Understanding the Minkowski Metric: Explained Step-by-Step for Beginners

[shounakbhatta]

Hello,

I don't know whether I have mentioned the subject line properly. Many times while reading over General Relativity I come across the following equation:


ds^2=dx1^2+dx2^2+dx3^2+dx4^2

=dx^2+dy^2+dz^2-c^2dt^2.


Now, my question from the above equation is:

(a) Are we putting the Pythagorean theorem into Minkowski Metric?
(b) dx1,dx2...are they four dimensions like, length, breadth, height and time?
(c) c is it the speed of light?
(d) How the factor -c^2dt^2 comes?
(e) What is the use of converting this to Minkowski metric?

Kindly explain me in step-by-step as I am new to this world.

Thanks

 

[yenchin]

A metric simply describes how to measure distance. In 2D space, the Pythagoras theorem says
[itex]s^2=x^2 + y^2[/itex]. In 3D space, it says [itex]s^2=x^2+y^2+z^2[/itex], but you know that already I assume. The infinitesimal version is simply [itex]ds^2=dx^2+dy^2+dz^2[/itex].

In Special Relativity, we are interested in both space and time {t,x,y,z}. But they don't have the same unit since t is time and the rest are spatial dimensions. Thus one uses {ct,x,y,z} instead. Note that ct has dimension of length. The metric is *defined* to be [itex]ds^2=-c^2dt^2+ dx^2+dy^2+dz^2[/itex]. This has a few nice properties: The metric gives zero for light ray, since light travels at speed c, i.e. [itex]c^2 dt^2 = dx^2 + dy^2 + dz^2 [/itex] is precisely the distance traveled by light in time interval dt. I hope this provides a nice picture of what is going on.

Note that in many relativity texts, one chooses the unit where [itex]c=1[/itex] so that the metric is simply written [itex]ds^2=-dt^2+ dx^2+dy^2+dz^2[/itex]. There are also convention in which one writes [itex]ds^2=dt^2- dx^2-dy^2-dz^2[/itex].

Does that help?

 

[shounakbhatta]

Hello Yenchin,

That was really a great reply. Actually, it cleared out lot of doubts that I have been carrying for a long time. Now, I still have certain questions. I would be thankful if you can please answer me:

(1) As you have told that metric is basically how to measure the distance. Now, in General Relativity we have different metrics like Schwarzschild metric and other metric. Does that mean that we have different type of metrics to measure different things in different co-ordinates?

(2) As you have mentioned the metric has been **defined**. Does that mean ds^2=-c^2dt^2 is not a derivation but an assumption?

It would be very kind of you if you please.explain a bit.

Thanks

 

[WannabeNewton]

The metric tensor is not a coordinate dependent quantity. It is something that assigns an inner product to each fiber of the tangent bundle of space-time. We express the metric in certain bases such as the coordinate basis when doing actual computations but the metric tensor itself is a purely geometric quantity. The different metrics you mentioned are different solutions to Einstein's equations and describe different space-time geometries.

 

[yenchin]

(2) As you have mentioned the metric has been **defined**. Does that mean ds^2=-c^2dt^2 is not a derivation but an assumption?

You can assign many types of metric on a given manifold [if you are not familiar with this notion, think of it as an abstract mathematical "space"]. A manifold a priori does not come with metric. For example the four dimensional space [itex]\Bbb{R}^4[/itex]. You can equip it with the usual Euclidean metric [itex]d = \sqrt{dx^2 + dy^2 + dz^2 + dw^2}[/itex], or you can equip it other the various other metrics, including the Minkowski metric, to define your geometry. It is not a derivation, but it is not just an "assumption", it really depends on what do you want to do; in relativity the theory is a theory of spacetime and we use a pseudo-Riemannian manifold to model the physics.

 

[shounakbhatta]

Hello,

Thanks for the reply. It cleared up certain concepts. I just want to ask you few more questions. I hope it is ok with you, as more discussions clear certain concepts and crops up new things:

(a) The other types of metric like Schwarzschild metric and others, are they for measuring distances at different co-ordinate system?

(b) Do we have only three types of metric, the Euclidean, the Lorenzian and Schwarzschild metric?

(c) The x,y,z,t (4 dimension) is measured in Lorentzian metric, right?

