Metric Tensor: What is it and how does it relate to Riemannian geometry?

[Dale]

in the spirit of the OP, what in the heck is length contraction and time dilation? What in the heck is going on, what mechanism can I picture that, allows/requires, the Minkowski terms of x,y,z, and t to automatically jigger themselves, talk to each other, trade increments, optimize around their Pythagorean norm ? or c? when moving from one Minkowski coordinate to another? I have no imago for this process and I can't for the life of me get what it implies (that moving objects carry around strictly different but automatically related clocks and rulers). It seems like it must be something going on that isn't nothing. And I feel like everyone is saying "nothing changes". Which really leaves me confused.

So I will ask you a very closely related question about Euclidean geometry. Hopefully it will give you a feel for the source of your confusion. First, let's set up the scenario:

I have a flat piece of paper and two identically constructed rulers (for simplicity let's make it 6" by 8"). On the piece of paper, I have two bold lines. The first is a black line that goes from the center of the bottom to the center of the top of the page. The other line is a red line which goes from the bottom left corner to the top right corner. On the page there are also two sets of 1" grid lines, one set is gray and it is parallel and perpendicular to the black line. The other set is pink and it is parallel and perpendicular to the red line.

I place one ruler along the black line and the other ruler along the red line. I note that on the black ruler the gray grid lines are separated by 1", but the pink grid lines are separated by 1.25". So on the gray grid the black ruler marks off grid lines correctly, but the red ruler is "length contracted". Similarly, I note that on the red ruler the pink grid lines are separated by 1", but the gray grid lines are separated by 1.25". So on the pink grid the red ruler marks off grid lines correctly, but the black ruler is "length contracted".

So what the heck is this, is it Euclidean length contraction? What mechanism has automatically jiggered the rulers so that they talk to each other, trade increments, and optimize around their Pythagorean norm? It seems to imply that different lines carry around strictly different but automatically related rulers. It seems like it must be something going on that isn't nothing.

 

[FactChecker]

Then things move along to gravity, and I start to lose the plot.

Gravity is another can of worms. I wouldn't tackle that yet if I were you

But a clock - any clock at all - is just a device that measures time, not time itself. ... pouring a cup of tea should not take longer to fill the cup than a person making tea and viewing the moving clock from earth.

You are not the only one who thought this. But what if you make a clock that measures time by how long it takes to pour tea? If that slows down, what then? No matter how hard scientists tried, they could not build a clock that measured the speed of light differently on the moving earth. Einstein's conclusion was that everything slows down and there is nothing that can measure the speed of light differently. And the math works out. To an outside observer of a very fast moving rocket ship, everything in the rocket ship is in slow motion. Their time has slowed down. But the people on the ship don't know it because everything in the rocket has slowed equally and looks normal to them.

 

[georgir]

In other words, in the spirit of the OP, what in the heck is length contraction and time dilation? What in the heck is going on, what mechanism can I picture that, allows/requires, the Minkowski terms of x,y,z, and t to automatically jigger themselves, talk to each other, trade increments, optimize around their Pythagorean norm ? or c? when moving from one Minkowski coordinate to another? I have no imago for this process and I can't for the life of me get what it implies (that moving objects carry around strictly different but automatically related clocks and rulers). It seems like it must be something going on that isn't nothing. And I feel like everyone is saying "nothing changes". Which really leaves me confused.

Well, I already mentioned this analogy but I can repeat it here with more details.
Imagine that you have two lines parallel to each other, and you want to measure the horizontal distance between them. What happens if your definition of "horizontal" changes? The distance you measure changes, without the lines themselves changing...
Lorentz transforms are just like rotating your definiton of "horizontal" (simultaneous). Your reference frame - the imaginery grid you use for measurements - changes and gives you different results. The measured objects stay the same.

