Modes of Time

[Alceste]

I really have to object to this.

The experiments completely verify GR and the associated equations that describe the motion of light and matter under gravity.

Now please tell me where the curvature of spacetime enters into the equations of GR. Show me specifically where this occurs because when I studied this I saw nothing of the sort. The experiments verify GR – and contrary to popular belief GR has nothing to do with the curvature of spacetime. The rubber sheet analogy is great for conveying the effects (such as gravity waves), but terrible at representing what the equations actually say.

Maybe you would have been better off reading the wikipedia page than taking that course.

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916... It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the four-momentum (mass-energy and linear momentum) of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.

 

[Magic Man]

I really have to object to this.

The experiments completely verify GR and the associated equations that describe the motion of light and matter under gravity.

Now please tell me where the curvature of spacetime enters into the equations of GR. Show me specifically where this occurs because when I studied this I saw nothing of the sort. The experiments verify GR – and contrary to popular belief GR has nothing to do with the curvature of spacetime. The rubber sheet analogy is great for conveying the effects (such as gravity waves), but terrible at representing what the equations actually say.

I'm confused. I already brought up the fact that GR and spacetime curvature have nothing to do with each other, which you seem to support. Where did Spinkles mention even mention GR?

 

[Mr Spinkles]

I really have to object to this.

The experiments completely verify GR and the associated equations that describe the motion of light and matter under gravity.

Now please tell me where the curvature of spacetime enters into the equations of GR. Show me specifically where this occurs because when I studied this I saw nothing of the sort. The experiments verify GR – and contrary to popular belief GR has nothing to do with the curvature of spacetime. The rubber sheet analogy is great for conveying the effects (such as gravity waves), but terrible at representing what the equations actually say.

I have to admit, I am very, very surprised and confused by your question. You have taken a course in GR, and I haven't, so maybe I am about to embarrass myself.....so please correct me if I'm wrong here.....but everything I have ever heard of GR and special relativity is all about the geometry of space(time).

Describing the motions of anything requires you to specify the geometry of space, I don't see how you can calculate position or velocity or anything if you don't specify this (I suppose you could things like energies). This is true in everything, classical mechanics, electromagnetism, etc. however normally the space is "flat" so we don't always explicitly deal with this.

In other words if I measure the position of two points in space according to my chosen coordinate frame, how do I calculate the distance between the two points? In classical mechanics, and electromagnetic dynamics and Special Relativity this is encoded mathematically by the metric tensor. In a "flat" 3D space (the usual kind) the distance S we are interested in is calculated by the Pythagorean theorem, S^2 = X^2 + Y^2 + Z^2 where S is the distance from the origin (0,0,0) and X,Y,Z are the coordinates of the point in question, and in this case the metric tensor is just an all-zero matrix with 1's along the diagonal (I'm ignoring the time-coordinate). In a different space the metric tensor would be different, perhaps with off-diagonal elements and the same point at X,Y,Z would NOT be measured to be distance S from the origin, as it was in the "flat" space according to the Pythagorean theorem.

Now, taking a cursory look at the Wiki articles on GR, apparently the solutions to Einstein's equations are metrics. Same with this U. Cal. Riverside intro. to GR:

General relativity explains gravity as the curvature of spacetime. It's all about geometry. The basic equation of general relativity is called Einstein's equation. In units where

, it says

(1)
It looks simple, but what does it mean? Unfortunately, the beautiful geometrical meaning of this equation is a bit hard to find in most treatments of relativity. There are many nice popularizations that explain the philosophy behind relativity and the idea of curved spacetime, but most of them don't get around to explaining Einstein's equation and showing how to work out its consequences. There are also more technical introductions which explain Einstein's equation in detail -- but here the geometry is often hidden under piles of tensor calculus.
​

So I am at a complete loss as to why you are saying it has nothing to do with the geometry of space. As I understand, you are ultimately solving for that little "g" character with two subscripts *edit: which is related to the big "G" * the spacetime metric, which is a second-rank tensor (a matrix) with four columns (t,x,y,z). This little guy gave me some headaches in classical mechanics and showed up again in electrodynamics and special relativity. However, when I took those courses g was given by its value for an assumed flat spacetime.

