Why two mass attracts each other?

Undefined, are you familiar with the notion of a tangent ? Are you able to picture a tangent vector at some point on a surface ?
I am just asking because I am attempting to come up with a better, more intuitive way to show what curvature in space-time actually means. It isn't an easy thing to show, because I never really know what concepts other people are familiar with, and which ones not

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Getting outraged by a certain use of language, without considering what it refers to, is not educating anyone.

The difference is obvious and not an issue to anyone's understanding. I mean, what's your story here? You think the average physics undergraduate can learn calculus, differential equations, linear algebra, group theory, matrix Lie algebras, and more, but can't count to four? Is that really what you believe?

In a literal sense, no. In a more abstract sense commonly used in the Minkowski spacetime view, where worldlines are parameterised in terms of an affine parameter and we define things like the four velocity and four acceleration, it does.

Er, yes? And...? Is this supposed to be surprising or profound?

Yes, you look at space and you see space. Just like if you look north you'll see what lies north and you won't see south. Because that's what you asked for by definition. The spacetime view doesn't change that and doesn't imply otherwise.

As I've tried to explain, what they actually think is more nuanced than that and some of the language (e.g. stuff moving "through spacetime") is meant figuratively and is not intended as a reference to the literal (v = dx/dt) Newtonian definition of motion.

That is not an accurate restatement of how it works in general relativity. In the mainstream understanding of general relativity, light and other test particles follow geodesic worldlines in spacetime. Geodesics are the closest equivalent to straight lines on a curved Riemannian manifold and to say that light and other test particles follow geodesics is to say that they are behaving in the simplest way they possibly can behave.

 

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No, I was rather thinking about the size of all of the content if the GR outcome were to be a big crunch. I know that is not the current universe we are in, which features accelerating expansion. The question is off topic and trivial, and I did post a Googled answer right after I posted the question.

 

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This doesn't address what I asked you, and your explanation of the geodesic equation is inaccurate. General relativity is a theory about reality and not reality itself, so the "map" is, in this instance, exactly what we're supposed to be talking about.

Then why don't you act like it? It is possible for a theory to make predictions that apparently work on a qualitative level but don't work on a quantitative level. That is one of the reasons we care about quantitative detail in physics, and why you should too.

Since I never said that mathematics is empirical evidence, you are again not addressing what I actually said.

If you'd read what I'd actually said, it's a bit different: mathematics, if used correctly, is what makes a theory falsifiable on a quantitative level. Do you care about falsifiability? If so, you should care about the math.

A second reason we use mathematics that I'll add here is precision. If you skip the mathematical details and derivations of a theory, you are far more likely to end up with misconceptions about it than if you don't skip those details. Do you care about understanding a theory correctly and harbouring a minimum of misconceptions about it? If the answer is "yes" then, again, you should care about the math.

Yes, and? You've jumped on a use of language instead of considering the substance. The metric coefficients \(g_{\mu\nu}\) are not a simple analogue of the Newtonian gravitational potential. No such quantity fundamentally exists in the formulation of general relativity. The \(g_{\mu\nu}\)s aren't even invariant quantities. They're very heavily coordinate-dependent.

If you want to call the metric coefficients "gravitational potentials", then go ahead. But that doesn't make statements like "a clock at higher gravitational potential" generally meaningful or well-defined.

Why? Why should that give me useful or accurate information? Do you think atoms are actually indivisible just because the etymology of the word says so?

This doesn't answer my question. Let's pick a particular metric coefficient as a specific example: g[sub]01[/sub] (or g[sub]tx[/sub]). What does this coefficient represent physically? How do I measure it? If g[sub]01[/sub] = 0.2, what does that mean, and how is that different from g[sub]01[/sub] = 0 or g[sub]01[/sub] = -0.2?

That is the general expression of the space time interval in an arbitrary coordinate system. You could have copied that from anywhere, including some of my own previous posts. What does "knowing" this expression have to do with understanding it and where it comes from?

It wasn't relevant because it is normally taken for granted in special relativity that we can work in globally inertial frames, where the t and x coordinates really do represent seconds and metres everywhere. This is generally not possible in GR. This is why, when learning GR, one of the first things you learn is how to do special relativity in arbitrary coordinate systems. This is one of the first new things you would learn from Einstein's 1916 paper for example, if you fit the definition of his target audience (already knowing special relativity and Minkowski's framework, for example).

 

I am not putting words in your mouth. If anything, I am pointing out the conspicuous absence of certain words that should be in your mouth. Words acknowledging that Einstein's original formulation of general relativity, as given in his 1916 paper for instance, is based on the principle of general covariance which he develops mathematically. Words then responding to that in some meaningful way.

