Invariance of Diff. Line Elems. in Hartle's Gravity: Intro to GR

[sophiatev]

TL;DR

I have questions about whether or not differential line elements for different geometries represent the same physical quantity, and about the Polar coordinate form of a differential line element in 2D Euclidean space.

In Hartle's book Gravity: An Introduction to Einstein's General Relativity he spends chapter 2 discussing some basic aspects of differential geometry. For example, he derives the expression for a differential line element in 2D Euclidean space:
dS^2 = (dx)^2 + (dy)^2 in Cartesian coordinates
dS^2 = (dr)^2 + (rdΦ)^2 in Polar coordinates
and also for a differential line element on the surface of a two-dimensional sphere of radius a:
dS^2 = a^2((dθ)^2 + sin^2(θ)(dΦ)^2)

My first question is, do these two line elements (for the plane and for the two-sphere) have the same physical length? Is the distance between two infinitely close points on the surface of any shape (sphere, egg, peanut) the same? Is this because at infinitesimal distances, curvature is non-existent?

My second question is it looks like the length of the line element for a plane, when expressed in polar coordinates, should increase at higher values of r. However, Hartle asserts that dS is a physically invariant quantity. Why does it not increase? Does it have something to do with the nature of multiplying by a differential, meaning that rdΦ is the same regardless of the value of r?

My third question is why a line element of the form
dS^2 = (dr)^2 + ((r + dr)dΦ)^2
is invalid. It seems to me that in this diagram he includes (forgive how faint the lines are)

The rdΦ is closer to the (r + dr)dΦ arc length anyway. Furthermore, since (r + dr) is slightly bigger, (r + dr)dΦ seems closer to a straight line than rdΦ . Does this again have something to do with the fact that at infinitesimal lengths, both are considered completely straight? Either way, I don't see why my formulation would be invalid. It clearly is because if we convert from Cartesian to polar coordinates, starting with the Cartesian form of dS, we end up with Hartle's form of dS in polar coordinates. But I don't see why my expression for dS in polar coordinates is wrong just based on the diagram he provides.

 

[Orodruin]

My first question is, do these two line elements (for the plane and for the two-sphere) have the same physical length?

What do you mean by this? The line element $ds$ is by definition the distance between two nearby points on the manifold. This obviously depends on the points in question. It is not a single quantity.

My second question is it looks like the length of the line element for a plane, when expressed in polar coordinates, should increase at higher values of r. However, Hartle asserts that dS is a physically invariant quantity.

The distance between any two points is the same regardless of coordinate system. This does not mean that points that differ by fixed amounts in the coordinates have the same distance between them regardless of where they are - they have completely different straight lines between them.

The rdΦ is closer to the (r + dr)dΦ arc length anyway. Furthermore, since (r + dr) is slightly bigger, (r + dr)dΦ seems closer to a straight line than rdΦ . Does this again have something to do with the fact that at infinitesimal lengths, both are considered completely straight? Either way, I don't see why my formulation would be invalid. It clearly is because if we convert from Cartesian to polar coordinates, starting with the Cartesian form of dS, we end up with Hartle's form of dS in polar coordinates. But I don't see why my expression for dS in polar coordinates is wrong just based on the diagram he provides.

It is irrelevant if you use r or r + dr. In the limit of very close points, dr becomes much smaller than r and can be safely ignored in comparison.

 

[PeroK]

On your first question there is a difference between coordinates and geometries. The Euclidean plane is a geometry and any two points have a well-defined distance between that must be independent of your choice of coordinates. If you change coordinates and get a different distance between two points, then you must have done something wrong.

The surface of a sphere is a different geometry. It's not the same as a Euclidean plane. The surface of a sphere has a finite total area and is a closed surface. The properties of lines and triangles on the surface of a sphere are different from those on a Euclidean plane. For example, the sum of the internal angle of a triangle on the surface of a sphere is greater than 180 degrees.

 

[sophiatev]

The distance between any two points is the same regardless of coordinate system. This does not mean that points that differ by fixed amounts in the coordinates have the same distance between them regardless of where they are - they have completely different straight lines between them.

Okay, this makes sense. So what you're saying is that at higher values of r, the distance between two points is indeed larger. I guess the reason this makes me uncomfortable is it seems that the Cartesian coordinate system has a higher granularity. The distance between two points in the Cartesian system remains constant, whereas the smallest distance we can express between two points in the polar system increases as r increases. It "feels" like there should be more Cartesian points than polar points as a result, which is of course completely nonsensical. But if the distance between the polar points increases at higher r, it seems like we should be able to "fit" more Cartesian points between them.

What do you mean by this? The line element dsdsds is by definition the distance between two nearby points on the manifold. This obviously depends on the points in question. It is not a single quantity.

I see now that this question is nonsensical since the distance between two points changes based on where those points are on the manifold.