The analogy with Euclidean space and rotation is not perfect of course... in it, you will measure the shortest distance between two lines when your "horizontal" is perpendicular to them. At all other angles, you get longer distances. In SR, it is the opposite - the distance is longest when the worldlines are perpendicular to the line of simultaneity (objects appear at rest) and shortens when they are at an angle (in motion). This is due to the non-euclidean nature of the time coordinate.

Similarly for time dilation - instead of measuring "horizontal" distance, you are measuring "vertical", and it changes depending on how you rotate your reference frame. It is shortest when the line between the two events is perpendicular, and gets longer at an angle, again opposite of what you can get from Euclidean space rotation.

Essentially, the measurements we make are a projection onto an axis, and the diagrams we can draw are like a projection onto a plane. A transformation that is very much like a rotation of the original space ends up appearing like a skewing in the projected space.

 

[harrylin]

[..] By what mechanism is the rate of aging changed in a person who gets accelerated? Or, the dilated 'clock' exactly how does it "go slower''. When acceleration occurs to something real, is something real happening to that thing or not? If not, then... I'm really really confused. If so, then what is it?
[..] what I really am trying to understand, or get a picture of is what in the heck can be happening to space and time as things, or processes (geometry?), in the transition from Newtonian space, which I understand it is flat Euclidean Space in the environment of invariant time - to the Minkowski space-time where only the relationship between of space and time is invariant, leading to length contraction and time dilation from acceleration or relative velocity.

In other words, in the spirit of the OP, what in the heck is length contraction and time dilation? What in the heck is going on, what mechanism can I picture that, allows/requires, the Minkowski terms of x,y,z, and t to automatically jigger themselves, talk to each other, trade increments, optimize around their Pythagorean norm ? or c? when moving from one Minkowski coordinate to another? I have no imago for this process and I can't for the life of me get what it implies (that moving objects carry around strictly different but automatically related clocks and rulers). It seems like it must be something going on that isn't nothing. And I feel like everyone is saying "nothing changes".

You can just as well find papers that claim that length contraction is "real". The problem here is that, similar to quantum mechanics, special relativity itself is based on physical principles about observations and there are at least two competing interpretations of "what really happens". Those have been present from the very start and for sure nobody can prove who is right so that it is a cause of endless (and after a while fruitless) debate. A summary posting based on the discussions on this forum is here:
https://www.physicsforums.com/threads/what-is-the-pfs-policy-on-lorentz-ether-theory-and-block-universe.772224/
You can find back some of the discussions by searching this forum for "block universe".

On a positive note, you can check out those models and if after reflection at least one of them makes sense to you, then you have found a mechanism that you can picture. :)

 

[Jimster41]

Thanks harrylin, looking around is good advice. I've only visited a couple of forums on this site, but I started out lurking in the "Beyond the Standard Model". It's really great. And it's largely about Quantum Theories of Gravity and Space-Time - whether and how the ruler really gets squished because it traded some millimeters for seconds - that it owed the clock, or whether it's just the angle we are holding the graph paper (I think the two cases are indistinguishable - and that's shocking). I came over here Iguess because I needed a gut check on whether I really had a usable understanding of SR and GR. Glad I did.

Along your lines of papers, one Mentor dude there 'Marcus' on that forum has this really great style of collecting links to current research papers discussing that subject, which IMHO underlies what this thread has, maybe inadvertently been about. My understanding of that fascinating discussion is that there is currently no complete theory that explains how spatial dimensions of space-time geometry and the time dimension of space-time geometry "flow together" dynamically, in transformations. They know how to calculate what happens beyond all necessary precision, but Quantum Mechanically no one knows precisely just yet, how Gravity, how the full dynamics of space-time work.