(d) The Schwarzschild metric helps us to measure around a spherically symmetric body, such as a planet, or a black hole etc, right?

Now, I have one confusion. The Lorentzian metric g, when taken to General Relativity, does that take a general form and becomes Schwarzschild metric?

Thanks.

 

[WannabeNewton]

(a) The other types of metric like Schwarzschild metric and others, are they for measuring distances at different co-ordinate system?

I would like to stress again that the metric tensor is an object that is by itself coordinate independent. We can express it in any allowed coordinate basis but the value given by the action of the metric tensor on elements of each fiber of the tangent bundle is independent of the coordinates; you are confusing the metric tensor with coordinate systems, the latter of which have no real physical meaning. However, coordinate transformations of the metric tensor are important in the sense that since it is coordinate independent, we can always pick a convenient or more revealing coordinate system to express our metric in, knowing we will get the same value regardless for the "infinitesimal displacements".

Here's a relatively interesting example for anyone interested in a neat coordinate transformation (this is problem 3.2 in Wald): consider a 2 dimensional manifold ##M## with metric ##g_{ab}## and associated derivative operator ##\nabla_{a}##. Let ##\alpha## be a scalar field on ##M## that is harmonic meaning it satisfies ##\nabla^{a}\nabla_{a}\alpha = 0 ## and let ##\epsilon_{ab}## be a 2-form (i.e. a tensor field such that ##\epsilon_{ab} = -\epsilon_{ba}##) satisfying ##\epsilon_{ab}\epsilon^{ab} = 2(-1)^{s}## where ##s = 0## if ##g_{ab}## is a riemannian metric and ##s = 1## if ##g_{ab}## is a lorentzian metric. ##\epsilon_{ab}## is also called the natural volume element on ##M##; note that the above definition implies ##\epsilon_{ij}\epsilon^{ab} = (-1)^{s}2\delta^{[a}_{i}\delta^{b]}_{j}##. Finally, consider another scalar field ##\beta## given by the equation ##\nabla_{a}\beta = \epsilon_{ab}\nabla^{b}\alpha##.

We first want to show that locally there always exists a ##\beta## that actually satisfies the above equation. That is, we must show that given the 1-form ##\omega_{a} = \epsilon_{ab}\nabla^{b}\alpha##, there always exists locally a scalar field ##\beta## such that ##\nabla_{a}\beta = \omega_{a}##. The most straightforward way to do this would be to show that ##\omega_{a}## is closed, meaning ##\nabla_{[a}\omega_{b]} = 0## because then the Poincare Lemma guarantees that locally ##\omega_{a}## is exact, that is locally there exists a scalar field ##\beta## such that ##\nabla_{a}\beta = \omega_{a}##, which is exactly what we want.

Note that if we can show ##\epsilon^{ab}\nabla_{a}\omega_{b} = 0## then we are done since this would imply ##\epsilon_{ij}\epsilon^{ab}\nabla_{a}\omega_{b} = \nabla_{[i}\omega_{j]} = 0##. Also note that since ##\epsilon_{ab}\epsilon^{ab} = 2(-1)^{s}##, we have that ##\epsilon^{ab}\nabla_{c}\epsilon_{ab} = 0## implying ##\epsilon_{ij}\epsilon^{ab}\nabla_{c}\epsilon_{ab} = \nabla_{c}\epsilon_{ij} = 0## identically. Proceeding with this, we have ##\epsilon^{ab}\nabla_{a}\omega_{b} = \epsilon^{ab}\nabla_{a}(\epsilon_{bc}\nabla^{c}\alpha) = \nabla_{a}(\epsilon^{ab}\epsilon_{bc}\nabla^{c}\alpha)## and this reduces to ##\epsilon^{ab}\nabla_{a}\omega_{b} = (-1)^{s + 1}\nabla_{a}(\delta_{c}^{a}\nabla^{c}\alpha) = (-1)^{s + 1}\nabla_{a}\nabla^{a}\alpha = 0 ## hence ##\nabla_{[a}\omega_{b]} = 0## as desired so locally there always exists a ##\beta## such that ##\nabla_{a}\beta = \epsilon_{ab}\nabla^{b}\alpha## as desired. Finally, note that ##\nabla^{a}\nabla_{a}\beta = \epsilon_{ab}\nabla^{a}\nabla^{b}\alpha = 0## because ##\nabla_{a}## commutes on scalar fields (torsion-free) so the second covariant derivative is symmetric in the two indices whereas ##\epsilon_{ab}## is anti-symmetric in the two indices so the contraction vanishes. Thus ##\beta## itself is harmonic.