In other words, given some physical scenario in which you make approximations about mass and energy, you use Einstein's equations to find the metric (or curvature) of space, and that lets you ultimately calculate things as simple as distances. And the metric of space is not flat in all but the simplest situations......the simple, practical consequence of this is you set up a coordinate system, you carefully measure the X,Y,Z position of some point, then you carefully measure its distance from the origin S, and you find the Pythagorean theorem is not accurate, S^2 does not = X^2 + Y^2 + Z^2. The same thing happens in curved 1D or 2D space, this is curved 3D space but it's difficult to picture it in your mind.

Honestly, I feel really awkward because I am supposed to be a physicist (*edit: someday anyway) and really know this stuff, but I have a gap in my education where I haven't studied GR, so please by all means tell me what it was you solved for/calculated when you studied it. At the UCR site they make some simplifying assumptions and then use Einstein's equation to derive the metric of flat spacetime, but of course flat spacetime is only the solution in that special/simple case.\

*edit: To be a little more specific: the Pythagorean theorem is a special case (flat space) of a more general case (non-flat space) : S^2 = G1*(X^2) + G2*(Y^2) + G3*(Z^2), in flat space G1 = 1 and G2 = 1 and G3 = 1, so you don't even write it out explicitly. But this is only true in flat space, that is, it is only true when the matrix g (the space metric) is all zeroes with only 1's along the diagonal. If this were always true (space is always flat) then Einstein's equation wouldn't make sense, G would simply be given by g (and I believe some tensor related to curvature) and would not depend on momentum or energy (contained in the tensor T).

 

[Mr Spinkles]

Question:
Is the singularity as described by the popular Big Bang model even a possibility when considered alongside what we materially empirically know of the universe we live in?

Yes.

Can something really materially be infinitely dense?

Your posts?

Sorry too tempting....

Okay in all seriousness, I am trying as hard as I can, I can think of no theoretical reason, or experimental evidence, that says something cannot be infinitely dense. Of course you could never prove something is infinitely dense even in principle, but the definition of a "particle" is something with some finite characteristics (charge, mass) and infinitely-small size, hence infinite density. The electron has been shown to be smaller than (insert ridiculously small number here) but you could always argue it might have some ridiculously small size to it. I am not aware of anything that is known to be made of finite-sized objects, all the fundamental particles are theoretically infinite and experimentally ridiculously small.

"Infinity" is a tricky concept but it is not magical or mysterious and anyone can understand it just like they can understand the concept of a negative number. I am going to tell you exactly how to understand the concept of "infinity" right now, seriously read this very carefully:

The key to understanding "infinity" is to realize that you could ask equally valid, equally difficult questions about all the other things you take for granted. To "understand" something is to know the definition and know the rules for manipulating it mathematically, and to be able to apply those rules to anticipate unknown outcomes, and to know the consequences of this for real physical situations. It also helps to bootstrap mental pictures and turns of phrase which are "accurate" as far as they go. If you can do these things, you "understand" in the only truly achievable sense of the word, even if you still feel uneasy. The unease will pass just like all those other concepts you take for granted....or it will increase for all of those other concepts!
​

Just think about negative numbers, for example. Can you ever truly, physically have a "less than zero" amount of something? How is that physically possible? I can imagine things like cars and dollars, but I can't imagine less than zero of them, except by using trickery. So I can imagine a Greed-colored bar above a line and a Red-colored bar below a line, and say the red bar is "negative" but that's just a mental game I'm playing to help my imagination, the red bar is just a representation, it is not the "negative thing" itself. Less than zero dollars is not truly imaginable as far as I know except by using tricks.

So you might say there is no such thing as a less-than-zero amount of something. Fine, but good luck trying to understand much of the physical universe. As we all know the concept of negative is mathematically well-defined and experimentally invaluable. We couldn't make sense of what we see when we look around in this crazy world without negative numbers. We have positive and negative charges on electrons and protons, for example. At the end of the day this is just the mathematically-rigorous and precise way of saying a "positively charged" thing attracts a "negatively charged" thing. And with "infinity" a similar story plays out, at the end of the day an "infinitely dense" thing is a particle. Negative numbers are just as good mathematically, and just as good as far as the experimental evidence is concerned, as the idea of positive numbers. The main drawback is that we can't *picture* it except by using tricks.

