General Relativity is pure Mathematics

[rogerl]

General Relativity Spacetime is pure Mathematics. But we live in a real world. How do we get coupled to Spacetime? We are real. Spacetime is math. Math just describe reality and doesn't affect it. So Spacetime only describe reality as metaphors. But how come physicists believe Spacetime is real. How can this model work when the manifold is just pure math and we who are real and solid can't interact with a manifold that has no physical form. Please explain. Thanks in advance.

 

[Dale]

General Relativity Spacetime is pure Mathematics.

No it is not: http://en.wikipedia.org/wiki/Tests_of_general_relativity

 

[bobc2]

General Relativity Spacetime is pure Mathematics. But we live in a real world. How do we get coupled to Spacetime? We are real. Spacetime is math. Math just describe reality and doesn't affect it. So Spacetime only describe reality as metaphors. But how come physicists believe Spacetime is real. How can this model work when the manifold is just pure math and we who are real and solid can't interact with a manifold that has no physical form. Please explain. Thanks in advance.

In addition to the excellent point made by DaleSpam with his reference, you might consider the math as an extension of normal everyday language--an extension providing more precision, clarity, etc., as well as providing for logical developments in the understanding of the world we inhabit. You could make a statement like, "That is a chair. It has four legs, a platform to sit on and a back to lean against." The mathematical language would extend the description of the chair (mass density, dimensions, modulus of elasticity, material strength in psi, mass matrix, specific heat, resonance frequencies, mode shapes, etc.).

Throughout the history of science we've developed pictures and models of things "real"--gradually including things more and more remote to our direct experience. At some point, as things get more and more remote from direct observation, you may prefer to transition to the realm of philosophy and religion to pursue your search for reality.

Indeed you may find a more satisfactory discussion of your question over in the philosophy forum, though your question is certainly worthy of consideration here. I'm not sure how optimistic any physicist is about prospects of a successful mapping of real objects (whatever "real objects" means) and events to a manifold and coordinate system of physics.

On the last day of class, my graduate philosophy of physics professor (who first obtained a PhD in physics before switching to philosophy) stated, "...physics never has and never will contribute anything to the understanding of reality." He also stated, "The room down the hall does not exist until I walk down the hall and open the door."

 

[revnaknuma]

general relativity is purely physics. observational and experimental evidences completely completely supports general relativity. it is a main-stream physics.
however, string theory is purely mathematics. no observational and experimental evidence. it is non-mainstream.

 

[rogerl]

general relativity is purely physics. observational and experimental evidences completely completely supports general relativity. it is a main-stream physics.
however, string theory is purely mathematics. no observational and experimental evidence. it is non-mainstream.

So how do you map real objects to a spacetime manifold? How are they coupled?
Quantum objects and spacetime?

 

[khemist]

a manifold is a description of a surface, sometimes real.

 

[Daverz]

So how do you map real objects to a spacetime manifold? How are they coupled?

You could use measuring sticks and clocks. Or a GPS. Or a sextant and a clock.

http://en.wikipedia.org/wiki/Surveying
http://en.wikipedia.org/wiki/Navigation

The first chapter of Misner, Thorne & Wheeler may be helpful on this topic.

The book Space, Time and Things is also good on these fundamental topics.

Quantum objects and spacetime?

We can still measure the position of a "quantum object" at a particular time. With a scintillation screen, for example.

 

[homology]

I wouldn't say that pure math is simply that which has no experimental verification. It seems math provides many models, many more than we need. Which we choose and why we choose them depend on physical motivations. Whether they are the correct models depends on experimental verification.

Rogerl, it would be like saying: accounting is pure arithmetic (a subset of pure math) how is it that we can balance our budgets (I mean individuals not governments :P).

Or trigonometry is pure math, how can we survey land create maps, etc.

So maybe your question is a more philosophical one: why does math seem so effective at describing the world? That's certainly an interesting question though not one that we're likely to figure out or agree on any time soon.