Now, what does all this have to do with coordinate transformations? Let's say we had originally expressed our metric in some coordinates ##(x^1,x^2)## so that ##ds^{2} = g_{\mu\nu}dx^{\mu}dx^{\nu}## where ##\mu,\nu = 1,2##. Let's choose the scalar fields ##\alpha(x^{1},x^{2}), \beta(x^{1},x^{2})## as coordinates. The coordinate transformation law for the metric tensor tells us that ##g_{\alpha\alpha} = \nabla^{\mu}\alpha \nabla^{\nu}\alpha g_{\mu\nu} = \nabla_{\mu}\alpha \nabla^{\mu}\alpha## and similarly ##g_{\beta\beta} = \nabla_{\mu}\beta\nabla^{\mu}\beta = \epsilon_{\mu\nu}\epsilon^{\mu\gamma}\nabla^{\nu}\alpha \nabla_{\gamma} \alpha = (-1)^{s}\nabla_{\mu}\alpha\nabla^{\mu}\alpha ##. Finally, ##g_{\alpha\beta} = \nabla_{\mu}\beta \nabla^{\mu}\alpha = \epsilon_{\mu\nu}\nabla^{\nu}\alpha \nabla^{\mu}\alpha = 0 ## again because we are contracting symmetric indices with anti-symmetric ones. Letting ##\Omega^{2} = \nabla_{\mu}\alpha \nabla^{\mu}\alpha##, under the coordinate transformation our line element takes the form ##ds^{2} = \pm \Omega^{2}(d\alpha^{2} + (-1)^{s}d\beta^{2})##. We call these coordinates harmonic coordinates. So given any lorentzian metric ##g_{ab}## on a 2-manifold, we can always locally put the metric in the form ##ds^{2} = \Omega^{2}(-dt^{2} + dx^{2})## i.e. we can always make a coordinate transformation to locally make the metric equivalent to the flat 1 + 1 lorentzian metric up to a conformal factor. I thought this was a cool result

(b) Do we have only three types of metric, the Euclidean, the Lorenzian and Schwarzschild metric?

The Schwarzschild metric is a Lorentzian metric. A Lorentzian metric on space-time is a pseudo-Riemannian metric of signature (-,+,+,+).

(c) The x,y,z,t (4 dimension) is measured in Lorentzian metric, right?

Yes.

(d) The Schwarzschild metric helps us to measure around a spherically symmetric body, such as a planet, or a black hole etc, right?

It is, to be more specific, for a spherically symmetric static body.

 

[pervect]

There are as many metrics as there are ways of giving events coordinates, i.e. an infinite number.

It might be useful to talk about metrics in a more familar context - the surface of the Earth. It is convenient to represent locations on the surface of a spherical Earth by two coordinates which we call lattitude φ and longitude ##\lambda##. This is not the only possible approach, but it's a very common one.

Suppose you have two points and we want to know the great circle distance (the distance on the Earth's surface) between them. You can look up an exact function for the "great circle" distance between two points on the Earth's surface, this goes like

http://en.wikipedia.org/wiki/Great-circle_distance

##s = 2\,\arcsin \left( \sqrt {\sin^2\frac{d\phi}{2}
+ \cos \phi_1 \cos \phi_2
\sin^2 \frac{d\lambda}{2} }
\right)##

This is rather complicated and doesn't illustrate the concept of a metric yet. The metric comes into play when we approximate the above, for "nearby" points. If you're familiar with taylor series, we can just use a multi-variable taylor series. The result then becomes the much simpler formula

##ds^2 = dφ^2 + cos^2 φ \, d \lambda^2 ##

If you compare this to the "Schwarzschild " metric, you'll see that it's rather similar, it's a quadratic equation, or bilinear form. Symbolically

## ds^2 = g_{φφ} d φ^2 + g_{\lambda\lambda} d \lambda^2 \quad g_{φφ}=1 \quad g_{\lambda\lambda} = \cos^2 φ ##

But there is no time involved, it's a "distance" metric, not a space-time metric. A little more on the time part later.

