually want to make physical predictions, one needs an understanding that reaches through to the underlying mathematics . Unfortuanttely if all one his is pictures and popularizations based on non-mathematical language, one may (and usually does not) not have enough understanding of a theory to actually get good predictions out of it - not even qualitative ones :(.

pervect,

I understand what you are saying about the importance of math in not only describing curves but all geometries, and I agree, in spite of being a bit "math challenged".

The short answer is math, which must be always wholly based in geometry, has become so abstract that we cannot easily picture the geometry with our intuition. Checks and balances no longer exist. Do we calculate additional mathematically correct epicycles to fit our hopes, our present intuitions, or do we not? What is the essence of our ability to derive essence?

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The long answer requires a few more words, just as do complicated equations. Mathematics offers us a handy shorthand of logic prose, but nothing is gained without compromise. Perhaps because of items like the evolvement of helio-centricity (that I gave above in #26), I implied the Master himself (Einstein) displayed a distrust (in post #15) of mathematical tricks, I assume basically because of the same reasons I, and perhaps the OP, find. Mother Nature cannot be fooled: Meaning humans can.

The danger with simple envisionment is we only get a "napkin drawing", but with math we get a "precision 3D (or more D's) blueprint" that may either reveal the impossible humorous lines to a rough-drawn proposed object, or confirm proof that all surfaces meet well. But even with perfect, beautiful math proof, we can still apparently get an inside-out object that blueprints perfectly, yet does not meet the criteria of sensible geometry.

In relation to this thread, and the OP whom we must not forget, the OP came back and stated in #4 that he was lost by the math descriptions and, I assume in frustration, wished to appeal to a geometric version. DaleSpam gave him one since his classic example can yet be pictured in the human mind. The difficulty to continue a sole picture mentality lay after, and still frustrates him when the OP starts a new thread on a more specific, but similar mission to understand without math. The OP's struggle to understand, with or without math, is central to our own, all of us, not just he and I.

Regarding your comment, "...Unfortuanttely if all one his is pictures and popularizations based on non-mathematical language, one may (and usually does not) not have enough understanding of a theory to actually get good predictions out of it - not even qualitative ones."
This might be true. But we have specifically evolved to, and are able to make many everyday predictions using "just thought pictures", and with just such thought experimentation, Einstein was able to first derive a predictive "napkin version" of Relativities and then apply math to form a final blueprint directly tied to this "picture" geometry. No one has matched that feat since. In my opinion, it does not work so well to draw an abstract blueprint first, a print that is the take-off of a previous abstract blueprint ad infinitum, and then try to figure out, and proclaim, what is it's worth... unless one is Van Gogh, or at least loves beautiful abstract art.

Thanks for responding, pervect.

Wes
...

 

[pervect]

pervect,

I understand what you are saying about the importance of math in not only describing curves but all geometries, and I agree, in spite of being a bit "math challenged".

The short answer is math, which must be always wholly based in geometry, has become so abstract that we cannot easily picture the geometry with our intuition. Checks and balances no longer exist. Do we calculate additional mathematically correct epicycles to fit our hopes, our present intuitions, or do we not? What is the essence of our ability to derive essence?

Epicycles, or as I would call them, Fourier series, are only part of the issue. The paths of planet(s), in some particular coordinate system, can be modeled with this technique. But this is just part of the big picture, which is space-time geometry. The "big picture" allows you not only to plot the orbits of planets, but to evaluate any sort of measurement (for instance distances as measured by radar time delays) that you might choose to make.

Without belaboring the details, for the coordinates you calculate via the epicylce/fourier series approach to also make predictions about distances and times, you need a way to translate from the coordinates you calcluate to the distances and times you can measure. The translation tool that does this is called the metric. To really understand the metric, one needs to understand the underlying geometry of special relativity, where distances and times are replaced by the Lorentz interval as the fundamental geometric objects.

But even with perfect, beautiful math proof, we can still apparently get an inside-out object that blueprints perfectly, yet does not meet the criteria of sensible geometry.

I would disagree in what you literally said. If the math works out right, it meets the criterion of geometry, at least in this area of application. "Sensible" is an odd word here, which I will interpret as self-consistent. The math may describe an unfamiliar or non-euclidean geometry, for instance the Lorentzian geometry of special relativity, but if the math works out, the geometry will be self-consistent. It might not match the reader's intuition of geometry, though. I'm afraid that I can't think of any simple cure for this - an intuition for an unfamiliar geometry can be developed, but it will take time and effort.The real test, though, comes not in just getting a theory that is self-consistent, though that's obviously necessary. The real test requires getting a theory that is not only self-consistent, but matches experiment.

In relation to this thread, and the OP whom we must not forget, the OP came back and stated in #4 that he was lost by the math descriptions and, I assume in frustration, wished to appeal to a geometric version.

