A baffling quote from Einstein, badly requiring explanation

[echoing song]

Subject: A Baffling Quote from Einstein, badly requiring explanation

All sources I've consulted indicate that Einstein reconceived gravity in General Relativity by discarding the prevailing, intuitive, Newtonian view of it as a 'force' accelerating objects, and daringly envisioned it purely as a warper of space-time where accelerated motion is inertial. The key insight galvanizing his grand epiphany emerged from his thought experiment where he considered a man floating in an elevator in the most remote space, far removed from all masses and forces, and a man in an elevator freely falling in the Earth's gravitational field. The almost total similarity of the two men's experiences led to the jettisoning of the idea of gravity as a force, since the first man was by definition (and by objective observation) not being acted upon by any forces.

And yet in Einstein's own book, “Relativity”, in the appendix where he discusses “Experimental Confirmation of the General Theory of Relativity”, the following appears: He has just described the size of the angle of deflection of a ray of light passing the sun, and then says, “It may be added that, according to the theory, half of this deflection is produced by the Newtonian field of attraction of the sun, and the other half by the geometrical modification ('curvature') of space caused by the sun.”

Notice that he DOESN'T say that the result is entirely caused by the curvature of space and that it is twice what would be caused by SUPPOSED Newtonian attraction. The wording, which he had many decades and opportunities to revise before his death (but didn't in any of the subsequent editions), clearly indicates that gravitational attraction (presumably by a 'force') is half of the explanation. How can this be, in light of the unanimous view that GR casts aside all notions of anything but deformation of space?

And one further question: Is it true that the deflection is EXACTLY twice the Newtonian prediction? If so, why EXACTLY twice? And does this apply to starlight bent by the sun no matter how far away the starlight is from the sun, or only to starlight essentially grazing the sun as it passes? Obviously, in most situations, GR modifications of Newtonian predictions don't involve a doubling, but only an infinitesimal alteration (as with the GPS satellites data). I presume that if one were doing GPS locating on the sun, the modifications, while greater than on the earth, wouldn't approach a doubling. So why then in the case of passing starlight is it of such large magnitude?

I eagerly await enlightenment on all these points.

 

[atyy]

http://www.einstein-online.info/en/spotlights/equivalence_deflection/index.html

 

[echoing song]

Thanks for the link atyy but it doesn't explicitly answer my primary question. The link simply shows that special relativity COULD be invoked to explain half of the deflection, but not that it SHOULD be. Is the curvature of space sufficient to account for only half of the deflection and therefore are all the accounts of GR wrong in claiming gravity is NOT to any degree acting like a Newtonian force and wrong in insisting that all effects are due entirely to the warping of space-time?

 

[atyy]

I'm not sure if Newtonian physics really predicts the bending of light. One can certainly get a formula for light bending from Newtonian physics, and it is a useful heuristic. However, it's not clear that the derivation is rigourous, because the speed of a body pulled by Newtonian gravity should change for energy conservation, but I don't know how this works out in Newtonian light bending.

All gravity in GR is the warping of space. It is a completely different theory from Newton's (not really, there is Newton-Cartan theory). However, in the weak field limit, the predictions of GR are almost the same as Newtonian gravity, in which case the GR prediction can be written as the Newtonian gravity prediction plus a small correction. If the corrections are so small that we feel happy ignoring them, then we get Newtonian gravity.

So I think Einstein was not strictly correct when he said half of the light bending is due to Newtonian gravity and half due to GR. I would say all of it is due to GR (as far as we know, until we get a working theory of quantum gravity), and it is GR which indicates the region of applicability of Newtonian gravity as an approximation to GR.

 

[Ich]

We had this question before, here's my answer at that time:

https://www.physicsforums.com/showthread.php?p=1633290

 

[A.T.]

Is the curvature of space sufficient to account for only half of the deflection and therefore are all the accounts of GR wrong in claiming gravity is NOT to any degree acting like a Newtonian force and wrong in insisting that all effects are due entirely to the warping of space-time?