Well do you understand the mathematics of general relativity or not? If you don't, how can you be certain that understanding it isn't important to understanding the theory itself?

No, the fact that the phenomenon he is referring to is coordinate-dependent is what counters that. Also the fact that a variable coordinate speed of light appears in certain circumstances in special relativity (in accelerating and rotating reference frames for instance) and thus isn't actually new to general relativity.

As I said, the theory developed in Einstein's 1916 paper is very recognisable to me as the same one I learned in university.

I figured you'd say something like that, despite the fact I explain exactly why you didn't address my point in [POST=3068299]post #340[/POST], specifically here:

All you are proving is that you aren't capable of understanding a simple argument unless it is thoroughly idiot-proofed for you. I understand the difference between space and spacetime just fine. I don't think I should have to say so explicity, or explain why I am referring to one and not the other, in every damn response in every damn post I write to you. You are supposed to be intelligent and honest enough to be able to figure that out for yourself. Or is that giving you too much credit?

No, Einstein described space (more correctly he should have referred to spacetime, but so be it) as something, and threw the word aether at it.

What about it? It's a picture. What does that tell me about its correctness for example? In this instance your graphic depends on an arbitrary coordinate convention.

I could draw a picture of the rubber sheet analogy for instance. Doesn't mean it's a good or accurate depiction of general relativity.

I don't know why you are pointing this out to me. Yes, coordinates are artifices. The same is true of coordinate-dependent quantities such as the metric components, the coordinate speed of light, and the "curving" of light rays. You are the very person who needs to learn this the most.

 

This doesn't address the criticisms I made of that paper. Here's the sequence of events so far:

1. I alluded to the fact that I had previously criticised that paper, and linked to the post where I had done exactly that.
2. You ignored the criticism and instead quoted something else I said.
3. I reposted my criticism in its entirety in such a way that you couldn't possibly miss it.
4. You ignored it again with "the paper is what it is" and the results of a Google search, as if the popularity of an idea were in any way evidence of its validity.

So just who is clutching at straws here?

 

So it's OK for you to arrogantly assert you understand general relativity and nobody else does, but it's not OK for anyone else to say exactly the same thing back at you? What kind of hypocrisy is this?

Incidentally, you missed the point I was making in [POST=3067899]post #276[/POST], which is that this game of amateur psychologist you play with me is every bit as unscientific as your references to gravitational time dilation etc. as evidence for your position. You fail not only because it's an ad hominem but more importantly because you do nothing to rule out alternative explanations for the same observation. There may be more than one possible explanation for the observation of time dilation, just as there may be more than one possible explanation for my behaviour in this thread.

Well you really don't understand the math, and I really have addressed your arguments already. Regarding the math, Einstein included mathematical developments in, for instance, his 1916 paper and clearly intended that his readership should pay attention to them. You have never given a good explanation for why you saw fit to skip those parts, especially given that 1) Einstein made the effort to include them, and 2) everyone who does study them keeps telling you they are important to understanding the theory.

Regarding the fact I've addressed your [POST=3066616]post #158[/POST], I can do better than just assert I have. Post #158 fails at basic scientific rigour for the following reasons that you consistently fail to address:

• You rely too much on argument from authority and quote mining.
• The quotes you pull up often don't say what you attribute to them. For instance, there is nothing in the passage from the Leyden address about curved spacetime being equivalent to inhomogeneous space. Nor does it actually say that inhomogeneous space should be understood as the foundation of general relativity rather than merely an aspect or consequence of the theory.
• Where we nowadays disagree with or downplay something Einstein said, we generally have good reasons for doing so. For instance, the coordinate speed of light is, as its name suggests, a coordinate-dependent quantity. We even have a variable coordinate speed of light in special relativity in, for instance, accelerating and rotating reference frames, so this isn't even a new feature of general relativity.
• The evidence you cite in your favour, such as gravitational time dilation, is already quantitatively accounted for in the mainstream understanding of general relativity and, consequently, does not exclusively support you.
• You often mischaracterise the mainstream understanding of general relativity that you object to, for instance in saying "light curves because spacetime is curved" when in fact light follows geodesic trajectories which are the closest thing in a curved spacetime to a straight trajectory. The geodesic equation is the curved spacetime version of Newton's second law. It is only particular coordinate descriptions of the trajectory of light that has it "curved".

Your post #158 simply isn't as convincing or damning as you keep saying it is.

 

Space-time is a 'real-thing' or a 'pseudo-thing which helps in visualising the mathematics of GR'?