It is irrelevant if you use r or r + dr. In the limit of very close points, dr becomes much smaller than r and can be safely ignored in comparison.

If this is the case, shouldn't I be able to use my expression for dS, dS^2 = (dr)^2 + ((r + dr)dΦ)^2 , to convert from Cartesian to polar and back? This doesn't work if I use y = rsinΦ and x = rcosΦ. Would I have to replace these by y = (r + dr)sinΦ and x = (r + dr)cosΦ, perhaps?

 

[PeroK]

Okay, this makes sense. So what you're saying is that at higher values of r, the distance between two points is indeed larger. I guess the reason this makes me uncomfortable is it seems that the Cartesian coordinate system has a higher granularity. The distance between two points in the Cartesian system remains constant, whereas the smallest distance we can express between two points in the polar system increases as r increases. It "feels" like there should be more Cartesian points than polar points as a result, which is of course completely nonsensical. But if the distance between the polar points increases at higher r, it seems like we should be able to "fit" more Cartesian points between them.I see now that this question is nonsensical since the distance between two points changes based on where those points are on the manifold.If this is the case, shouldn't I be able to use my expression for dS, dS^2 = (dr)^2 + ((r + dr)dΦ)^2 , to convert from Cartesian to polar and back? This doesn't work if I use y = rsinΦ and x = rcosΦ. Would I have to replace these by y = (r + dr)sinΦ and x = (r + dr)cosΦ, perhaps?

You're embarking on a senior undergraduate level course in GR, but you've never met polar coordinates before?

As an absolute minimum prerequisite for Hartle, you ought to work through everything on here:

http://tutorial.math.lamar.edu/

Or a reputable book on calculus and vector calculus.

Moreover, you'll need a solid understanding of SR. Hartle will simply race through these topics as a brief revision and to get you familiar with his notation. You can't learn SR from Hartle just as a prelim to tackling GR.

 

[Orodruin]

Okay, this makes sense. So what you're saying is that at higher values of r, the distance between two points is indeed larger.

No, this is not what I am saying. You are missing very important qualifiers to that statement. The distance between two points whose coordinate differences are the same is larger at larger r as long as the coordinate difference in the angular coordinate is non-zero.

The distance between two points in the Cartesian system remains constant

Again, missing important qualifiers. The distance between two pre-determined points is the same regardless of the coordinate system. It is an invariant.

If this is the case, shouldn't I be able to use my expression for dS, dS^2 = (dr)^2 + ((r + dr)dΦ)^2 , to convert from Cartesian to polar and back? This doesn't work if I use y = rsinΦ and x = rcosΦ. Would I have to replace these by y = (r + dr)sinΦ and x = (r + dr)cosΦ, perhaps?

No. You should throw away the dr^2 dphi^2 term from your expression. It is irrelevant for expressing the line element.

 

[sophiatev]

No, this is not what I am saying. You are missing very important qualifiers to that statement. The distance between two points whose coordinate differences are the same is larger at larger r as long as the coordinate difference in the angular coordinate is non-zero.

Sorry, yes. So the distance between r + dr, Φ + dΦ is larger at larger r as long as dΦ is non-zero. This is where my confusion about "granularity" arises from. The distance between two points with the same coordinate difference (r + dr, Φ + dΦ) increases at higher r. The same is not true for the Cartesian system. As a result, it feels like at higher r, we should be able to express more points using the Cartesian system than the polar. If the distance between (r + dr, Φ + dΦ) increases while the distance between (x + dx, y + dy) remains the same, it feels like we should be able to fit more (x, y) than (r, Φ) between the smallest granularity coordinate difference we have for polar coordinates, (r + dr, Φ + dΦ). From what I understand, we can fit an infinite amount of points between (r + dr, Φ + dΦ) in both coordinate systems, it just feels like one infinity is larger. I'm sorry if that question made no sense, it's difficult for me to express what I'm thinking. Thank you so much for your help thus far.

 

[Orodruin]

So the distance between r + dr, Φ + dΦ

No. The "distance between $(r,\phi)$ and $(r+dr,\phi+d\phi)$".

As a result, it feels like at higher r, we should be able to express more points using the Cartesian system than the polar.

No. There are as many points in the interval (0,2) as there are on the entire real line.

 

[PeroK]

Sorry, yes. So the distance between r + dr, Φ + dΦ is larger at larger r as long as dΦ is non-zero. This is where my confusion about "granularity" arises from. The distance between two points with the same coordinate difference (r + dr, Φ + dΦ) increases at higher r. The same is not true for the Cartesian system. As a result, it feels like at higher r, we should be able to express more points using the Cartesian system than the polar.

There is a paradox of sorts if you choose a point "at random" in the unit circle and ask what is the probability that it lies in the circle of radius $1/2$. If you choose the point using Cartesian coordinates in the obvious way, then the probability is $1/4$. And, if you choose the point using polar coordinates in the obvious way then the answer is $1/2$.