- curvature of space-time explains all gravitational effects and 100% of the light bending. It can be split up into:

- curvature of time : accounts for Newtonian gravity, gravitational time dilation, 50% of the light bending​

- curvature of space : accounts for orbit precession, the other 50% of the light bending​

http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/principal_of_equivalence_and_general_relativity/curved_spacetime.html"

 

[echoing song]

Thank you Ich for the link and A.T. for the link and the comments—I found them very instructive. It's odd but I never before encountered an explicit division of space-time curvature into 'curvature of time' with specific effects and 'curvature of space' with another, separate set of effects—together accounting for the whole.

Am I understanding you (and the links) correctly in concluding that the commonly experienced effects of gravity—falling objects, the Earth's orbit around the sun-- are NOT in fact due in any significant degree to the curvature of space but rather to the curvature of time? Exactly how does that work with regard to, say, an object falling off a table? How do you explain that—not in terms of a force accelerating the object, but by gravity curving time? And why do the two approaches yield essentially identical results in ordinary, everyday circumstances?

 

[A.T.]

Am I understanding you (and the links) correctly in concluding that the commonly experienced effects of gravity—falling objects, the Earth's orbit around the sun-- are NOT in fact due in any significant degree to the curvature of space but rather to the curvature of time?

Yes, but keep in mind that this separated consideration, is artificial. The dominating role of the time dimension curvature for commonly experienced effects of gravity, is just due to the slow speeds of objects. Light is affected by warped time in the same amount as by warped space.

Exactly how does that work with regard to, say, an object falling off a table? How do you explain that—not in terms of a force accelerating the object, but by gravity curving time?

No force means a straight path in space time: http://www.physics.ucla.edu/demoweb/demomanual/modern_physics/principal_of_equivalence_and_general_relativity/curved_time.gif" shows it. More here:
http://www.relativitet.se/spacetime1.html
http://www.adamtoons.de/physics/gravitation.swf

And why do the two approaches yield essentially identical results in ordinary, everyday circumstances?

Well, if they wouldn't match everyday observation, they would hardly qualify as physical theories.

 

[Karl G.]

I was surprised to learn that Newtonian theory predicts light being curved by gravity; the most famous confirmation of GR is light deflection, and to learn classical, Newtonian gravitational fields could deflect light came as something of a shock. (Although I am fully aware that GR predicts the deflection to its correct extent).

 

[echoing song]

Now I'm really puzzled!

I can accept abandoning the concept of gravity as a force in order to resolve what would otherwise be intolerable contradictions in the two-elevator scenario discussed in my original post (above), especially when you add to the scenario the element of a ray of light crossing each of the elevators. The light will travel a straight path from the perspective of both of the men inside the elevators, but to an outside observer (at rest with both elevators before the second elevator starts free-falling in Earth's gravity) the ray of light moving horizontally across the elevator in remote space (no forces or masses) will be seen by the outsider to follow a straight path, but the ray will be observed to follow a curved path by the outsider looking at the freely falling elevator. In order to have a logically coherent explanation accounting for the experiences of the two men inside the elevators (where no forces are seen to be acting) as well as the two observations of the outsider (light moving on a straight path in one case, a curved path in the other) you need 'curvature' of some kind, to promote inertial motion.

But now it turns out that it's not curvature of space, but of time!

What does that even mean?

How can 'curvature of time', a semi-abstract concept (curvature) on top of a REALLY abstract, totally non-physical concept (time) produce a physical effect in the real world—namely acceleration of a material object!??

If it were curvature of space, I could understand it—space not only being physical but allowing for this easy analogy: constant motion in a straight line in flat space is inertial, contrary to Aristotle, needing no application of force to sustain it, so, similarly, accelerated motion in curved space would be inertial, needing no force to initiate it.

But no, I'm told it's due to the 'curvature of time', not space! What in heaven's name does that actually signify?

I clicked on the links A.T. provided (the adamtoons one didn't work) and there's nothing beyond a graphic presentation of various 'straight line' paths on curved surfaces depicting what happens to various objects thrown directly upwards at different velocities, and the reader appreciates how nicely the concept of the curvature of time REPRESENTS outcomes occurring in the real world. But EXPLAIN the real world?? Present a meaningful model of the real world?? Not in the least. On the contrary—both of the two possible physical mechanisms (force and curvature of space) for a meaningful model have been expressly eliminated as causative agents. What remains, the 'curvature of time', an abstract, non-physical entity, doesn't even hint at a physical mechanism for changing the velocity of a material object.