This is a very serious drawback, it might be that we are kidding ourselves about negative numbers and we missed some positives-only alternative to understanding the universe which would be much simpler and conceptually-pleasing. That's certainly one possibility.

Or, it could be that negative numbers "exist" in some sense but Darwinian evolution did not see fit to give us the capacity to picture them in our brains.

The third option, the one I like best, is that EVERYTHING we think about and imagine is just a tool we have invented (or inherited) to understand reality, including the regular concepts like flat 3D space and the steady flow of time. We ASSUME for simplicity's sake that these concepts "exist" in the real-world as well, and we drop this assumption when the experimental evidence forces us to. We PREFER intuitive explanations (like flat 3D space) whenever possible, and adopt non-intuitive explanations (curved space) when the facts force us to.

So what I'm saying here is not that infinities "exist" or "do not exist", I'm just saying you could ask the same annoying questions about irrational numbers, negative numbers, and on and on.

 

[Alceste]

I've never quite figured out why people seem to be uncomfortable with infinity. I'm much less comfortable with the finite. I once dated a guy who said something like "the human mind can't conceive of infinity". I told him he must be wrong about that, since mine can't conceive of anything but, and he wouldn't believe me. It didn't work out.

But is it really so difficult?

 

[themadhair]

You have taken a course in GR, and I haven't, so maybe I am about to embarrass myself.....

You’re not embarrassing yourself. We both are familiar with the same mathematical constructions and, if asked to crunch them, would derive the same results.

Describing the motions of anything requires you to specify the geometry of space,

I don’t believe this to be true. You have to specify the geometry of the phenomenon being described and not the medium through which it travels (something I’m not happy with calling ‘spacetime’ if I’m honest). Maybe I’m looking at this too much as a mathematician but, to me, I don’t see how the relevant metrics describing the phenomenon of gravity can be extrapolated to be a representation of spacetime.

I don't see how you can calculate position or velocity or anything if you don't specify this (I suppose you could things like energies).

But the axes we specify don’t necessarily correspond to reality. The four dimensional coordinates we use are purely for measurement – and any metrics are representation of phenomena’s with reference to that coordinate system. Notice that I didn’t mention the medium through which those phenomena travel, since mathematically that is not what is being represented by the equations.

Stuff on distance metrics.

A distance metric is our construction. To use your example of ‘flat’ space – the Pythagorean isn’t the only distance metric we could use. Any metric that satisfies the relevant conditions (symmetry, triangle inequality, positive definiteness, etc.) could be used. What determines the accuracy of a given metric is its agreement with the phenomena being measured – NOT necessarily the geometry of the medium.

For example, imagine a particular phenomenon was restricted to the surface of a sphere in 3D space. You could use a 2D coordinate system (ignoring the 0 and 360 problem) or a 3D coordinate system. In either case the particular metric used depends on the geometry of the phenomenon and not the medium (the medium being flat 3D space).

In other words, given some physical scenario in which you make approximations about mass and energy, you use Einstein's equations to find the metric (or curvature) of space, and that lets you ultimately calculate things as simple as distances.

Not strictly true this, let me try to explain. You seem to be implying that distance is a ‘fixed’ quantity which isn’t the case. In any given coordinate system there is a tremendous range of metrics that can be used to accurately map it. As long as the isomorphism between metrics hold (or in the case of 3D and 4D space the metrics all map perpendicular vectors to perpendicular vectors) then any such metric can be used. The determining factor for metric suitability is the phenomenon’s geometry and not the medium’s.

To use the same spherically restricted phenomena floating in 3D flat space I used earlier, consider the two distance metrics we have at out disposal. We could use the 2D coordinate system (which measures geodesics along the surface) and associated isomorphisms, or we could use the 3D metric. Which one of these metrics would most accurately map the phenomenon? The 2D one would be more accurate despite the medium being 3D.

Stuff about using distance measurements to verify curvature.

But, in experiments, we don’t actually measure distances in the way you do in the above description. Measuring the phenomena of increased mass due to increased velocity, for example, isn’t measuring a distance of spacetime but a distance of the phenomena’s geometry. The closest analogy to what you propose is taking distance measurements of the earth’s surface and verifying the curvature from that – but the experiments done to verify GR don’t do this and are measuring the phenomena rather that spacetime.