So, our metric is just a function that converts changes in coordinates to changes in distances - for nearby points.

You can think of a metric as also representing how distorted a map made using your coordinates will be, if we draw it on a flat sheet of paper.

In the example above, we can see that near the equator, the metric is diagonal, indicating no distortion. Near the poles, ##g_{\lambda\lambda}## goes to zero, indicating large distortions.

We could choose different coordinates (a polar projection) to make the poles appear less distorted - but then the equator would be distorted. We will never be able to represent a curved surface undistorted on a flat piece of paper.

Note that if we choose different coordinates, we get different values of the metric components. The values of the metric components tell us how distorted are maps will be, but they aren't anything physical in and of themselves, by choosing different coordinates we will have different metric component values.

The above is how spatial metrics work. Let's go back to the space-time metrics.

One of the things one learns in special relativity is that time, and distance, are not fundamental. What's fundamental is the "Lorentz interval" ds, which is giving by the Minkowskii metric.

If you want to know more about the Mnikowskii metric, you'll need to learn more about special relativity.

It's tough to say a lot about GR without learning more about SR first. One thing we can say is that the metric coefficients for space-time tell us how distorted our representation is, just as they did in our simpler example on the Earth.

Metrics that approach the Minkowskii metric (or the various spherical forms of it) represent undistorted space-time.

i.e
##ds^2 = dx^2 + dy^2 + dz^2 -c^2 dt^2## is a cartesian form of the Minkowskii metric
##ds^2 = dr^2 + r^2 d \theta^2 + r^2 cos^2 \theta d \phi^2 - c^2 dt^2 ## is a spherical form of the same metric.

As the metric coefficients get to be significantly different from their Minkowskii values, our representation of space-time is becoming more distorted. Unless we know more , we can't say for sure whether this distortion represents an odd choice of coordinates, or whether it reflects effects that are due to actual space time curvature. We can always change coordiantes to make the distortion go away at anyone point, but in the presence of actual curvature, we can't choose coordinates to make it go away everywhere.

If you compare the Schwarzschild metric to the spherical form of the Minkowkii metric, you can see that it's similar, but become very distorted near r=2m, where some of the components vanish, and others become infinite, indicating infinite distortion.

It turns out that this is a coordinate distortion, similar to the way that we had vanishing metric coefficients near the poles of the Earth with our lattitude-longitude coordinates. But it takes quite a bit of math to demonstrate this.

 

[shounakbhatta]

Hello Pervect,

Thank you very much for the detailed reply. I can understand the following things. Please correct me if I am wrong:

(a) A metric measures distortion over a surface, a manifold.
(b) ds is the Lorentz interval
(c) The line element in Euclidean space is: ds^2=g11 (dx1)^2+g22(dx2)^2, where g is the metric tensor.

Now, I want to ask you the Euclidean metric, in the matrix format gives (1 0 0,0 1 0,0 0 1). I want to know, are there any simultaneous equations which give rise to the values mentioned in the matrix?

I can well understand the example that you have provided in the cartesian form of Minkowski metric and the spherical form of Minkowski metric.

 

[WannabeNewton]

http://en.wikipedia.org/wiki/Riemannian_metric#Riemannian_metrics
I don't think it can be made any clearer considering it has been repeated multiple times already.

 

[pervect]

Hello Pervect,

Thank you very much for the detailed reply. I can understand the following things. Please correct me if I am wrong:

(a) A metric measures distortion over a surface, a manifold.
(b) ds is the Lorentz interval
(c) The line element in Euclidean space is: ds^2=g11 (dx1)^2+g22(dx2)^2, where g is the metric tensor.

I should clarify a few things of what I said, though you quoted me correctly.

a) The distortion isn't the distortion of the mainfold, per se. The distortion is in the map you draw. Making the map is technically a projection operation. The metric gives you the information on how badly the physical shape of the actual surface is distorted when you project it on to a flat piece of paper so that the coordinates form a cartesian grid.

c) ds^2=g11 (dx1)^2+g22(dx2)^2 [+ g12 dx1 dx2] is the general line element for a 2 dimensional Euclidean space.