I must admit that WN (in my opinion) sometimes goes a bit overboards on the math, assuming a great deal of specialized knowledge in the recipient. But it's a hard problem to try to match the right response to the right poster. The more sophisticated math responses can be vary valuable to the right poster, even as they fly over the head of others.

 

[Wes Tausend]

epicycles, or as i would call them, Fourier series, are only part of the issue. The paths of planet(s), in some particular coordinate system, can be modeled with this technique. But this is just part of the big picture, which is space-time geometry. The "big picture" allows you not only to plot the orbits of planets, but to evaluate any sort of measurement (for instance distances as measured by radar time delays) that you might choose to make.

What you say is conventionally true, and I agree in that sense, but...
I'm sorry, I misled you when I spoke of epicycles. I meant them in a metaphorical sense rather than literal. In other words, an "epicycle" in a helio/geo-centric system can be a metaphor for any superfluous mathematics required to describe a needlessly complicated system, and, with a little frantic searching, one can sometimes add metaphorical "epicycles" indefinately to give a preferred theory bootstrap logic-support that is hard to detect.

The later helio-centric model, against a supposed static background, was supposedly simpler mathematically, but more importantly, simpler to imagine, or picture from afar. Had not Copernicus and Galileo fortunately preceeded Newton, he, wrapped up in the latest actual epicycle, may never have formed his laws of motion and gravity and our relativity science would still be (a mere) 500 years behind. That far-flung heavenly bodies would be hurling around violating inertia would be of no consequence without Newton's Laws and would likely be explained away in otherwise fashion. Do not be so sure that someone else, rather than these three men, would have surely discovered these consecutive pre-relativity laws so soon. Even seasonal star parallax, a seeming proof of helio-centricity, may have gotten assigned it's own "epicycle" in more stagnant circumstances. I hope it makes more sense why I used epicycles as a "bad example of science"* and how nature fools us to construct them.

But even with perfect, beautiful math proof, we can still apparently get an inside-out object that blueprints perfectly, yet does not meet the criteria of sensible geometry.

i would disagree in what you literally said. If the math works out right, it meets the criterion of geometry, at least in this area of application. "sensible" is an odd word here, which i will interpret as self-consistent. The math may describe an unfamiliar or non-euclidean geometry, for instance the lorentzian geometry of special relativity, but if the math works out, the geometry will be self-consistent. It might not match the reader's intuition of geometry, though. I'm afraid that i can't think of any simple cure for this - an intuition for an unfamiliar geometry can be developed, but it will take time and effort.

I use the word ,"sensible", as in meaning both h/g-centric "picture" systems cannot be preferable aesthetically. The word "Self-consistant" is not a bad synonym. The geometry of geo-centricity and helio-centricity are quite different by intuitive visual perspective, but the pre-relativistic math very similar, because they are in fact, the same mathematical solar system. A hunter could calculate a travel path, shot from orbiting earth, to another orbiting solar system body, to:
* either be a path to intercept one another, both in solar orbit, as a distant observer viewed the sun from a solar pole, without significant star rotation (helio-centric),
* or be a path observed from a supposed non-sun-orbiting earth, to pierce the other non-sun-orbiting planet mid-epicycle (geo-centric).

the real test, though, comes not in just getting a theory that is self-consistent, though that's obviously necessary. The real test requires getting a theory that is not only self-consistent, but matches experiment.

There may be a problem defining the exact difference in h/g-centrics. The physical laws are the same in either reference system, so what will be different? Yes, the discovery of seasonal parallax with distant stars seems proof enough after the fact, but the observation could almost certainly be labeled a new exciting epicycle if one did not know better beforehand. There may no conclusive mathematical proof today between h/g-centric systems other than strong circumstantial evidence and, I believe, our usual sincere belief that GR superceeds the Mach Principle.

i must admit that wn (in my opinion) sometimes goes a bit overboards on the math, assuming a great deal of specialized knowledge in the recipient. But it's a hard problem to try to match the right response to the right poster. The more sophisticated math responses can be vary valuable to the right poster, even as they fly over the head of others.

That is ok, and I understand. I worked for a railroad and they have their own jargon built over 100 years. There is a language barrier to laymen in modern science primarily because of advanced math. A picture is worth a thousand words, but only half a good equation. I am not being critical of WN, as it is his noble intent to say much with few words and there is no other way to do it considering the deep concepts discussed here. As ever the devils advocate in this thread, I am biased to envisioning geometry foremost because that is what my quirky mind excels at compared to my meager other skills.

That, and frankly, I believe we just may someday do the helio/geo-centric debacle again concerning who, or what, is at rest or some other semi-deceiving intuition starting us off on the wrong track however true it must seem at first. Not the sun vs Earth this time, but perhaps matter vs energy, the latest canon to run the gauntlet. As you say, pervect, "an intuition for an unfamiliar geometry can be developed, but it will take time and effort". I can only say it has.

I want to thank everyone, especially the OP, DaleSpam, Nugatory, WannabeNewton and yourself, pervect. It has been a very enlightening and stimulating discussion for me and I hope others as well.