 

[Naty1]

But how come physicists believe Spacetime is real.

Spacetime is as "real" as energy, forces, or mass...but "real" has no universally agreed upon meaning. But physicsts believe spacetime is an entity just as they believe electromagnetic radiation or gravity exists.

I like this comment:

It seems math provides many models, many more than we need. Which we choose and why we choose them depend on physical motivations. Whether they are the correct models depends on experimental verification.

So GR can be thought of as a model, a model of our unierse, because it explains a lot of what we observe about space, time and gravity, for example...and many tests confirm it's validity. But we know something is missing because it does not mesh with quantum theory...another model, or theory, to reconcile these is quantum gravity...still being developed.

 

[Fredrik]

I completely agree with the claims in the OP (Edit: See my comments in post #15), and I think it's a very good question. I've been thinking about these things off and on for several years now, and I still haven't found a really satisfactory answer. As far as I know, there are no answers to be found in the literature either. These things are completely ignored in physics books. I even looked for answers in a couple of "philosophy of science" books, but I didn't find anything.

Mathematical structures like smooth manifolds are obviously "pure mathematics". (To me this means that they have mathematical definitions, nothing more, nothing less). Physics on the other hand, is science, so a theory of physics must be falsifiable. To be falsifiable, it has to make predictions about results of experiments. So if we're going to use a piece of mathematics (like a smooth manifold with a metric and stress-energy tensor that satisfy Einstein's equation) in a theory of physics, we need a set of rules that tells us how to interpret the mathematics as predictions about results of experiments.

These rules are definitely not pure mathematics. They are statements about real-world objects. They can't be derived from the mathematics, and must be postulated. The fact that a theory isn't falsifiable without an adequate set of rules means that the rules should be considered part of the definition of the theory.

It might seem that it should be straightforward and easy to write down a set of rules that tells us how to interpret the mathematics of GR as predictions about the results of experiments, but it's not easy for any theory. In GR, I think the most obvious statement to include in the rules is this one:

A clock measures the proper time of the curve in spacetime that represents its motion.​

I suppose that this should be preceded by a few statements that tell us that motion is represented by timelike curves in spacetime, and explains the idealizations we make (e.g. when we pretend that a clock is a point particle).

There's a major problem with statements like the one above. What's a "clock"? We can't define it mathematically, because that would completely mess up what we're trying to do, which is to specify how to interpret the mathematics as statements about the real world. This can't be done by eliminating references to the real world.

So the best we can do is to define the term by writing a set of instructions that tells you how to build the type of device that we would like to call a "clock", and say that what you get when you follow these instructions is called a "clock". This is where we run into another major problem: The best set of instructions that we can write today can't be understood by someone who doesn't already understand something about relativity and quantum mechanics!

This looks circular, but there's a way to avoid circularity. I'll quote myself from another thread.

I think the only way to describe this process is in terms of a hierarchy of theories. [...] You start with the definitions of the purely mathematical parts of a collection of theories (say pre-relativistic classical mechanics, SR and QM). Postulate a correspondence between mathematical observables and measuring devices in any way you can. You can e.g. define the term "clock" by a describing an hourglass or something, and define a "second" by saying that it's the time it takes a certain amount of sand to run through. A few such definitions is enough to define "version 1" of pre-relativistic classical mechanics and start using it to make predictions.

Experiments will show you that you're on the right track. So now you have a reason to believe that the theory says something useful. One of the things it tells you is that the swings of a pendulum take roughly the same time. So you redefine a "second" to be the time it takes a specific pendulum to swing away and back, and you define "version 2" of the theory with the term "clock" defined by a description of how to build a pendulum clock. This way you can continue to define new versions of the theory, each one more accurate than the previous version.

You do the same to your other theories, including QM. At some point, you will see that to go from version n to version n+1 of pre-relativistic classical mechanics, you will have to use a version of QM(!), because it's the predictions of (some version of) QM that justify the new definition of a second that we're going to use in version n+1 (a statement about radiation emitted from a cesium-137 atom). At this point we define the term "clock" by a description of how to build an cesium clock, and we won't be able to do that without using earlier versions of several theories, including pre-relativistic classical mechanics and QM.

