@ChrisWhite Well I'm quite sure there are a whole lot of statistics on how education affects pay. I'm not sure if this is a good measure of actual value added as different educational backgrounds will often lead to different jobs, as well.
@0celo7 I don't think my differential geometry is where it needs to be. Wald is great but I feel like I'm not getting as much out of it as I would if I were more comfortable with certain diffgeo concepts
@0celo7 another is my underatanding of lightlike vs nulllike vs spacelike. Like I read in Hartle that if Bob stays still and Alice moves in the x dimension and comes back. They meet at rhe same spacetiem point. But Bobs path is longer. And that doesn't make sense to me.
Like on paper Alice's path is longer. I can fins the diagram
@Danu I left academia after my PhD not because there weren't positions available (post docs at least were easy to come by), but rather because I was just bored with the work itself. When I was younger I didn't think that would happen, but it did. So... all the more reason to focus on more than just one thing, I suppose, especially if you don't have a lot of experience doing research and publishing it yet.
@0celo7 wow, you're right. Okay, I keep forgetting about the non-positive definite. See, for me, although correct, its counterintuitive because we day light is moving yet it travels 0 length curves right because of equal space and equal time. Doesn't that seem counterintuitive like....how can something move and yet travel no distance? Because if light moves I unit space and 1 unit time, you get zero for the interval
Yeah! This is where I am confused. Good. So how can an object move from p to p' and travel along a 0 length curve. Are u saying we don't use length? But people always talk about proper time lengths... Don't they?
@StanShunpike The proper time is defined as the line integral $$\tau=\int_\gamma d\tau$$ along the path $\gamma$. This is equal to the time measured by the particle traveling along $\gamma$ in its own rest frame.
@StanShunpike It is (modulo a sign and a factor of $c$ or two) the time in the rest frame.
It's just the time elapsed in the rest frame, not really a distance.
Ok. The general idea of flow is that it is a one-parameter family of diffeomorphisms generated by a vector field $X$. The integral curves of $X$ are of course $\dot\gamma =X$. The flow is defined by $\phi_t(x)=\gamma(t)$. The flow operator $\phi_t$ is just a map from the point $\gamma(0)$ to $\gamma(t)$ along the integral curve.
@KyleKanos lmao chip kelly. did they make the playoffs? no right, they lost to Dallas at the end of the season.
@0celo7 So if I am at a point m with a vector defined there, and I act on that point with the flow operator and push it forward to point m', is the vector from point m now at m'
Oh, you subtract the non-operated on one from the operated on one.
That makes sense. That's awesome. I've wanted to learn that for ages. Coool
Alright, let's see if I can get this notion of length correct. So if I have a point p on my manifold, don't I have to travel some length to get to some other point o'? Or am I being too Euclidean here....and not thinking manifold enough. I just don't see how the metric $ds^2$ could be 0 and yet travel between to points...hang on I think Wald has a section on this
Yeah, page 59
He says "Thus, the labels at $t = x^0, x^1, x^2, x^3 of a given event do not have any intrinsic meaning"
Is this related to why I see a contradiction between.....on the one hand, traveling between points on a manifold....and on the other....having distances whose interval is zero
@StanShunpike I have to go now. Homework. Note that this is an artifact of having a metric that is not positive definite. It is perfectly possible to Wick rotate, i.e. $t\rightarrow -it$ and then work in $E^4$.
In Wick rotated space there is no such thing as null.
@0celo7 I still don't think that guy answered my question. He explained how different particles can only travel along certain intervals. But that didn't explain how we could measure a curve on a manifold and have the spacetime interval distance traveled be 0 and yet move from one point to another. I know it must work somehow, but I still don't get it.
0 distance should imply same point. That's like having a line connecting two points and shrinking it to 0. Same point then.
from a one dimensional space to a zero dimensional one
I mean, @StanShunpike , you're thinking of a metric space, which is the good meaningful sensible extension of euclidean geometry
The crux of special relativity is that invariant interval, so...
actually if you google "pseudometric space" you get en.wikipedia.org/wiki/Pseudometric_space for which the very first sentence is... "is a generalized metric space in which the distance between two distinct points can be zero."