The most important thing I have learned here is that, in the essence, our temporary theories are all built upon axioms, which are all inevitably built upon intuition. Next, proper logic flawlessly proceeds accordingly. If a theory, any theory such as geo-centricity or otherwise, ever should stumble or be incomplete, the flaw will be, if not improper logic, then because of prior faulty intuition, and be found lurking in the raw axiom.

Many students and faculty are on this forum, class is out and I wish all a happy holiday.

Thanks,
Wes

 

[D H]

Regarding, "the exception of the deception of men", the best example to the contrary I can think of is the story of Ptolemy. He mapped our solar system with predictive math so well, that I understand NASA still uses a version today for space shots

That is sheer nonsense.

Perhaps you are thinking of the use of geocentric coordinates for vehicles orbiting the Earth, selenocentric coordinates for vehicles orbiting the Moon, Saturn-centric coordinates for vehicles orbiting Saturn, etc. That isn't anything like Ptolemy's system.

and we our clocks and calendars.

And that too is nonsense.

 

[pervect]

I don't think it's quite nonsense, but I don't think "epicycles" are terribly significant, either.

Any path—periodic or not, closed or open—can be represented with an infinite number of epicycles.

This is because epicycles can be represented as a complex Fourier series; so, with a large number epicycles, very complicated paths can be represented in the complex plane.[23]

Wiki gives this amusing video as an illustration - the [23] above:

So since any path can be represented with epicycles the fact that we can represent orbits with them doesn't say much about the underlying physics.

 

[micromass]

So since any path can be represented with epicycles

I'm kind of doubting that, even if the path is continuous. But this is off-topic.

 

[Wes Tausend]

regarding, "the exception of the deception of men", the best example to the contrary i can think of is the story of ptolemy. He mapped our solar system with predictive math so well, that i understand NASA still uses a version today for space shots

that is sheer nonsense.

Perhaps you are thinking of the use of geocentric coordinates for vehicles orbiting the earth, selenocentric coordinates for vehicles orbiting the moon, saturn-centric coordinates for vehicles orbiting saturn, etc. That isn't anything like ptolemy's system.

and we our clocks and calendars

and that too is nonsense.

D H,

You have done me a great favor in removing a bad reference of mine just recently, so I don't know whether I dare continue here in good conscience.

If I have offended you, or disparaged NASA in any way, I apologise. The statement was meant to be, not nonsense, but striking... to illlustrate Ptomlemy's accomplishment as an early map-maker and demonstrate that, in math alone, Ptolemy did not differ much from the Copernicus system. IOW, the later intuitive Copernican geometry model is almost all the improved early difference from a math standpoint. Granted, the latest versions are much more accurate, but Copernicus did well for the naked eye.

When I read the NASA analogy long ago, it seemed self-evident, perhaps mistakenly, that a Ptolemy version (much improved) is still used today. My first thought was that, I'll be darned, a version of the crude Ptolemy math resides in my Meade "finder" telescopes. With a built-in standard clock and calandar, the low priced telescopes are capable of determining where Mars is in the sky on any given day and accurate enough to automatically capture the planet within the field of view. From there one can zero in on the planet more dead center, which is what I suppose corrective retro-rocket engines do on a similarily calculated pre-aimed space shot. It seems to me that the initial rocket aim need not be much more accurate than my simple "finder" telescope, as some sort of guidence system will be used for inevitable trajectory errors anyway.

My imagination is that Ptolemy mathematically mapped the heavens on the supposed inner two-dimensional sphere well enough to roughly predict where the two dimensional coordinates of, say, Mars would be on a certain day.

Hunters naturally lead their target intuitively. Ptolemy may well have dreamed that if he could only shoot an arrow with enough might to not fall to the ground, and know how long it would take to get to Mars, that he could aim the arrow at the very patch of sky that his math predicted Mars would coincide with during flight. The arrow would ostensibly arrive on that exact hour and day and therefore hit his target.

When we think of my suggested simpler portion of space shots, not involving more complicated temporary paths for escape or landing gravities, I believe the Ptolemy Arrow is essentially what we still do. I would embrace being corrected, as I am a NASA fan. My DVR is cluttered with video such as, http://natgeotv.com.au/tv/death-of-a-mars-rover/. It made me sad to see Spirit lose her fight and I am saving it to show my grandchildren.

Thanks,
Wes
...

 

[Wes Tausend]

I don't think it's quite nonsense, but I don't think "epicycles" are terribly significant, either.



Wiki gives this amusing video as an illustration - the [23] above:

So since any path can be represented with epicycles the fact that we can represent orbits with them doesn't say much about the underlying physics.

Thanks for the video, pervect. A bit of humor is always welcome, especially if it includes a lesson in science.

Micromass is probably correct, we have gotten way off topic and it is my fault. I was just taken by the concise answer that DaleSpam gave in #3, and how instinct and axioms are nearly everything in understanding science in it's most basic form.

Wes
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