So the process of refining the correspondence between mathematical observables and measuring devices involves a large number of steps, and it's also clear that theories aren't refined in isolation from each other.

To go from the nth level in the hierarchy to the (n+1)th, you just write down a set of instructions on how to built a (n+1)th level measuring device that can be understood and followed by someone who understands the nth level theories and has access to nth level measuring devices. There's nothing circular about this.

The rule stated above tells us how to use clocks (real-world objects) to measure proper time (a mathematical property of a curve). At the very least, we need a second rule that tells us how to use some kind of length measuring device to measure the proper length of spacelike curves. If we e.g. decide to use rulers, we have to deal with the practical problem that rulers are deformed by acceleration and gravity. In principle, we can attach accelerometers to many different parts of the ruler, and make our length measurement axiom a statement about how to use a ruler with a bunch of accelerometers glued to it to measure proper lengths in situations where the accelerometers all read zero.

But what's an accelerometer? You could e.g. measure acceleration by attaching clocks to opposite ends of a hollow cylinder made of a solid material and having them exchange information using light signals inside the cylinder. If they stay synchronized, we're not accelerating (in that direction at least). But this raises a number of new issues, including how to determine if your hollow cylinders would be straight if they were in free fall in a region with negligible tidal forces, or if they just look straight right now because they're under the influence of acceleration and/or gravity.

As you can see, this is quite complicated, and one must be very careful to avoid circularity.

 

[Dmitry67]

Physics IS mathematics (c) Max Tegmark

 

[Fredrik]

Reality may be mathematics, but physics is the process of finding new theories and testing the accuracy of their predictions.

 

[Dale]

Hi Fredrik, as much as it pains me to do so, I am going to have to disagree with you here. I think that your support of the OP's stance that GR is pure mathematics is untenable. In fact, you seem to run into the contradiction here:

There's a major problem with statements like the one above. What's a "clock"? We can't define it mathematically, because that would completely mess up what we're trying to do, which is to specify how to interpret the mathematics as statements about the real world. This can't be done by eliminating references to the real world.

If you cannot eliminate references to the real world then it is not reasonable to assert that it is pure mathematics.

Now, if the OP had said "Differential geometry is pure Mathematics" I would have agreed completely. Differential geometry is the mathematics of GR, but GR is more than just differential geometry, and that "extra stuff" is precisely the part that you mention above which ties the math to the real world. That part is not purely mathematical. So while the claim that "Differential geometry is pure mathematics" is OK, the claim that "General Relativity is pure Mathematics" is wrong.

 

[Nabeshin]

Hi Fredrik, as much as it pains me to do so, I am going to have to disagree with you here. I think that your support of the OP's stance that GR is pure mathematics is untenable. In fact, you seem to run into the contradiction here:If you cannot eliminate references to the real world then it is not reasonable to assert that it is pure mathematics.

Now, if the OP had said "Differential geometry is pure Mathematics" I would have agreed completely. Differential geometry is the mathematics of GR, but GR is more than just differential geometry, and that "extra stuff" is precisely the part that you mention above which ties the math to the real world. That part is not purely mathematical. So while the claim that "Differential geometry is pure mathematics" is OK, the claim that "General Relativity is pure Mathematics" is wrong.

I'm going to agree here with you, DaleSpam, but am acknowledging that perhaps I haven't grasped the full breadth of your long post Fredrik.

It seems to me the distinction can be best seen in the following: Differential geometry is mathematics, and this will tell us what the geodesics on a given manifold are. So if we're just finding geodesics on manifolds, maybe it's the manifold corresponding to schwarzchild, we're just doing mathematics. But when I say "I'll restrict myself to Lorentzian manifolds, and I'll have that particles move on timelike geodesics" all of a sudden I'm doing physics. I've restricted myself to a subclass of mathematical theories which I have hypothesized correspond to observable reality.