The way I see it, first you use special relativity and all those weird experiments to establish that yes, you can consider such an invariant interval. From the invariant interval you get lorentz transformations and all that nice stuff
which is all just trying to understand what you've defined when defining the invariant interval
"geometry" means that metric here
"Why do we take that [pseudo]metric" is met by experiments in special relativity. "Why is the value zero" is met by "because we took that [pseudo]metric"!
@NeuroFuzzy Okay, so I have a spacetime interval. this interval is invariant for all reference frames. Is this interval measuring the distance between point on my spacetime manifold?
@DavidZ Okay, I define distance as.....in the metric space way.
WAIT
Let me get a specific definition so I am not vague this time.
If M is my space
and d is my distance function
d: M x M --> R
where R is the real numbers
So,
(1) $d(x,y) \geq 0$
(2) $d(x,y) = 0$ iff $x =y$
(3) $d(x,y) = d(y,x)$
(4) triangle inequality
That's what I think of distance as. But the question is, does this actually work for the structure of spacetime? If not, what do we have to relax or add in order to make it work?
@StanShunpike You're mixing up two different definitions there: one is the metric space definition, and the other is the enumerated set of properties (1)-(4) in your last message
At least where pseudometric spaces (like Minkowski space) are concerned.
They're the definition of a distance function on a metric space. But in order for a space to be a metric space, the metric has to be positive definite.
Correct. So by definition it doesn't work here. So what I am asking then is, does that mean we lose our ability to define distance? or do we have to change what we consider distance to be to one of a pseudometric space?
When the metric isn't positive definite (e.g. Minkowski space), you can still define a distance function from the metric, but it doesn't satisfy those four properties.
Or you can define a distance function that satisfies the four properties, but then it's not related to the metric.
This is why I asked how you defined "distance": If you want a definition of distance that satisfies the four properties, i.e. one which matches your intuitive sense of what a distance is, then you cannot use the definition that comes from the Minkowski metric.
@StanShunpike I guess we generally call it the "interval"
Okay, let me try this approach. On a spacetime manifold, can I move between points? By move I mean like with proper time. That is, if I am at point $p$, I can move to point $p'$. Assuming that's true, there must be some way of measuring how far I went. Is that what the "interval" does?
@StanShunpike Yes, exactly. My point is that you need to decide on a definition of distance before you can meaningfully talk about how far one point is from another, but the reason you're confused is that you haven't decided on a definition of distance to use. You're picking pieces of two incompatible definitions.
I'm guessing you mean, if distance traveled is zero then you haven't moved to a different point? Yeah, you'll have to dispense with that notion if you want to use the definition of distance that emerges from the Minkowski metric.
Okay. So that idea doesn't work anymore. Hence, you said those axioms of a metric space no longer work.
Hmm....well, at the very least, I am learning my personal definition of distance is quite inadequate at the moment for the things I seemed to be interested in.
Are there multiple common ways of defining distance? I mean, correct me if I am wrong, but for Euclidean spaces we usually have one way of defining distance right? For Minkowski space, do we have multiple ones? What besides the spacetime interval could one use?
In my experience, when people talk about distance in Minkowski space they usually mean the difference in spatial coordinates between two events. This is reference-frame dependent, of course.
Yes, I may be running into that. I also don't know if there's some issue with the fact that...some of the math texts I read only address Riemannian manifolds...and those involve distance functions I think in the original 4 point definition I gave.
For instance, you can talk about the proper time elapsed for a particle moving on a (timelike) geodesic, or the change in affine parameter between two points on a lightlike geodesic (given a particular parametrization of course), and these things correspond in some intuitive sense to what we consider a "distance" so people may call them "distance".
e.g. "the particle moves a certain distance along the geodesic"
but this is not the sort of distance that satisfies the properties (1)-(4) of a distance function.
So this is a notion of distance that does or does not satisfy the 4-point metric space definition? Presumably not since you said it doesn't work on Minkowski space.
But it is a "function" and in quantifies the "distance" in this other sense people use regarding proper time
I couldn't tell whether you were being just thought provoking or literal/serious when you said "maybe we don't need to measure distance at all?" If the path influences the distance, then almost certainly distance wouldn't be very useful. That would be like saying if I walk 5 paces east and 5 paces north, then the point I arrive at is different then if I walked 5 paces north then 5 paces east.