Are we, God forbid, in a quantum mechanics situation where the best minds exhort us not to even try to picture the true nature of fundamental particles, just accept the magnificently accurate equations as the best representation of reality possible? A melancholy thought. I can't believe the wave function of my own existence will collapse with no deeper understanding of these matters.

 

[Dale]

How can 'curvature of time', a semi-abstract concept (curvature) on top of a REALLY abstract, totally non-physical concept (time) produce a physical effect in the real world—namely acceleration of a material object!??

When you meet contradictions like this you should check your assumptions. Perhaps curvature and time are not so abstract or non-physical after all. Or perhaps the acceleration is not so physical as you assumed.

I do find it odd that you assume time is non-physical but space is physical. I would think that if you assumed space was physical that you would assume time was also, and I would think that if you assumed time was non-physical that you would assume space was also. How did you come to your "space but not time" idea?

 

[echoing song]

To DaleSpam:

I find it odd that you find it odd that I assume time is non-physical but space is physical.

To me, the intuitive way of viewing space (never even suggested to be illegitimate, as far as I know) is as a structure or framework for matter and energy, a playing field on which particles and their aggregations can cavort and make all kinds of interesting trouble. As such, it certainly would have a physical nature, from a theoretical standpoint.

And empirically, too, it does—the success of GR's predictions, involving curved space, as opposed to those based on flat space, has demonstrated differently-shaped spaces have disparate real world effects. That stakes a powerful, and I would say irrefutable, claim to physical reality and agency for space.

But time? Some philosophers of science with far more sophisticated viewpoints than mine question its conceptual meaningfulness, some working physicists suggest past, present, and future are purely human constructions and everything is really happening at once. I don't go that far, but I think it's eminently reasonable to view time as real but not capable of having a direct, physical impact upon material objects. To me, the statement, 'The curvature of time caused the water balloon to fall with increasing speed from the second story onto Paris's decolletage' is as utterly devoid of meaning as Paris herself. In fact, the notion of 'curvature', as applied to time, I see as purely metaphorical, or as a way of graphically delineating a mathematical concept—not for an instant as a literal truth. On the other hand, 'curvature of space' I indeed conceive as literally, physically true, even if not visualizable. (I grasp that in Riemannian geometry the special stipulations made for antipodes in order to make them conform to Euclid's first postulate makes even a SURFACE in curved space not properly visualizable by human beings, let alone three dimensions.)

So, yes, DaleSpam, I do categorize time and space very differently, and I feel I'm justified in doing so. But I'm always ready to be 'set straight' by my superiors, and I will assume a properly chastened tone if I am.

 

[Dale]

To me, the intuitive way of viewing space (never even suggested to be illegitimate, as far as I know) is as a structure or framework for matter and energy

I agree, and you can say the same of time, they are on completely equal footing. Furthermore, you cannot have energy nor waves without time, and at the most fundamental level what we call "matter" is made up of waves.

 

[JesseM]

And empirically, too, it does—the success of GR's predictions, involving curved space, as opposed to those based on flat space, has demonstrated differently-shaped spaces have disparate real world effects. That stakes a powerful, and I would say irrefutable, claim to physical reality and agency for space.

GR's predictions do not involve curved space, they involve curved spacetime. GR predicts that objects follow geodesics in curved spacetime, analogous to geodesics on curved spatial surfaces like the surface of a sphere, but different in the sense that whereas a geodesic in curved space between points is the path with the shortest length (at least shortest relative to other 'nearby' paths), a geodesic in curved 4D spacetime would be the path with the greatest value of proper time (time as measured by a clock taking that path through spacetime, and again, greatest relative to 'nearby' paths rather than greatest of all possible paths through spacetime which intersect those points). The successful predictions of GR about the path of objects which you mention are not based on looking for the shortest path through a curved space, they're based on looking for spacetime geodesics in curved spacetime, with the curvature of spacetime given mathematically in terms of the "metric", and the Einstein field equations telling you the mathematical relation between the distribution of matter/energy and the metric.