So I'm curious, I notice in your post you mention the stipulation that particles move on timelike geodesics, but you seem to gloss over it and further discuss the definition of a clock. This seems to me (perhaps) to be more fundamental than the clock discussion, but again it's possible I simply am afraid of falling too far down the rabbit hole discussing the nature of space and time at the most fundamental levels...

Edit: In light of Fredrik's response below, I think this post might be moot and just chalked up to mis-communication and a vague OP...

 

[Fredrik]

the claim that "General Relativity is pure Mathematics" is wrong.

Yes, it is, but did anyone make that claim?

What he actually said was "General Relativity Spacetime is pure Mathematics" (my bold). This sentence is a bit weird, but I can't interpret it as saying "GR is pure mathematics". If that's what he meant, then why is the word "spacetime" in there at all? To me that sentence means "The spacetime of GR is pure mathematics". The spacetime of GR is just a smooth manifold equipped with a metric, so I think he seems to be saying that a subset of the mathematics of GR is pure mathematics. I can't disagree with that.

 

[Fredrik]

I'm going to agree here with you, DaleSpam, but am acknowledging that perhaps I haven't grasped the full breadth of your long post Fredrik.

I think the only disagreement is about what the OP meant.

So I'm curious, I notice in your post you mention the stipulation that particles move on timelike geodesics, but you seem to gloss over it and further discuss the definition of a clock. This seems to me (perhaps) to be more fundamental than the clock discussion,

Those things are certainly fundamental too. The main reason why I didn't say more about them is simply that I still haven't found an entirely satisfactory way to deal with them. The reason I had a lot to say about definitions of terms like "clock" is that I recently figured out those things myself. (The discussion in the thread that I took those quotes from helped me see things more clearly).

 

[Nabeshin]

I think the only disagreement is about what the OP meant.

Haha yes, I think so too after reading your last post.

Those things are certainly fundamental too. The main reason why I didn't say more about them is simply that I still haven't found an entirely satisfactory way to deal with them.

Ah, ok. It's tricky, especially for me in your discussion of what a clock is... I could easily see one getting confused and going nuts thinking too much about these questions. But that's for another thread!

 

[atyy]

As you can see, this is quite complicated, and one must be very careful to avoid circularity.

Why is circularity to be avoided?

 

[mistergrinch]

Richard Dawkins has pointed out that our minds can easily formulate questions in the English language that have no meaningful answers, such as “what is the color of jealousy?” This is a good thing to keep in mind in these kinds of philosophical discussions of physics. The brilliant thing about physics is that the truth is in the math, not the words, and if we forget that we’ll just confuse ourselves. The philosophy professor mentioned in a previous post had it backwards; it’s philosophy that has nothing objective to tell us about the world, whereas physics allows us to actually calculate useful facts! The desire for some "deeper meaning" than physics is probably just our minds playing tricks on us, like trying to find the color of jealousy...

 

[atyy]

The technical answer to the OP, going a different route from clocks is that the gravitational field is not the manifold. The gravitational field is simply a tensor field on the manifold with a certain gauge structure. Other forces such as the electromagnetic force, and matter such as electrons, are represented by other fields. The interaction between the gravitational field and the electron field is represented by a term in the Lagrangian, just as the interaction between the electric field and electrons is represented by another term in the Lagrangian.

All the philosophical issues raised by Fredrik remain eg. what is an electron? what is an electric field?

 

[Fredrik]

Why is circularity to be avoided?

If a madman points a gun at you and says that he'll shoot you in the face if you don't raise your right hand (and only your right hand), you will probably be glad that this isn't the only definition of "left" and "right" you've seen:

1. Your right hand is the one with the thumb on the left.
2. Your left hand is the one with the thumb on the right.

Circularity isn't necessarily a disaster. It can be made harmless by an additional statement. Someone can grab your right hand and say "This is your right hand". I just think it would be hard to see which statements we really need and which ones we don't if we allow circularity.