Honestly, I feel really awkward because I am supposed to be a physicist (*edit: someday anyway) and really know this stuff, but I have a gap in my education where I haven't studied GR, so please by all means tell me what it was you solved for/calculated when you studied it.

The extent of my GR course was deriving the Schwarzschild solution after deriving the Einstein Field equation. If it helps I hate tensor calculus as much as you do. As I have said, I don’t see how these equations, which describe phenomena, can be extrapolated to be representations of spacetime.

 

[Mr Spinkles]

The definition of spacetime is the set of points t,x,y,z.
The definition of flat spacetime is a metric in which the Pythagorean theorem is satisfied.

(This of course encapsulates all the coordinate frames we are normally familiar with, such as XYZ coordinates or spherical coordinates. Usually we assume a flat metric and then choose any of these coordinate frames at our convenience, because it's only distances that are physically significant, and all observers in these frames agree distances between points in space are conserved by translation, i.e. the Pythagorean theorem is true).

The definition of non-flat spacetime is a metric in which the Pythagorean theorem is not satisfied.
The solutions to Einstein's equation are metrics which may or may not satisfy the Pythagorean theorem.

So Einstein's equation says spacetime in general is not flat.

Where am I going wrong?

 

[Magic Man]

I've never quite figured out why people seem to be uncomfortable with infinity. I'm much less comfortable with the finite. I once dated a guy who said something like "the human mind can't conceive of infinity". I told him he must be wrong about that, since mine can't conceive of anything but, and he wouldn't believe me. It didn't work out.

But is it really so difficult?

Actually, yes. I think that only proves that you aren't human, but that probably comes as no surprise to you (I know it doesn't to me).

Infinity or eternity is not something our minds can really grasp. It's more than just a really long time. Trying to think about living in Heaven for eternity used to scare me and I couldn't fully understand it. I have to say that I do find it hard to believe that you can actually picture or wrap your head around infinity.

 

[Mr Spinkles]

The four dimensional coordinates we use are purely for measurement – and any metrics are representation of phenomena’s with reference to that coordinate system.

Are you sure you aren't confusing "metric" with "coordinate system"?

 

[themadhair]

Where am I going wrong?

I think you are wrong when you claim that Einstein’s equations are representations of spacetime rather than representations of phenomena with reference to a given coordinate system. We don’t use the equations of GR to map spacetime, we use them to calculate phenomena with reference to a given coordinate system.

When I used the example of a phenomenon restricted to a spherical surface in a 3D space I hoped to illustrate that the metric used to map the phenomenon would fail the Pythagorean test you propose without the encapsulating spacetime necessarily failing that same test. To make the claim that GR’s equations are mapping spacetime really doesn’t follow. Mapping the interaction of light and matter – yes. Mapping spacetime – not that I can see.

Are you sure you aren't confusing "metric" with "coordinate system"?

I’m pretty sure what I wrote is correct. Different coordinate systems give different metrics though, but these metrics should be isomorphic to each other. In GR the metric is not determined by spacetime – but by the interaction of light and matter.

 

[Arkwort]

Where as I have control over moving in space (I can stand still or move [left, right, forward, backward, up, down, or a combination]),
I haven't control over moving in time (I always move forward in time - I can't stop time).

If however I fell out of a plane I'd fall down to earth.
So is there a big mass at the end of time that I'm falling towards at a constant speed?
And why can't I look back or forward along the time dimension (has it got something to do with the speed of light)?

Arkwort :sleep:

 

[Alceste]

Actually, yes. I think that only proves that you aren't human, but that probably comes as no surprise to you (I know it doesn't to me).

Infinity or eternity is not something our minds can really grasp. It's more than just a really long time. Trying to think about living in Heaven for eternity used to scare me and I couldn't fully understand it. I have to say that I do find it hard to believe that you can actually picture or wrap your head around infinity.

The trick is to grasp it on its own without trying to attach it to anything, like time, or space, or a piece of string, or whatever. I mean, I can't literally form a mental picture an infinite number of ping pong balls, but I can easily conceive of the ping pong balls beyond the fuzzy edges of my ability to imagine going on and on forever.