Here's a link to a new thread he just started on things to watch for in 2015.
https://www.physicsforums.com/threads/six-themes-for-qg-in-2015-developments-to-watch-for.791643/

I have downloaded and printed a number of the papers he lists. I stare at them on the train and try to get through the first calculation chain. Marcus does a great job of organizing them in such a way that you can sort of hear the discussion at the level of the "abstracts" at the start of each papers, and he comments on them to aid understanding. I think only a tiny number of people in the world can actually really follow them well. But there is some great back and forth in the forum (from fans in the seats, some very knowledgeable). I may have embarrassed myself a couple of times. But they are good at ignoring you if you aren't asking good questions, or if they have no idea what the answer is I suppose. Most of mine were not good questions. But I did learn about "Pachner Moves" and I do have a cartoon of space-time as Coral Reef that is growing through the assembly of Tetrahedrons via Pachner Moves - a scheme related to the Energetic Causal Set and/or Spin Foam models of quantum space-time. There is literally a movie of Pachner moves with tets that someone did in one thread. Very cool. However, I wish it meant more to me...

I'm not trying to say that's a good way to learn, especially not if you have other options. But there's no hard rule against it (is there?)... and for some, for me at least, it's important and fun. Better than giving up... better than coming home from work and working on problem sets. ;-).

One thing I have fantasized about - for people like me who need to, want to learn, in context - and are coming over from Natural Philosophy and are just not adept at Mathematical Physics. I would love for someone to invent an on-line document viewer that has contextual wiki-ness around mathematical symbols, equation speciation and history. So if you are staring at a hard core symbolic expression, and you have a general idea of how it works, or what it's trying to be precise about conceptually, but are only weakly familiar with some symbol, or understand all of it but one important symbol, you can just hover over it and pop up a snippet of mathematical dictionary/encyclopedia. Even if it was just an annotation of two or three of the main statements in cutting edge theory... I for one, would have such a better view of the game. But that's work to build, and who would pay (Maybe the National Science Foundation, I don't know).

Anyway, back to trying to sharpen my bowling ball understand of SR.

Well, I already mentioned this analogy but I can repeat it here with more details.
Imagine that you have two lines parallel to each other, and you want to measure the horizontal distance between them. What happens if your definition of "horizontal" changes? The distance you measure changes, without the lines themselves changing...
Lorentz transforms are just like rotating your definiton of "horizontal" (simultaneous). Your reference frame - the imaginery grid you use for measurements - changes and gives you different results. The measured objects stay the same.

The analogy with Euclidean space and rotation is not perfect of course... in it, you will measure the shortest distance between two lines when your "horizontal" is perpendicular to them. At all other angles, you get longer distances. In SR, it is the opposite - the distance is longest when the worldlines are perpendicular to the line of simultaneity (objects appear at rest) and shortens when they are at an angle (in motion). This is due to the non-euclidean nature of the time coordinate.

Similarly for time dilation - instead of measuring "horizontal" distance, you are measuring "vertical", and it changes depending on how you rotate your reference frame. It is shortest when the line between the two events is perpendicular, and gets longer at an angle, again opposite of what you can get from Euclidean space rotation.

Essentially, the measurements we make are a projection onto an axis, and the diagrams we can draw are like a projection onto a plane. A transformation that is very much like a rotation of the original space ends up appearing like a skewing in the projected space.

A couple of questions about your example here.
I follow when you say that a rotation "causes no change" - Is it precise to say that the Minkowski metric is invariant under rotational transforms? Is it correct to say that the Minkowski Metric is invariant under all Lorentz transforms? I want to say that. But is it correct?

When you say "non-euclidean nature" do you mean the minus sign. I was wondering whether it is precise to say that the Minkowski Metric is or isn't Euclidean, and/or Pythagorean. I certainly looks like Pythagoras, but as I think you are highlighting there is that minus sign. Is there any other sense in which it can't be considered Euclidean

 

[Jimster41]

I just checked out DaleSpam's FAQ on Block Universe vs. Lorentz Ether philosophy.
I can understand the need for some discipline in a forums like this and my own crackpot-ness is a pretty hard problem for myself, with respect to myself. So I can sympathize with the challenge the moderators and mentors face here. It's very much appreciated for my part, the work they do.