 

[atyy]

(and only your right hand)

 

[Dale]

the claim that "General Relativity is pure Mathematics" is wrong.

Yes, it is, but did anyone make that claim?

it is the title of the thread, word for word.

 

[atyy]

I like Fredrik's sequence of theories. It's very much in line with the thinking that ALL our theories are wrong, and all we have are effective theories. So it doesn't really matter whether or not any of our theories are even mathematically consistent. Then the question becomes why is Nature so kind as to let us understand her by inconsistent, partial theories?

However, does the sequence of theories mean that whoever does have the final theory must use circularity?

 

[Rap]

General Relativity Spacetime is pure Mathematics. But we live in a real world. How do we get coupled to Spacetime? We are real. Spacetime is math. Math just describe reality and doesn't affect it. So Spacetime only describe reality as metaphors. But how come physicists believe Spacetime is real. How can this model work when the manifold is just pure math and we who are real and solid can't interact with a manifold that has no physical form. Please explain. Thanks in advance.

I agree with bobc2 - mathematics is a language - If we rewrite your question substituting "Shakespeare's play Macbeth" for "spacetime" and "English" for "math", we get:

"Shakespeare's Macbeth is pure English. But we live in a real world. How do we get coupled to Shakespeare's Macbeth? We are real. Shakespeare's Macbeth is English. English just describes reality and doesn't affect it. So Shakespeare's Macbeth only describes reality as metaphors. But how come people believe Shakespeare's Macbeth is real (i.e. contains some truth)? How can this play work when the play is just pure English and we who are real and solid can't interact with a play that has no physical form."

I hope this gives some insight into the confusion. Spacetime is a mathematical construct that we use to gain insight into reality, just as Macbeth is an English play that we use to gain insight into the human condition. The analogy is not perfect, but mathematics cannot be divorced from physics, it is the languange of physics. Understanding mathematics is like understanding the rules of English grammar. Understanding the concepts and laws of physics is like understanding English vocabulary and English literature. Hopefully, both will give you a glimpse of "reality".

 

[Fredrik]

it is the title of the thread, word for word.

Wow. I didn't even notice that. The statement in the title is definitely wrong, but I can see how a GR book might give someone the impression that it's not.

So it doesn't really matter whether or not any of our theories are even mathematically consistent.

In an inconsistent mathematical theory, every statement is true, so there's no way to use inconsistent mathematics in a theory of physics.

However, does the sequence of theories mean that whoever does have the final theory must use circularity?

I think a truly final theory would have only one interpretation rule: "This branch of mathematics is reality", or even "All of mathematics is reality" (as Dmitry67 already suggested). That doesn't make it circular, but it probably means that it has very little predictive power.

 

[petergreat]

The physicist Eugene Wigner has wondered about the same question. See the Wikipedia article
http://en.wikipedia.org/wiki/The_Un...ral_Sciences#Hamming.27s_follow-on_to_Wigner"
with philosophical discussion from many thinkers.

 

[Phrak]

General Relativity Spacetime is pure Mathematics. But we live in a real world. How do we get coupled to Spacetime? We are real. Spacetime is math. Math just describe reality and doesn't affect it. So Spacetime only describe reality as metaphors. But how come physicists believe Spacetime is real. How can this model work when the manifold is just pure math and we who are real and solid can't interact with a manifold that has no physical form. Please explain. Thanks in advance.

Well done, youall.

The keyword in the above is Metaphor. In other words: you are all idiots, physicists are all idiots, and you have no idea what you are doing, but just waving your hands metaphorically around in a dim-witted attempt to impose your interpretation of reality on others, generating arcane incantations in the form or equations no humanities professor could ever hope to understand. Therefore, science must and should be wrong, or these illustrious intellectuals lose their proper sense of superiority over the lowly, misdirected (hated) science cast.

