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

[ghwellsjr]

That is like saying that you need another beer in the morning to resolve a hangover.

That type of "paradox" is the "hangover" caused by the pedagogical overindulgence in reference frames. It is better to avoid it altogether, IMO.

Yes, but I doubt that you're going to succeed at that and if you did, things would be awfully boring around here.

 

[Dale]

I doubt that you're going to succeed at that

Me too.

 

[Jimster41]

I think I'm getting a better understanding! You guys need more beer!

I think I got into trouble here because this is more of a classroom than I thought, and you all are heads down on the difficult calculus of SR. And being just a dopey Natural Philosopher who doesn't actually need to pass a test in this stuff (thank the lord, because I could not). Frankly I was more interested in connecting with the OP around the thrilling and bizarre implications of SR (which I might add I have fought long and hard to try and grasp). I thought she (the original OP) was expressing an emotional reaction to the ontology of the theory and the idea of time dilation... as well she should IMHO.

That whining aside, I'll also humbly accept that the most important thing to me here is to come away with a more correct understanding. And if I am really picturing something so incorrectly that it is giving me false pleasure to imagine it... then I would absolutely rather avoid that hangover.

So if I could - Take the road example. If I draw an x-y plane on graph paper and the x-axis is some number of graph-paper boxes long (in reference to the road's length) and the y-axis is time, and some number of graph-paper boxes long. Then I accelerate the road. What happens to my x and y axes? More importantly what happens to my road?

I can imagine one answer might be that If I stay on the road, nothing. But if I am standing there looking at the road as it zooms off, something.
I think I'm also worried though about what is actually happening to the road even if as I stand on it while it zooms away, nothing seems to be.

I know I would feel a force right, my inert mass being dragged into acceleration?

 

[pervect]

That whining aside, I'll also humbly accept that the most important thing to me here is to come away with a more correct understanding. And if I am really picturing something so incorrectly that it is giving me false pleasure to imagine it... then I would absolutely rather avoid that hangover.

So if I could - Take the road example. If I draw an x-y plane on graph paper and the x-axis is some number of graph-paper boxes long (in reference to the road's length) and the y-axis is time, and some number of graph-paper boxes long. Then I accelerate the road. What happens to my x and y axes? More importantly what happens to my road?

It rather depends on the details of how you accelerate the road. So it's better to accelerate a pointlike observer than a road, because it's easier to specify what accelerating a point means than accelerating a road.. And it's even better not to worry about exactly how you accelerate an observer, but talk about the experiences of two observers with different velocities, and to describe the experience of each observer as they compare observations rather than to worry about how they accelerate.

So if you accept this approach and want to follow it out in more detail, I can point you at some references that might improve your understanding. If you want to do things "your way", well I don't think you'll be improving your understanding much.

 

[Dale]

Take the road example. If I draw an x-y plane on graph paper and the x-axis is some number of graph-paper boxes long (in reference to the road's length) and the y-axis is time, and some number of graph-paper boxes long. Then I accelerate the road. What happens to my x and y axes? More importantly what happens to my road?

I can imagine one answer might be that If I stay on the road, nothing. But if I am standing there looking at the road as it zooms off, something.
I think I'm also worried though about what is actually happening to the road even if as I stand on it while it zooms away, nothing seems to be.

I know I would feel a force right, my inert mass being dragged into acceleration?

I am not sure what you mean exactly. Your description of drawing on the graph paper is excellent. What you describe is called a spacetime diagram. It is one of the most important tools in relativity.

However you lost me as soon as you talked about accelerating the road. Nothing you do in your spacetime diagram will have any physically measurable effect on the road. So I am not sure what you intended to describe.

All I can think of to say is to explain that in relativity there are two different concepts of acceleration. The first is called proper acceleration. It is the physical acceleration measured by an accelerometer. The second is called coordinate acceleration. It is the change in coordinate velocity over coordinate time.

Proper acceleration is a coordinate independent thing. All reference frames agree. Coordinate acceleration obviously depends on your coordinates and has no physical consequences.

 

[Jimster41]

As I think about how to get my question out, I realize I am falling down into the question of just exactly how an accelerometer works. I know it registers g-force but what the heck causes G-force? Inertia like a damped spring. Ok, what is dampening the spring? it's mass. But what gives it mass? The gravitational field. But what is that? Space-time curvature. The geometry of --- space and time you say? But what cause that geometry to curve or whatever? Mass and/or acceleration. Oh, but I thought stuff got mass from gravitational field? Acceleration though, now that's interesting. How does acceleration curve the gravitational field? I mean how does acceleration mess with space-time geometry?

Or maybe along the lines of the original post. 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?

 

[Dale]

How does acceleration curve the gravitational field? I mean how does acceleration mess with space-time geometry?