 

[echoing song]

To JesseM:

My gosh, it was only earlier this very day that I was introduced, in this very forum, to the notion that curvature of time and curvature of space were separable entities, at least in a practical sense, each with its own domain. The esteemed A.T. in entry # 6, precisely demarcated the respective realms of the curvature of time and space. I quote:

“curvature of time: accounts for Newtonian gravity, gravitational time dilation, 50% of the light bending

curvature of space: accounts for orbit precession, the other 50% of the light bending”

This view was corroborated by others, implicitly and explicitly, and never challenged or dissented from. Not being well-enough informed on the subject to have a real view myself, I accepted what seemed to be the consensus. Now, just as I was struggling to get my shaggy head around the concept of 'curvature of time', you, JesseM , of mythic status, gently rebuke me with the words, “GR's predictions do not involve curved space, they involve curved spacetime”,etc. essentially the antithesis of the notion avidly promoted today in this forum.

So I ask you pointedly, is it your position that, for example, a falling object's acceleration in everyday circumstances is not due to the curvature of time, but of spacetime, and Einstein's distinction, in the quote that started all this, was wrongheaded, i.e. his speaking of half of the deflection being due to the Newtonian field of attraction of the sun and half due to the curvature of space was mistaken?

 

[A.T.]

The esteemed A.T. in entry # 6, precisely demarcated the respective realms of the curvature of time and space. I quote:
“curvature of time: accounts for Newtonian gravity, gravitational time dilation, 50% of the light bending
"curvature of space: accounts for orbit precession, the other 50% of the light bending”

You are quoting me out of context. Here is what I wrote;

- curvature of space-time explains all gravitational effects and 100% of the light bending. It can be split up into:

- curvature of time : accounts for Newtonian gravity, gravitational time dilation, 50% of the light bending​

- curvature of space : accounts for orbit precession, the other 50% of the light bending​

And also:

Yes, but keep in mind that this separated consideration, is artificial...

So I ask you pointedly, is it your position that, for example, a falling object's acceleration in everyday circumstances is not due to the curvature of time, but of spacetime,

Time is part of spacetime. If time is curved so is spacetime. There is no contradiction here.

 

[JesseM]

A.T., when you say the light bending can be split up into 50% curvature of time and 50% curvature of space, does that depend on some specific way of slicing up spacetime into a series of spacelike surfaces (a specific 'foliation' of spacetime)? In other words, are you really talking about "curvature of surfaces of contant t" and "curvature of the t coordinate" for an arbitrary coordinate system with a timelike t coordinate? Also, leaving aside physics and talking just about differential geometry, if you had a purely spatial manifold (say, a curved 3D surface), and you had some coordinate system on that manifold (like x,y,z coordinates), and you wanted to calculate geodesics based on the spatial curvature, would it be possible to split up the curvature here into two parts as well (like 'curvature of surfaces of constant z' and 'curvature of the z-coordinate')?

 

[Dale]

the notion that curvature of time and curvature of space were separable entities, at least in a practical sense, each with its own domain

Consider a sphere. You can say that the sphere itself is curved. You can also talk about curvature in the north-south or east-west directions. Neither of these statements are contradictory, but the curvature of the sphere itself is the "big picture".

Now, consider two longitude lines. Longitude lines are geodesics which means that they are "straight" (i.e. they never turn and they are always the shortest path between any two points on them). However, because the sphere is curved longituge lines are parallel at the equator and intersect at the poles despite the fact that they are "straight" everywhere inbetween.

Similarly with gravitation in GR. In a spacetime diagram an inertial object's worldline is a straight line (i.e. a geodesic). Two objects which are at rest wrt each other have parallel worldlines.

If we consider north to be the positive time direction and east to be the positive space direction, then we can talk about spacetime diagrams in a curved space. Longitude lines, being geodesics, can therefore represent the worldline of an inertial object. Two longitude lines at the equator represent two objects initially at rest wrt each other. As you go north they get closer, despite the fact that they are geodesics. So two objects initially at rest wrt each other can accelerate towards each other despite the fact that both are inertial.