That said, every point in the plane maps to exactly one and only one pair of Cartesian coordinates and one and only one pair of polar coordinates. (Except the origin, of course, where $r = 0$ and $\phi$ is not well-defined.)

 

[pervect]

Summary:: I have questions about whether or not differential line elements for different geometries represent the same physical quantity, and about the Polar coordinate form of a differential line element in 2D Euclidean space.

In Hartle's book Gravity: An Introduction to Einstein's General Relativity he spends chapter 2 discussing some basic aspects of differential geometry. For example, he derives the expression for a differential line element in 2D Euclidean space:
dS^2 = (dx)^2 + (dy)^2 in Cartesian coordinates
dS^2 = (dr)^2 + (rdΦ)^2 in Polar coordinates
and also for a differential line element on the surface of a two-dimensional sphere of radius a:
dS^2 = a^2((dθ)^2 + sin^2(θ)(dΦ)^2)

My first question is, do these two line elements (for the plane and for the two-sphere) have the same physical length? Is the distance between two infinitely close points on the surface of any shape (sphere, egg, peanut) the same? Is this because at infinitesimal distances, curvature is non-existent?

"Physical distance" is a bit vague, though my personal answer is "yes", I think of the distance as defined by differential geometry as being "physical".

I wouldn't say curvature is "non-existent", but it doesn't show up directly and simply in the line element. I think that the notion of how curvature shows up and is defined is probably beyond the scope of what I want to write about, though I will say that the Riemann curvature tensor is a useful and general description of curvature, and that the Riemann curvature tensor can be computed from the line element in a process vaguel aken to differentiation. At the moment, though, we are trying to understand the line element, we can't progress to discussing the curvature tensor until the line element is understood. Plus, I'm not going to try and summarize the entire textbook chapter discussing the Riemenn curvature in a single short post, though I'm willing to try and answer questions about it if the topic arises in the future.

Let's consider the sphere more carefully. Suppose we have some general function that expresses the distance between two points on a sphere that works in general, i.e. it's not necessarily infinitesimal. We have to assume that we have some notion of what "distance" is that is well defined enough for us to calculate to proceede further.

Given that we have a well-defined and agreed-upon notion of distance, we can proceed further. We'll start with defining a coordinate system. I'll use $\theta$ and $\phi$ as the coordinates, though with a few adjustments you could use longitude and lattitude if you prefer.

Using our coordinates, we have two coordinates $\theta$ and $\phi$ that define the initial point on the sphere, and two coordinate deltas, $\Delta \theta$ and $\Delta \phi$ that describe the second point on the sphere.

Then in general the distance will be some ungainly function of $\theta$, $\phi$, $\Delta \theta$ and $\Delta \phi$ that gives the distance between two points.

If you try and linearize this function around $\Delta \theta$ and $\Delta \phi$, so that you can consider what happens as the delta's get small, you find that a linear approximation is not good enough, the second order approximation is the lowest order that works. Furthermore, adding terms of third order or higher doesn't matter if one takes the limit of low deltas. So, we approxomate the distance as a quadratic form of $\Delta \theta, \Delta \phi$. And then we replace $\Delta \theta$ with d$\theta$ in the limit, and $\Delta \phi$ with d$\phi$.

Note that the quadratic form is allowed to include $\theta$ and $\phi$ (and functions of $\theta$ and $\phi$) as coefficients of the quadratic form. It turns out that for the specific coordinate choice we made, we need. $\sin^2 \theta$ as one of the coefficients.

My second question is it looks like the length of the line element for a plane, when expressed in polar coordinates, should increase at higher values of r. However, Hartle asserts that dS is a physically invariant quantity. Why does it not increase? Does it have something to do with the nature of multiplying by a differential, meaning that rdΦ is the same regardless of the value of r?

The point here is that coordinate choices are just human choices. The physical world doesn't come equipped with a coordinate grid, that's a human invention for communication purposes. They are basically just labels. So we can do physics with cartesian coordinates, and we can do physics with polar coordinates, it makes no difference, we should get "the same" answer, just with different labels. This is the notion of covariance.

Then the point is that the choice of labels shouldn't matter to anythign physical, and our mathematical structure should reflect this.

Perhaps you are not onboard the idea that coordinates are merely labels. If you are not onboard with this concept, it's something you should think about further.

I think this is enough for now, so I'll stop here.

 

[sophiatev]

Thank you so much for everyone's help, I think I have more of a grasp on this.

 

[haushofer]

Switching between coordinate systems to describe a tensor is like switching between languages to describe an object (where the dictionaries have a 1 to 1 relation!). The object doesn't care about the language you use (unless you're into some extreme variant of the Sapir-Whorff hypothesis :P ).

For a scalar like ds this means that the numerical values doesn't change. "Three parachutes" remain 3 parachutes, even if you call them "treie deadonderdeldoekjes", as we do in my native language.