It doesn't mess with the spacetime geometry at all. What proper acceleration does is curve the world line of the accelerating object.

Think back to the spacetime diagram. You had position on one axis and time on the other. So if you have a "point" object then it would have a position along the road at each moment of time. In other words, it would trace out a line on your spacetime diagram. This is called a worldline.

An accelerometer attached to the object would read 0 everywhere that the world line was straight. The proper acceleration would be nonzero everywhere the world line was bent, with the accelerometer reading being inversely proportional to the radius of curvature.

 

[Jimster41]

I think I'm definitely puzzling over the case where The accelerometer reads non zero. So if I picture the twin carrying the clock at the origin and drive them up and out to the right curving until they are moving at a 45 deg angle. Is that the "null" line? speed of light. Does the clock go slower? The twin age less? Than if they had stayed in straight vertical line up from the origin and their accelerometer had stayed at zero the whole time?

Just an aside, I also realize that many posts ago I was confusing an "accelerometer" attached to one clock carrying twin and a "tensiometer" stretched between an accelerated and non accelerated clock carrying twin.

I guess I would have said that "curving the world line" was messing with spacetime geometry somehow. Maybe I am over using the phrase curved spacetime. But I thought that was what SR did, say that space time geometry was flexible, and it had to be to preserve the measured speed of light for any worldline? Is that really saying it way too wrong?

I appreciate the help here man. I know it's above and beyond the call and there's only so much you can do. Don't feel compelled. I'm not going to drown. Just having fun trying to understand dang.

 

[georgir]

RE: accelerometers
They work by basically measuring stress forces internal to the device.
So, they can not measure a uniformly applied force, one that accelerates all parts of the device equally and so does not cause any stress. Pretty much the only such force we know of is gravity. In GR, it is not regarded as a force at all, so it is a valid generalization to say that accelerometers measure all kinds of "real" forces.

Of course if we ever discover other ways to create a uniform acceleration force field, that generalization would no longer be valid. It is really a question of semantics, we may as well consider that new force field a kind of gravity too...

 

[pervect]

I am not sure what you mean exactly. Your description of drawing on the graph paper is excellent. What you describe is called a spacetime diagram. It is one of the most important tools in relativity.

My reading was that the poster had a space-space diagram (x and y), not a space-time diagram (no t), presumably not having realized the significance or need for putting time on his diagram. So he'll still need a bit of motivation as to why to include time to get a space-time diagram. If / once he does include time in his analysis, he'll be close to getting the idea of the Lorentz transform.

The rest of your thoughts match mine.

 

[georgir]

RE: curved spacetime
There is no curved spacetime in SR. Transforming coordinates from one reference frame to another may seem weird, but still is not the same as curved spacetime in GR. There are a few points to this:
- It is not a physical transformation process, just a "point of view" thing.
- It does not change anything about the spacetime itself. In SR, spacetime is not an Euclidean space, but a Minkowski space. The difference is in how the time component is considered to be imaginery (or its square is negative) when calculating distances. This is impossible to visualize properly, so when we draw our charts, it seems like we really are skewing it, but in fact, no distances change by this transformation. It is very similar to a rotation of an Euclidean space - would you say a rotation is actually transforming anything?
- Even when we look at the Euclidean representation we often use to visualize it, we have a linear transformation. It transforms a flat spacetime into a flat spacetime. There is still no "curvature" to speak of.

So in SR, you have this "normal" spacetime, completely flat, but still a bit weird to visualize and understand because a) it is 4D and b) its 4-th dimension contributes to distances negatively.
You can have all sorts of accelerated motion in it of course, and the "worldline" of such motion would simply be a curved line instead of a straight one.

Curved spacetime is introduced in GR because we take a special subset of these curved lines that are possible in SR - the ones caused by gravity only - and decide to treat them as if they are "straight". Well, not quite straight, just the straightest possible thing on a curved "surface". Our "surface" here is the already weird 4D spacetime from SR, so it gets even weirder here and I give up :p

 

[georgir]

RE: distance (spacetime interval) vs lorentz contraction.
I expect the next thing that might confuse you and could be your next question. Why did I say that no distances change by applying Lorentz transformation, when it clearly leads to well-known effects like lorentz contraction, which is precisely a change in distances?

The point is that in Minkowski geometry, the analogue to Euclidean distance is what we call a spacetime interval. Instead of just sqrt(dx^2+dy^2+dz^2), we have sqrt(dx^2+dy^2+dz^2-dt^2). This "distance" does not change, no matter what reference frame we measure x, y, z, t in.

On the other hand, length - the distance between two worldlines at a given moment - changes depending on reference frames, because the definition of a "moment" changes. Still, nothing physically changes, but we measure the distance at a different angle.