In the above analogy, you can certainly separate the curvature of the sphere into north curvature and east curvature components. For a longitude line, only the curvature in the north direction is important, but for a great circle going northeast the curvature in the east direction is just as important.

I hope the analogy helps.

 

[A.T.]

A.T., when you say the light bending can be split up into 50% curvature of time and 50% curvature of space, does that depend on some specific way of slicing up spacetime into a series of spacelike surfaces

The whole idea of this splitting is crude and more of a historical and didactic value. For computations it makes more sense to consider space-time a one entity, as you suggest.

What the OP refers to, is the historical context: Newton modeled gravitation for slow moving objects (v << c), which advance mainly trough the time dimension. Einstein initially tried the mimic exactly that (inaccurate) Newtonian gravity using curved spacetime, so he concentrated on the warping of the time dimension. He just later realized, that the space dimensions must be warped too, for mathematical reasons. And that this has a large effect on fast objects, like photons, doubling the amount of light bending.

The problem with learning a theory developed over years from the original papers and statements, is that you get confused by the misconceptions and overcomplicated views, the developer had in mind at a certain time point. The final result is often much easier to understand.

 

[DrGreg]

In the above analogy, you can certainly separate the curvature of the sphere into north curvature and east curvature components. For a longitude line, only the curvature in the north direction is important, but for a great circle going northeast the curvature in the east direction is just as important.

I'd have to disagree with this. To account for longitude lines converging, both the north curvature and the east curvature are necessary. Consider a cylindrical planet where the longitude lines are straight (in 3D) and the latitude lines are all circles. No converging of parallel lines occurs at all on this planet, and in fact the 2D-surface geometry is identical to the geometry of a flat Earth (apart from the non-local fact that you can circumnavigate the planet, but that doesn't count as "geometry" but topology). Technically, the curvature of the cylindrical planet is zero, depite appearances.

In terms of "manifold geometry", talking about curvature in a single dimension (e.g. "curvature of time") is meaningless. It's curvature of spacetime.

 

[atyy]

I think there's some condition like asymptotic flatness (or something) for the Newtonian limit to work. Sorry, am very hazy about this, but it might be interesting for whoever's interested to look into.

 

[Enoy]

This guy confuses me. Can someone be so kind to tell me that this guy is wrong!

 

[echoing song]

I hope you gentlemen don't mind my returning briefly to a more fundamental matter; I believe I understand how the issue of 'curvature of time' as a causative agent, which has provoked my latent rebellious spirit in my last couple of posts, can be dealt with in a way that is faithful to both the mathematics involved and our intuitive sense of causation and physical reality. Tell me if the following is valid:

Spacetime is a mathematical construct that reflects both the literal, physical curvature of space in the presence of masses and the figurative, mathematical curvature of time, with this latter curvature an unavoidable consequence of being one coordinate of a system that has a physically curved component (space). Everyday gravitational events (falling objects, orbiting planets), though they are represented by movements almost entirely along the time axis of spacetime, OCCUR IN THE WAY THEY DO because of the overall curvature of spacetime, which in turn is FUNDAMENTALLY DUE TO THE PHYSICAL CURVATURE OF SPACE.

As DaleSpam was illustrating (please see Entry #18 for the full context and I'm not saying he in any way endorses my viewpoint), 2 separated objects initially at rest with respect to one another at the equator may accelerate towards one another gravitationally over time, purely inertially, as their figurative lines of longitude, originally parallel, gradually converge. My addition is that although this movement is essentially solely along the north-south time axis of spacetime, THIS GRAVITATIONALLY-INDUCED CONVERGENCE ONLY OCCURS BECAUSE OF THE OVERALL CURVATURE OF SPACETIME, WHICH ITSELF IS THE CONSEQUENCE OF THE LITERAL, PHYSICAL CURVATURE OF SPACE. Therefore, to speak of such inertial motion as due to the curvature of time is superficially true but deeply misleading. I believe these paragraphs put it into the proper perspective.

I think the preceding nicely reconciles the disparate views, but feel free to eviscerate my formulation if your scientific conscience so dictates.