I want to respect the rules of this Forum (SR and GR) and I will refrain from further discussing theories of Quantum Gravity. The test-ability at quantum scales problem is real as understand it. Though I was under the very vague impression that efforts to conceive of experiments to get more information are bearing fruit, as the theories are actively developed, but I may be pretty wrong about that. And there are forums for that subject. I got involved here because I felt the OP was in need of some support for what I felt was a good question, or a good reaction to the strangeness of the implications of SR... (that time and space are part of the same thing, whatever it is and it's hard to imagine how they could possible mingle) and her

But a clock - any clock at all - is just a device that measures time, not time itself. To me time is the way that humans experience events occurring in sequence.

A person standing next to the clock on a horizontally moving spaceship and pouring a cup of tea should not take longer to fill the cup than a person making tea and viewing the moving clock from earth. The light beam may have to move longer between bounces but the tea should not take longer to hit china just because it's traveling horizontally, regardless of where you're seeing it from.

"... events occurring in sequence" metaphor sounded a bit like an Energetic Causal Set description of QG to me - one that is speculated about by professionals.

 

[georgir]

I follow when you say that a rotation "causes no change" - Is it precise to say that the Minkowski metric is invariant under rotational transforms? Is it correct to say that the Minkowski Metric is invariant under all Lorentz transforms? I want to say that. But is it correct?

Indeed, when a quantity is unchanged by a transform, it is called invariant. Pythagorean distance (Euclidean metric) is rotation and translation invariant. Spacetime interval (Minkowski metric) is translation-invariant and Lorentz-invariant. But it is not rotation-invariant. If you look at a subset of the rotations around planes parallel to the time axis, then yes, it is invariant to those. 4D rotations are a messy business :p Note that this subset of rotations is included in the general meaning of the term "Lorentz transformation", even though introductory materials focus on a single spatial dimension where you can only have rotation-free transformations, or "boosts".

When you say "non-euclidean nature" do you mean the minus sign. I was wondering whether it is precise to say that the Minkowski Metric is or isn't Euclidean, and/or Pythagorean. I certainly looks like Pythagoras, but as I think you are highlighting there is that minus sign. Is there any other sense in which it can't be considered Euclidean

The Minkowski metric is not Euclidean, precisely because of that minus sign. The specific coefficients in the two metrics are the very definitions of those two terms.
http://mathworld.wolfram.com/MetricTensor.html

 

[Jimster41]

That is a great page.

The Minkowski metric is not Euclidean, precisely because of that minus sign. The specific coefficients in the two metrics are the very definitions of those two terms.
http://mathworld.wolfram.com/MetricTensor.html

That's a great page. Hard, but not so hard that I'm bouncing off it. Thanks. This was especially helpful since references to "Riemann" geometry have been hitting me in the head for awhile."When defined as a differentiable inner product of every tangent space of a differentiable manifold [PLAIN]http://mathworld.wolfram.com/images/equations/MetricTensor/Inline35.gif, the inner product associated to a metric tensor is most often assumed to be symmetric, non-degenerate, and bilinear, i.e., it is most often assumed to take two vectors http://mathworld.wolfram.com/images/equations/MetricTensor/Inline36.gif as arguments and to produce a real number

such that

Associative

Distributive

Distributive

Commutative (So commutative is about sequence?...)
Note, however, that the inner product need not be positive definite, i.e., the condition

with equality if and only if

need not always be satisfied. When the metric tensor is positive definite, it is called a Riemannian metric or, more precisely, a weak Riemannian metric; otherwise, it is called non-Riemannian, (weak) pseudo-Riemannian, orsemi-Riemannian, though the latter two terms are sometimes used differently in different contexts. The simplest example of a Riemannian metric is the Euclidean metric

discussed above; the simplest example of a non-Riemannian metric is the Minkowski metric of special relativity"

I notice that I can copy paste right from it into this...kindof sweet. I learned the algebraic properties at one point like every 10th grader, but man talk about a waste of time trying to teach me that then... Now I'm interested. Now it's all algebraic notation I guess. Very disappointing to know the math I learned has been--- sort of superseded. Not that I really learned it.