Therefore, it is important that science be reduced to metaphor. This weak attach was probably second hand, by a humanities student. There's no mystery behind the motives of humans as long as we remember that we lie, cheat, steal, deceive each other and ourselves, develop worshipful respect for authority figures, and more.

For more psycho fun by a group that smokes way too much pot for inspiration, do a google search on "Science as Metaphor" for plentiful results.

 

[rogerl]

After some research in sci.physics newsgroup. I think there is a way to visualize General Relativity spacetime as it interacts with matter. For a start, let's accept that gravity is a curvature of space and time. I found the following from numerous posters in the newsgroup:

Mr. Hansen said "A particle sitting stationary in space is still going forward in time,
and the curvature sort of tells you how much the particle is deflected
into a spatial direction when it does that."

Mr. Hoova said (in sci.physics google newsgroup): "Now According to GR to each point in space and at each instant time we can assign to it a 4x4 quantity called the metric. It is the metric that determines how objects move like to each point in space and to each instant of time we can assign the value of an EM field (called the EM 4 vector) and it is that that determines how charged particles move. It turns out however the metric is a fundamental geometric entity."

(back to me Roger) So as I am sitting in this chair. The GR metric working is in each point in space and at each instant of time controlling the flow of space and time and binding matter to the curvature of the combined spacetime. Now add the following rules:

"GR consists of a number of assumptions one of which is particle follow geodesics (which
is logically equivalent to the principle of maximal time which is logically equivalent to Newton's first law)."

And

"the most striking is that GR has a very remarkable property. It consists of two parts: the EFE's that describe how space-time is curved and that particles follow geodesics. Einstein and his collaborators at the time showed that the EFE's all by themselves imply particles follow geodesics. This was a startling discovery - no other theory has the field equations
dictating how particles move. So we can say the basis of GR is the EFE's which follow from no prior geometry which in turn imply gravity is space-time curvature and particles follow geodesics."

(back to me Roger) Now back to me sitting in this chair. So my position is deflected into a spatial direction that is the Earth due to my every point in space and instant of time following the principle of maximal time and because Einstein EFE rule reality which makes my particles follow geodesics which makes my chair get stuck to the floor or producing this thing called gravity.

Do you guys agree with the above description of how to couple spacetime differential geometry with actual objects?

Furthermore, it is argued that to explain the physical mechanism for why triangles obey
Pythagoras's theorem is the same as asking what is the physical mechanism for General Relativity. Answer. It just is. Do you agree with this too?

Everything I mention seem intuitive and can explain General Relativity isn't it? Did not what I mentioned explained it clearly Fredrik? You said "I still haven't found a really satisfactory answer". Is not everything I mentioned above a satisfactory answer?
Or did I miss something you can see clearly.. pls. comment on everything I mentioned above as I want to understand what you are coming from. Maybe your question is akin to asking why Triangle follow Pythagorean theorem or does it has something to do with why the differential geometry just can't get coupled to actual space where we move and instant of time where we pass by?

 

[haushofer]

I completely agree with the claims in the OP (Edit: See my comments in post #15), and I think it's a very good question. I've been thinking about these things off and on for several years now, and I still haven't found a really satisfactory answer. As far as I know, there are no answers to be found in the literature either. These things are completely ignored in physics books. I even looked for answers in a couple of "philosophy of science" books, but I didn't find anything.

...

But Fredrik, isn't this the old question "what is the relationship between mathematics and reality"? Is mathematics inspired by reality, or should it completely be derivable from reality? Is there some platonic notion of mathematics, or is it just human construction, in such a way that it is the most effective way of describing nature? (Like Wigner pondered in his "The unreasonable effectiveness...")

About the philosophy of GR: I read some papers of John Norton in the past. Maybe you already checked, but perhaps he has written some things about your questions :)

 

[harrylin]

General Relativity Spacetime is pure Mathematics. But we live in a real world. How do we get coupled to Spacetime? We are real. Spacetime is math. Math just describe reality and doesn't affect it. So Spacetime only describe reality as metaphors. But how come physicists believe Spacetime is real. How can this model work when the manifold is just pure math and we who are real and solid can't interact with a manifold that has no physical form. Please explain. Thanks in advance.