When I say it's difficult to imagine anything but infinity, I mean I can't grasp the concept of beginnings and endings. In my mind there is always something before the "beginning" and after the "end" - those points seem to me to be arbitrarily selected on an infinite timeline. I even think of the "Big Bang" as a point in an eternal, cyclical story, as if the universe is breathing (expand, contract, expand, contract, ad infinitum). If it is born and dies, then it's born of another universe and feeds into yet another universe after its death. And at the "edge" of it (if there is an "edge") there must be yet another universe, or perhaps the opposite edge of our own again, or something.

It's much harder for me to conceive of nothing. I don't mean "space", I mean nothing. To get a grip on beginnings and endings, you need to be able to believe that nothing can actually exist, since that's what lies outside the parameters of our edges and beginnings and endings. Most people picture the "nothing" before the Big Bang as empty space and endless time, right? But that's wrong - that's not "nothing". That's empty space and time.

Could be I'm not human any more though. I ate a lot of strange stuff I found out in the fields. I've probably turned into some kind of Pixie.

 

[Kilgore Trout]

Could be I'm not human any more though. I ate a lot of strange stuff I found out in the fields. I've probably turned into some kind of Pixie.

As long as you haven't turned into Frank Black, you should be fine.

 

[Magic Man]

It's much harder for me to conceive of nothing. I don't mean "space", I mean nothing. To get a grip on beginnings and endings, you need to be able to believe that nothing can actually exist, since that's what lies outside the parameters of our edges and beginnings and endings. Most people picture the "nothing" before the Big Bang as empty space and endless time, right? But that's wrong - that's not "nothing". That's empty space and time.

Oh, I'm with you there. "Nothing" is in the same class as infinity. I still can't imagine how you can actually picture or grasp infinity or eternity.

 

[Alceste]

Oh, I'm with you there. "Nothing" is in the same class as infinity. I still can't imagine how you can actually picture or grasp infinity or eternity.

"The infinite" is the default setting for someone who can't picture "nothing", and vice versa. If you can't grasp either, then what have you got?

 

[Magic Man]

"The infinite" is the default setting for someone who can't picture "nothing", and vice versa. If you can't grasp either, then what have you got?

I disagree. We're programmed to think in terms of "something", meaning not nothing and not infinity or eternity.

 

[Alceste]

I disagree. We're programmed to think in terms of "something", meaning not nothing and not infinity or eternity.

If you take your "something", and in your own mind it goes on forever without ever bumping up against a "nothing", what have you got? I say infinity / eternity - wouldn't you agree? I also tend to think of individual, seemingly limited "somethings" as little pieces of the infinite "something" - I can't see the trees for the forest, as they say.

 

[Magic Man]

If you take your "something", and in your own mind it goes on forever without ever bumping up against a "nothing", what have you got? I say infinity / eternity - wouldn't you agree? I also tend to think of individual, seemingly limited "somethings" as little pieces of the infinite "something" - I can't see the trees for the forest, as they say.

But that's the thing. It's too hard to imagine it going on forever. I can imagine things with finite shapes and sizes, but not things that have no shape or size, i.e. "nothing", or things that have infinite size.

 

[Alceste]

But that's the thing. It's too hard to imagine it going on forever. I can imagine things with finite shapes and sizes, but not things that have no shape or size, i.e. "nothing", or things that have infinite size.

Fair enough. I believe you - you have trouble imagining infinite things or nothings. Do you believe me - that I have no trouble with "infinite", but can't do "finite"? I figure we are each the ultimate authorities on the workings of our own minds.

I just finished Pratchett's "Hat Full of Sky" - there's a character in it (Granny Weatherwax) who says that when somebody tells her not to think of a pink rhinocerous, she has no trouble at all not thinking of one. Eventually the main character (Tiffany) deduces it's because she doesn't know what a rhinocerous looks like. So maybe it's not something extra I've got, but something missing - namely, the concept of things being finite. (Or even the concept of things being "things").

 

[Mr Spinkles]

I think you are wrong when you claim that Einstein’s equations are representations of spacetime rather than representations of phenomena with reference to a given coordinate system. We don’t use the equations of GR to map spacetime, we use them to calculate phenomena with reference to a given coordinate system.