 

[Jimster41]

RE: distance (spacetime interval) vs lorentz contraction.
I expect the next thing that might confuse you and could be your next question. Why did I say that no distances change by applying Lorentz transformation, when it clearly leads to well-known effects like lorentz contraction, which is precisely a change in distances?

The point is that in Minkowski geometry, the analogue to Euclidean distance is what we call a spacetime interval. Instead of just sqrt(dx^2+dy^2+dz^2), we have sqrt(dx^2+dy^2+dz^2-dt^2). This "distance" does not change, no matter what reference frame we measure x, y, z, t in.

On the other hand, length - the distance between two worldlines at a given moment - changes depending on reference frames, because the definition of a "moment" changes. Still, nothing physically changes, but we measure the distance at a different angle.

Thanks guys, esp Georgeir. That was helpful for me. I feel like I'm getting to the point of correction. I'm looking things up in the books I have here also, to try and maximize the effect of this opportunity to learn in question mode. which for me it's about the only way it works.

I gather now that I have been incorrectly posing my question in terms of how relative velocity or acceleration curves or distorts "space-time". This is incorrect because Minkowski's metric is Euclidean and flat (is that right?). Lorentz transformations are linear. This what ghwellsjr was especially objecting to I think. I really didn't get that clearly until now.

So I hope I'm making the correction when I say that 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". Which really leaves me confused.

I have to own that I've peeked at the sun a couple of times, like a dumb kid as it were, by flipping through stuff on space-time quantization (Conversations about Verlinde's Entropic Gravity theory and a book that is way over my head on LQG), and so - I may be seeing spots. But the fact that there is "time dilation" that somehow communes with "length contraction", and the fact that time is inextricably caught up with Entropy (as I understand it - perhaps incorrectly), and the fact that Entropy is another one of those things that bothers the crap out of me, and I have spent a lot of time trying to figure out how to carry around an understanding of what it means - all conspire to make the curiosity/excitement/frustration factor around picturing just how space-time adjusts itself in the Lorentz transform in Minkowski space I guess, nearly infinite.

I hope I am making progress in my understanding, and precision here. Thanks for being patient.

 

[Dale]

I think I'm definitely puzzling over the case where The accelerometer reads non zero. So if I picture the twin carrying the clock at the origin and drive them up and out to the right curving until they are moving at a 45 deg angle. Is that the "null" line? speed of light.

Yes, a 45 degree line on a spacetime diagram is called a null line and represents something moving at the speed of light. So a massive object can come very close to a 45 degree line, but not exactly.

I guess I would have said that "curving the world line" was messing with spacetime geometry somehow. Maybe I am over using the phrase curved spacetime. But I thought that was what SR did, say that space time geometry was flexible, and it had to be to preserve the measured speed of light for any worldline? Is that really saying it way too wrong?

Think about the spacetime diagram idea that you mentioned earlier. You can mark out positions and times in a grid on a piece of paper, as you proposed. You can use those positions and times to draw the worldlines of a few objects of interest on your paper, and so forth. None of the grid lines or worldlines that you draw on the paper change the fact that the piece of paper is flat. Regardless of how curved the lines are, the paper remains flat.

Now, you can delete the grid lines and leave the worldlines. What remains is the "invariant" geometry. Even without the grid lines you can do things like measure lengths of the lines (proper time), angles between lines (relative velocity), and curvature of the lines (proper acceleration). All of those geometric measurements correspond directly to something physical. You can put in new gridlines, you can use gridlines at an angle to the original ones, or even curved grid lines, none of that will change any of the invariant geometric measurements.

The one difference between geometry on your flat piece of paper and in spacetime is that distances on the piece of paper are given by the Pythagorean theorem ##ds^2 = dx^2+dy^2+dz^2## and in flat spacetime they are given by the Minkowski metric ##ds^2=-dt^2+ dx^2+dy^2+dz^2##

I appreciate the help here man. I know it's above and beyond the call and there's only so much you can do. Don't feel compelled. I'm not going to drown. Just having fun trying to understand ****.

No worries. This type of discussion is exactly the purpose of Physics Forums, so I am glad that you are asking. Personally, I wish we had more similar discussions, not less.

 

[ghwellsjr]

My reading was that the poster had a space-space diagram (x and y), not a space-time diagram (no t), presumably not having realized the significance or need for putting time on his diagram. So he'll still need a bit of motivation as to why to include time to get a space-time diagram. If / once he does include time in his analysis, he'll be close to getting the idea of the Lorentz transform.

He said the y-axis was for time:

If I draw an x-y plane on graph paper and the x-axis is some number of graph-paper boxes long (in reference to the road's length) and the y-axis is time, and some number of graph-paper boxes long.

 

[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.