As per wiki space-time is just a "mathematical model". So, it may not be the real thing. Ref following wiki quote:

Many believe that in reality "time" does not exist. In reality only mass, energy, space and their relative motions exist. A clock is also a relative motion of mass or energy. So, combining 'space' and 'time as indicated by a local clock' is equivalent to combining 'space' and 'space with motion'. Ref following wiki quote:

 

General Relativity is a huge body of work, and is still evolving as more and more math is fitted into the scheme of things, at least that is a layman's perspective. To me, for any one person to fully grasp all that it consists of is a monumental feat. To some, it becomes more than just the math and geometry, it becomes the most plausible explanation of reality they can conceive of because it mathematically explains all of the angles and curves that we observe in motion from any perspective. When it becomes reality to a person, they lose a certain perspective themselves because the "how" of GR must still be uncovered.

Markus has acknowledged that, and though he is certain that geometry curves spacetime, from his response to me earlier about the tentativeness of all science, and the possibility of a quantum gravity unification of the inconsistencies between the major theories, I think he just doesn't want to acknowledge an alternative until there is some evidence or a preponderance of opinion, maybe

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He'll probably refute me, and in any case, you are making some good points which have to be considered.

 

I am not certain what you are getting at. All of physics are "mathematical models" - Newton, Maxwell, Schroedinger, Dirac, Hawking etc etc. Maths is just a langauge to describe what we see in the world around us. It is obvious that maths is not the same thing as reality, like the word "tree" is not the same thing as the tree itself, but that is not restricted to GR, this is true for all physics.

At the end of the day all we can do is extract predictions from our models, and compare those to what we actually observe. There is no other way to do physics.

So time does not exist, but motion does ? That is a self-contradictory statement.

 

I am not certain whether you are aware of this or not, but there actually are viable "alternatives" and extensions to GR in existence, in fact there is a good number of them. I am personally very interested in them as well, and include them in all my self-studies. For example, personally, I think it would make good sense to permit torsion in addition to curvature into the field equations, giving you Einstein-Cartan theory. That model is in perfect accordance with all observations; however, it makes additional predictions ( e.g. gravitational spin-orbit coupling, non-existence of singularities etc ) which have not been tested or observed, so at the moment I see no reason to abandon GR in favour of any other model. I will, however, be the first person to rejoice if it ever happens, because I am naturally curious

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I want to know whether "spacetime" is real like mass, energy, space and their relative motions or it is something imaginary(not real) but good for visualising mathematics of GR.

Maths deal with real things as well as imaginary things.

The word "tree" corresponds to the "real tree" itself and it does not correspond to the "model of a tree". Is it the same case with the word "space-time"?

Mathematics is good for making predictions. Visualisation is good for making an image to follow the mathematics. Whether this image corresponds to reality or not, that is the question.

A clock exists. "Time" is as indicated by the "motion" of a clock. "Time" is not same as "clock".

 

Hansda, here is a post by Dinosaur, from another thread, that I believe does a pretty good job of explaining what spacetime is. I believe it speaks to your question above better than anything I would or could put together.

 

Markus, we do experience all 4 dimensions, we just experience them in a classical and very Newtonian context.., from where we "are" right now. Time itself is an abstract concept or label describing our sequential experience of change. In your earlier analogy,

it is "us" riding that crack in the ice sheet. We perceive the changing 3D space around us, as we move along the crack...

This is a very classical and Newtonian experience, and is one of the root reasons why in these lay discussions, a full understanding of "spacetime" is difficult to convey.

Without the math, language is built around classical everyday experience. It works very well for descriptions of the Newtonian mechanics of everyday experience, but is not well suited to describing the conceptual dynamics involved in GR and spacetime. That's really where an understanding of the math fills in.., and adds definition.., to what is often otherwise interpreted within the context of our everyday classical experience.

The point is, we do perceive and experience all 4 dimensions of spacetime, we just do so in a classical manner. Without the time component—the perception of change—the other three dimensions, of space would have no meaning.

 

Consider the following wiki quote:

It may so happen that in "non-relativistic classical mechanics" space is static, that's why space and time are separated but in "relativistic contexts" space becomes dynamic and that's why "time" can not be separated from "space".

 

Yes, I pretty much agree with what you are saying.

 

Yes, that is actually one of my main points in this discussion. In GR you cannot separate time from space; in other words there is only space-time. Unfortunately a certain participant on this thread has chosen not to acknowledge this very simple concept.

 

He is also saying that space-time is a "model" and not "real".

 

Yes, just like all physics are "models" and not "real".

 

Don't mind hansda,

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he's got some sort of memory loss.

In one thread on Newton's Cradle, I would answer his question, then he would ask the exact same question five posts later.