Calculate phenomena.....like what? The Schwarzschild solution to Einstein's equation you mentioned is a metric. A metric of spacetime. And it's not flat no matter what coordinate system you choose to represent it in. Schwarzschild Black Hole -- from Eric Weisstein's World of Physics

When I used the example of a phenomenon restricted to a spherical surface in a 3D space I hoped to illustrate that the metric used to map the phenomenon would fail the Pythagorean test you propose without the encapsulating spacetime necessarily failing that same test.

Okay so in your example, there is this "encapsulating spacetime" above and beyond the surface of the sphere. The sphere is just a curved 2D space restricted in a higher, flat 3D space. Okay. But the space we live in, which is curved, is a 3D (technically 4D) space. Its curvature is specified by the spacetime metric tensor, g (which is independent of the coordinate system you choose--see below) which essentially gives us rules for calculating distances and so forth between events A and B, which are points in spacetime by definition. Now, you can argue that, actually, there is some higher space that is the "real" spacetime, and it is indeed flat.....but if it is above and beyond the 3 spatial and 1 time coordinate physicists know and love, and we can't measure it, its very existence is mere speculation, much less its flatness vs. its curvature. Meanwhile, we do need words to describe the 4D space we can actually measure and that word is "spacetime" and it is a curved space.

I’m pretty sure what I wrote is correct. Different coordinate systems give different metrics though, but these metrics should be isomorphic to each other. In GR the metric is not determined by spacetime – but by the interaction of light and matter.

No no, the metric metric in tensor form is an abstract object, just like a vector in its abstract form, it is its own, independent thing and unaffected no matter what coordinate system you choose. Like a vector, only its representation changes when you express it explicitly in terms of a chosen coordinate system. The abstract metric tells you the curvature of space, the coordinate system is just any convenient way of measuring points in that space.

So for example, the representation of the flat metric in Cartesian t,x,y,z coordinates is a 4x4 zero matrix with -1,1,1,1 along the diagonal. The representation of the same metric in spherical t,r,phi,theta coordinates is different but "the metric" is the same in both cases, only its representation is different. In both cases, given the flat metric you will always find that (spatial) lengths are conserved by translation (for example) no matter what convenient coordinate system you choose. That is a flat space by definition, if the metric is different and lengths aren't conserved by translation, it is a curved space, by definition. Spherical or cylindrical or any other kind of curved coordinate axes have nothing to do with the curvature of the space one is talking about.


*edit: The other thing I wanted to say:

You say we're measuring "phenomena" and the "interaction between mass and light", not spacetime. Okay, I see what you are saying. Sure, you could say that light and matter follow these weird paths around each other due to an interaction between them, and this whole business of fooling around with "spacetime" is just a mathematical trick for capturing this interaction. However, you could say the same thing about light itself. "Light" is just a propagating wave of the electromagnetic field. And what does the EM field really tell us? Well, what it really says is how charges interact with each other. So you could say there really is no such thing as the EM field, and therefore no light, it is all just charges interacting with each other, and the mathematics of the EM field is just a trick to capture this interaction. This was in fact the position of many physicists before the discovery of propagating EM waves. Even now, there is still nothing "wrong" with this picture, but there are drawbacks: 1) Impossibly complicated interactions are vastly simplified by considering the EM field to be a "real" thing; 2) In some cases things like pressure, energy, momentum, etc. would seem to disappear and reappear during the interaction....where did the energy go? To preserve conservation of energy, etc. you have to consider that missing energy being stored "in" the EM field itself (transported by propagating waves). So there's nothing "wrong" with either picture as far as I know (charges affect the EM field affects other charges, VS. charges just affect each other directly). A similar story plays itself out, I believe, with GR. There's nothing necessarily wrong with either picture (mass affects spacetime affects light VS. mass affects light directly). But one picture is useful and predictive and the other is not. The real clincher may be gravity wave detectors, which are trying to detect propagating waves of spacetime, which are predicted by GR, just as 19th century physicists detected propagating waves of the EM field (radio waves, etc) which was predicted by Maxwell's equations.....I made my own momentous contribution to this effort during an internship when I successfully demonstrated the inferiority of a newly-proposed signal recognition algorithm.