Like on the surface of the Earth, if you go 3000 miles north and then 3000 miles west, you wind up at a different point than if you go 3000 miles west and then 3000 miles north
which in turn is different than going 3000 miles north, turning left by 90 degrees, and going 3000 miles in that new direction
@StanShunpike I was kind of going for both, actually. Distance is just not such a useful concept when working with arbitrary pseudometric spaces.
I mean, now that you mention it, I feel like I'm starting to see how limiting that Euclidean viewpoint can be because there are plenty of examples without even considering spacetime of non-Euclidean geometries where path influences distance.
It's like a the wrong paradigm that becomes exaggerated in its inadequacy for when you consider spacetime.
But without distance, how do I know where I am on the manifold? What are our tools for defining location in spacetime?
If you are working with polar coordinates in Euclidean space (or you can make this work for other kinds of spaces as well), then that label happens to be the same as the distance from the origin.
But you don't need to have any notion of distance to use a number as a label.
Because in many differential geometry situations I've come across, people talk about tensors and how if they are invariant in one system, they are invariant in all.
and I never understood how you could test all possible coordinate transformations.
to know if they were invariant.
But maybe if I just think about them as labels and not like distances....or at least pursue that train of thought, I might find a better answer to this confusion I've had.
Perhaps. I think what you're talking about might be getting into something else a little bit, but it's useful to think about.
Here's something to think about: if you allow $r$ to just be a label, then you can easily deal with a space that has a circular or spherical region cut out of it
You're right that is a good example. It's got the same geometrical intuition, but without the luxury of relying on that central point as the means for defining everything in the system.
Instead, you have to accept it for what it is. Coordinates not distance.
This is very relevant for black holes, incidentally. You can't readily define $r$ as the spatial distance from the singularity because the space inside the event horizon is so distorted. But it's straightforward to use a different procedure to come up with a label that you can call $r$ for any sphere of points centered on the hole.
Ah! That's why I've been having trouble with that too.
I couldn't visualize the geometries.
And it's made it hard to get started learning about them.
It's funny because MTW make it sound like coordinate-invariant approaches to everything is a more useful way to think about many concepts in GR, yet here we sit talking about many applications of coordinates. Really, of all the tools, it seems like distance is the tool that loses the most value as a determination of where you are.
Now, um, what happened here? John Rennie used "possible duplicate" as an "other" off-topic close reason because he couldn't actually use the duplicate close reason because the other question doesn't have an upvoted/accepted answer? And nobody thinks this is a wrong use of the other field?
Well, the duplicate is oddly closed as "homework" although it definitely isn't homework like - it's a conceptual question, just one that is easily answered by typing the phrase into a search engine of your choosing.
What is drift velocity? And why in some books it is expressed as drift speed and not drift velocity?Are these different ?
Does it mean that the electron will have extra velocity opposite to the direction of electric field i.e
thermal(random) velocity + drift velocity=Total velocity of electron,...
Why do you even close a question for a reason that doesn't even match it? Obviously because you think it's a bad question, so why did no one downvote it?
@KyleKanos There was a problem about a spinning mirror. I voted for closing, but also I wanted to give some advice to the OP, just because I saw that he/she is going to work too hard. But the question disappeared. Not only it is no more in the unanswered list, but neither is it in the questions list. Do you have any idea if it was migrated? I don't know the address of the question, and didn't have even time to post my comment. I would appreciate help.
@KyleKanos although I voted for close, because this is the rule, I feel sorry for just working hard without even knowing where to leads the work?
The "recently deleted" 10k tool is too limited:
it only includes a fixed number of deleted posts (45 apparently?)
it only displays posts in a certain time-span (haven't seen anything older than 1h)
it includes spam posts etc., making it hard to filter out deletions for review
it doesn't allow t...
@KyleKanos yes, you are right. The recently deleted queue is empty. Strange! But, tell me, you, the users with very big reputation points, also don't see self-deletions?
What is drift velocity? And why in some books it is expressed as drift speed and not drift velocity?Are these different ?
Does it mean that the electron will have extra velocity opposite to the direction of electric field i.e
thermal(random) velocity + drift velocity=Total velocity of electron,...