 

[JesseM]

Spacetime is a mathematical construct that reflects both the literal, physical curvature of space in the presence of masses

I don't think spatial curvature can be called "physical" in the sense of being coordinate-independent, because you have an infinite variety of ways of slicing up spacetime into spatial slices in different coordinate systems (each surface being a set of events with the same t coordinate in that system), and I would imagine (though I'm not sure) that whatever definition of spatial curvature you use, each slicing would say a different thing about the way space is curved in the surface that passes through some particular event, and hence a different thing about the shape of spatial geodesics in that surface. In contrast, no matter what coordinate system you use on spacetime, you'll always get the same answer about the question of what the spacetime geodesics are supposed to look like (i.e. what set of physical events they pass through).

 

[A.T.]

Spacetime is a mathematical construct that reflects both the literal, physical curvature of space in the presence of masses and the figurative, mathematical curvature of time,

Everything in a physical theory is a mathematical abstraction. Your philosophical distinction between "physical curvature" and "mathematical curvature" has no influence on the objective results and is therefore irrelevant. As already stated here, space is not more physical than time. All that matters in physics is: Can we measure it?

 

[JesseM]

Everything in a physical theory is a mathematical abstraction. Your philosophical distinction between "physical curvature" and "mathematical curvature" has no influence on the objective results and is therefore irrelevant. As already stated here, space is not more physical than time. All that matters in physics is: Can we measure it?

In physics we do distinguish between "physical" coordinate-invariant facts and coordinate-dependent ones, though. Am I right in thinking there is no coordinate-invariant notion of a spatial geodesic in GR, unlike with spacetime geodesics which are coordinate-invariant and thus give a coordinate-invariant way to define the curvature of spacetime with differential geometry?

 

[Dale]

echoing song, you still seem stuck on this unwarranted assumption that time is somehow less "real" than space.

 

[echoing song]

To A.T. (By the way, sorry if you felt I took your comments out of context.)

You say: All that matters in physics is: Can we measure it?

I say: Physicists and non-physicists often desire to understand things in ways that are non-mathematical/quantatative, in fact in ways that TRANSCEND the merely mathematical (a phrase that I realize is anathema to many in this forum and may result in my excommunication). In fact, the deeper sense of things that models provide (crude and even somewhat distorting though they may, of necessity, be) often galvanizes the development of the ideas that only LATER are expressed mathematically. Indeed, Einstein turned to 'measuring' long after he had conceived of all the key ideas of both special and general relativity NON-MATHEMATICALLY. If in 1895 you'd heard that your teenage neighbor Albert was imagining riding alongside light beams, you'd have been as dismissive of his future as Minkowski, right?

To DaleSpam: (By the way, it's quite touching the way you're trying to save my scientific soul)

I fully accept time as real, but why must it have the same status as space? Isn't THAT as unwarranted an assumption as you accuse me of making?

I'm striving to take the idea that everyday gravitational events, like objects falling, are due, in a certain sense, to 'the curvature of time' and render it meaningful—in my personal universe, time can't accelerate objects or have literal curvature, but space, through its curvature of spacetime, can accomplish both. So seeing time's curvature as just one figurative, mathematical axis of a spacetime fundamentally curved by space's own mass-induced actual physical curvature makes sense of it all for me.

 

[Dale]

I fully accept time as real, but why must it have the same status as space? Isn't THAT as unwarranted an assumption as you accuse me of making?

No, it has more than a century's worth of experimental data to support it.

 

[atyy]

Equation 39.10. A few pages before they say the coordinates are chosen to be as globally Lorentz as possible, and in which the solar system is approximately at rest, and that this provides a 3+1 split to make GR look like Newton. http://books.google.com/books?id=w4Gigq3tY1kC&printsec=frontcover#PPA1076,M1

 

[atyy]

In physics we do distinguish between "physical" coordinate-invariant facts and coordinate-dependent ones, though. Am I right in thinking there is no coordinate-invariant notion of a spatial geodesic in GR, unlike with spacetime geodesics which are coordinate-invariant and thus give a coordinate-invariant way to define the curvature of spacetime with differential geometry?

Well, that's true in Minkowski space, and Minskowski space is a vacuum solution of GR, so it's true in GR.