Of course a description of observations of reality cannot affect it - but why would we want our descriptions to affect reality?

General relativity is just like other theories of physics, a theory about physical measurements, as done with, among other things, time and length standards. Spacetime is a mathematical description of a standard measurement system that consists of clocks and rulers (or equivalent). This is explained by Einstein in the introduction of his 1916 paper:

- http://www.Alberteinstein.info/gallery/gtext3.html

Near the end of it he also predicted consequences for the "real world".

He also partially answered your question in a popular account:

"rays of light are propagated curvilinearly in gravitational fields."

"the general theory of relativity enables us to derive theoretically the influence of a gravitational field on the course of natural processes"

- http://www.bartleby.com/173/22.html

Harald

 

[Fredrik]

Do you guys agree with the above description of how to couple spacetime differential geometry with actual objects?

Most of that sounds good. This one is wrong: "no other theory has the field equations dictating how particles move". (The guy who wrote that probably knows GR, but was sloppy with this statement). Classical electrodynamics has field equations (Maxwell's) that tell us how particles move. What makes GR special is that the most important field is part of what defines a "spacetime". A spacetime isn't a manifold, it's a pair (M,g) where M is a manifold and g is a metric. The metric is a special kind of tensor field on M. The electromagnetic field is another, but it's not considered part of the spacetime structure. Instead it contributes to the stress-energy tensor, which has a relationship with the metric described by Einstein's equations.

Compare with special relativistic classical electrodynamics. Here we have a specific spacetime (M,η). The metric is always η, and the electromagnetic field is just a tensor field on this particular spacetime. Nothing can change the metric in this theory.

In both SR and GR, the electromagnetic field is what determines how charged particles move, in particular how their motion deviates from geodesic motion. In SR, the motion of charged particles will change the electromagnetic field, but not the metric, so the geodesics will remain the same. In GR, the change in the electromagnetic field induces a change in the metric as well, so the geodesics do not remain the same. (However, it would take an absurdly strong electromagnetic field to change the geodesics noticeably).

Everything I mention seem intuitive and can explain General Relativity isn't it? Did not what I mentioned explained it clearly Fredrik? You said "I still haven't found a really satisfactory answer". Is not everything I mentioned above a satisfactory answer?

Those quotes aren't precise enough for me. It's great that they have informed you that that motion is represented by curves in spacetime and that the motion of a test particle in free fall is represented by a geodesic, but they don't make it perfectly clear what the theory says about results of experiments.

 

[bobc2]

Example:

We have a wonderful model of a 4-dimensional universe populated by 4-dimensional objects. At face value special relativity would seem to imply that those 4-D objects are just that--objects frozen in time so to speak--motionless. Yet, we always speak of observers moving along their world lines at the speed of light. If the objects, including 4-D spaghetti-like bundles of neurons extending millions of miles along the 4th dimension, are not actually moving as 4-D objects, then what is doing the moving? And what is the physical significance of the imaginary i that is often attached to the 4th dimension? And where are all of the observers really located at a particular instant of time (they don't share the same simultaneous 3-D cross-sections of the 4-D universe)? Is there some universal synchronized time for all consciousnesses? Or does a consciousness and "NOW" experience exist at all points along the 4-D world lines for every observer?

It's questions such as these that cause physicists (probably most of them) to consider those 4-D objects as mathematical constructs and not real physical 4-D objects. So, with these kinds of mathematical constructs (and we haven't even thrown Quantum Field Theory into the mix) how does mathematics connect to reality (unless, as someone has noted, you go along with Max Tegmark)?

That's why I suggested that at some point along the pursuit of reality rogerl may not find the physics forum a satisfactory place to continue his pursuit. Pursue it here for awhile, yes--but eventually the discussion is not appropriate for this forum (check the rules).