 

[echoing song]

To DaleSpam:

The century's worth of experimental data you refer to just means that as a mathematical coordinate of spacetime of course time should have been given the equal status it was in fact given. But I'm talking about its role in the MODEL (at least what I see as the only reasonable model) generating our understanding of what underlies it all physically—there it's the kid brother, just tagging along.

DaleSpam, please, for one instant put aside your preconceptions on this topic and consider this: Don't you feel it's the curvature of space in the presence of masses that is the juggernaut that controls everything else in GR, and specifically, that FORCES a curvature upon time? THAT'S why I think the model of reality we conjure up must give space's curvature the decisive role.

Here's an analogy: DaleSpam, who do you consider to be the more important figure in deciding the gender of a baby, the mother or the father? Based on your reasoning thus far, you would say they have equal status, each contributing a vital chromosome. But I would say the father, since it is solely his contribution that makes a difference—if he provides a Y it's a boy, an X it's a girl. Once his contribution is made EVERYTHING ELSE IS DETERMINED. Once space is curved, everything else is determined. Time is as powerless as the mother to change things. Now, from the technical viewpoint of a geneticist, both parents' contributions are equal, just as from the technical viewpoint of a mathematician, both space and time are on an equal footing as coordinates. But from the perspective of someone looking more deeply, one is the guiding force, the Determiner, and the other just a taken-for-granted corollary. Yes, the X chromosome supplied by the mother must do what it does in order for the baby's gender to be properly established, but what it does is completely imposed upon it by the father's X or Y contribution. Similarly, time must play its role in order for spacetime to function, but the role is that of obedient servant, or really chattel slave, everything driven by the curvature of space's relentless whiplash.

 

[atyy]

THIS GRAVITATIONALLY-INDUCED CONVERGENCE ONLY OCCURS BECAUSE OF THE OVERALL CURVATURE OF SPACETIME, WHICH ITSELF IS THE CONSEQUENCE OF THE LITERAL, PHYSICAL CURVATURE OF SPACE.

Sometimes it works to think like that, but not always. See the comments before and after Eq 4.77 in Gourgoulhon's http://arxiv.org/abs/gr-qc/0703035.

And then you also need the spatial curvature to evolve in time. Eq 4.78 and following.

 

[Dale]

The century's worth of experimental data you refer to just means that as a mathematical coordinate of spacetime of course time should have been given the equal status it was in fact given. But I'm talking about its role in the MODEL (at least what I see as the only reasonable model) generating our understanding of what underlies it all physically—there it's the kid brother, just tagging along.

I don't understand your comment here. In normal usage a theory's "model" is the mathematical framework used by the theory to make experimental predictions. In the case of relativity the mathematical model is the Minkowski geometry of spacetime. You appear to separate the two somehow. So what do you mean by "the MODEL" and which theory's model are you referring to where time is not on the same footing as space?

Don't you feel it's the curvature of space in the presence of masses that is the juggernaut that controls everything else in GR, and specifically, that FORCES a curvature upon time?

No. Go back to the sphere analogy. You can have a sphere where the east curvature is equal to the north curvature, you can also have a prolate spheroid where the north curvature is less than the east curvature or an oblate spheroid where the north curvature is more than the east curvature. The two are not always equal, but only in the case of specific symmetry. Extending the analogy to spacetime, the time and space curvature are equal in the specific case of the Schwarzschild metric but not in general.

 

[feynmann]

Thank you Ich for the link and A.T. for the link and the comments—I found them very instructive. It's odd but I never before encountered an explicit division of space-time curvature into 'curvature of time' with specific effects and 'curvature of space' with another, separate set of effects—together accounting for the whole.

Am I understanding you (and the links) correctly in concluding that the commonly experienced effects of gravity—falling objects, the Earth's orbit around the sun-- are NOT in fact due in any significant degree to the curvature of space but rather to the curvature of time? Exactly how does that work with regard to, say, an object falling off a table? How do you explain that—not in terms of a force accelerating the object, but by gravity curving time? And why do the two approaches yield essentially identical results in ordinary, everyday circumstances?

Per Schutz's book, <Gravity from the Ground Up>, "All of Newtonian gravitation is simply the curvature of time" See this link: http://www.gravityfromthegroundup.org/pdf/timecurves.pdf