Now, I want to ask you cthe Euclidean metric, in the matrix format gives (1 0 0,0 1 0,0 0 1).

I want to know, are there any simultaneous equations which give rise to the values mentioned in the matrix?

If we consider the metric:
[tex]
\left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{array} \right)
[/tex]

if your coordinates are x,y,z, the line element corresponding to this metric would be dx^2 + dy^2 + dz^2. It would be a flat three dimensional Euclidean space. It's not a set of simultaneous equations, it's the matrix representation of a "quadratic form"

In matrix notation, it looks like this:

[tex]
\left( \begin{array}{c}
dx & dy & dz \end{array} \right)
\left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{array} \right)
\left( \begin{array}{ccc}
dx\\
dy\\
dz \end{array} \right) = \left( \begin{array}{c}
dx & dy & dz \end{array} \right) \left( \begin{array}{ccc}
dx\\
dy\\
dz \end{array} \right) = dx^2 + dy^2 + dz^2
[/tex]

 

[yenchin]

Hello Pervect,

Thank you very much for the detailed reply. I can understand the following things. Please correct me if I am wrong:

(a) A metric measures distortion over a surface, a manifold.
(b) ds is the Lorentz interval
(c) The line element in Euclidean space is: ds^2=g11 (dx1)^2+g22(dx2)^2, where g is the metric tensor.

Now, I want to ask you the Euclidean metric, in the matrix format gives (1 0 0,0 1 0,0 0 1). I want to know, are there any simultaneous equations which give rise to the values mentioned in the matrix?

I can well understand the example that you have provided in the cartesian form of Minkowski metric and the spherical form of Minkowski metric.

By the way, what is your background in mathematics? Are you reading certain GR book? We may be able to suggest suitable texts for further reading. I think you need to firmly grasp the idea of metric spaces and differential geometry of surfaces before discussing general relativity.

 

[WannabeNewton]

I think you need to firmly grasp the idea of metric spaces and differential geometry of surfaces before discussing general relativity.

I agree with this whole-heatedly. It is very hard to give rigorous accounts of something if the background is lacking and it is also very hard to hand wave something as important as metric tensors; "intuitive" but highly non-rigorous, non-mathematical explanations will only take you so far.

 

[shounakbhatta]

Hello,

To be very honest with you all, my background of mathematics is not very strong. I am doing a self-learning from Special Relativity, graduating to learn GR.

Thanks for the help.

 

[yenchin]

Hello,

To be very honest with you all, my background of mathematics is not very strong. I am doing a self-learning from Special Relativity, graduating to learn GR.

Thanks for the help.

That does not help us to suggest a suitable text... could you be more specific about your current level of mathematics, i.e., what have you already learned? Calculus? Differential equations? Linear Algebra?

 

[shounakbhatta]

Yes. Calculus, differential eqn.and linear algebra I have done but not into differential geometry or topology.

 

[yenchin]

OK. I think if you are ambitious, you are quite ready to learn some differential geometry.

Some people prefer to directly learn abstract manifold theory, but I think it is good to get some geometric intuitions by first learning 3-dimensional differential geometry of surfaces. A standard text of this is Do Carmo's "Differential Geometry of Curves and Surfaces ". Depending on your interest, you don't really have to go over the whole book though. You can then pick up Riemannian geometry. Do Carmo has a book on that too. But my favorite differential geometry text is John M. Lee's "Riemannian Manifolds: An Introduction to Curvature". His book on smooth manifold is also good. Of course GR is really about semi-Riemannian geometry [though physicists like to refer to it as merely 'Riemannian'], so you would want to also refer to texts like O'Neill's Semi-Riemannian Geometry With Applications to Relativity.

For GR, again, there are many choices. If you prefer to see the "physics first" approach, Hartle is a popular choice. If you are more mathematically inclined, go with Carroll. I personally don't like Hartle's book, and first learned GR via the little book of Foster & Nightingale. You should of course explore many texts and see which one suits you the best.

By the way if you are more mathematically-minded, you may want to check out the following book for the structure of special relativity [it won't teach you much physics, but it is a superb math book]: The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity .

 

[VantagePoint72]

Some people prefer to directly learn abstract manifold theory, but I think it is good to get some geometric intuitions by first learning 3-dimensional differential geometry of surfaces. A standard text of this is Do Carmo's "Differential Geometry of Curves and Surfaces ". Depending on your interest, you don't really have to go over the whole book though. You can then pick up Riemannian geometry. Do Carmo has a book on that too. But my favorite differential geometry text is John M. Lee's "Riemannian Manifolds: An Introduction to Curvature". His book on smooth manifold is also good. Of course GR is really about semi-Riemannian geometry [though physicists like to refer to it as merely 'Riemannian'], so you would want to also refer to texts like O'Neill's Semi-Riemannian Geometry With Applications to Relativity.

I realize you prefaced by saying it would just help develop intuition, but I think it should be emphasized that none of this is really necessary. Any standard intro GR text, of which you mentioned a couple, will teach all the necessary differential geometry in the first few chapters. Nothing more than some very basic multivariable calculus and a bit of linear algebra are necessary to get started. More important than prior experience with differential geometry is, I think, a solid foundation in special relativity, going as far as the covariant formulation of electrodynamics. It is the simplest example of a classical field theory and I think it's much more valuable for developing the sort of intuition needed to get through GR. Unfortunately, I don't have a good SR book to recommend since when I was an undergraduate and took relativistic electrodynamics the professor wrote his own course notes, guided by Landau and Lifgarbagez (which is not an acceptable stand-alone text for a self-guided student). For GR, though, I especially like Carroll's book.

 

[Fredrik]

But my favorite differential geometry text is John M. Lee's "Riemannian Manifolds: An Introduction to Curvature". His book on smooth manifold is also good.

I just want to add that the curvature book is not an introductory text on differential geometry. The other one is, but since Lee wrote the curvature book before the introductory book, he didn't include anything about curvature in the introductory book. Both books are great. Everyone who wants to learn differential geometry for GR should get both.

 

[WannabeNewton]

I realize you prefaced by saying it would just help develop intuition, but I think it should be emphasized that none of this is really necessary. Any standard intro GR text, of which you mentioned a couple, will teach all the necessary differential geometry in the first few chapters. Nothing more than some very basic multivariable calculus and a bit of linear algebra are necessary to get started. More important than prior experience with differential geometry is, I think, a solid foundation in special relativity, going as far as the covariant formulation of electrodynamics. It is the simplest example of a classical field theory and I think it's much more valuable for developing the sort of intuition needed to get through GR.

Learning math from a physics book is a terrible idea no matter what physics subject it is, period.

 

[VantagePoint72]

Learning math from a physics book is a terrible idea no matter what physics subject it is, period.

That is clearly a matter of taste.

 

[WannabeNewton]

That is clearly a matter of taste.

If only that were true but hey no one is making someone learn math properly from a math text. If someone wants to "learn" it second hand from a physics text then that is up to them. There was an old thread in the GR section on topology in GR and this thread served as a great example of why not learning math from an actual math book will come back to bite you.

 

[Nugatory]

That is clearly a matter of taste.

"Learning math" ≠ "Learning to use math"
Are we all talking about the same thing here?

 

[VantagePoint72]

"Learning math" ≠ "Learning to use math"
Are we all talking about the same thing here?

Whether we're all talking about the same thing depends on whether or not you missed the fact that I said, "Any standard intro GR text, of which [yenchin] mentioned a couple, will teach all the necessary differential geometry in the first few chapters." A book like Carroll's does an entirely competent job of teaching the math and teaching how to use the math in close succession. Differential geometry is, of course, more broadly useful in physics and so there is nothing wrong with studying it more in depth from dedicated textbooks. Such study would also be expected to bring an enhanced understanding of some concepts in relativity. My point was simply that it is not necessary to do so before studying GR since modern textbooks almost invariably present the subject in a mathematically self-contained way. In my experience, this is how a considerable majority of physics students—excluding those doing a double major with mathematics—encounter the machinery of differential geometry for the first time.

I have found this to be an effective approach. However, as I have acknowledged, that is a matter of taste. What is not a matter of taste is the claim that it is necessary to study differential geometry before a book like Carroll, Schutz, or Wald is accessible. That is not an opinion; it is just factually wrong. Going beyond that, my opinion is that it is not only unnecessary, there's no particular reason why it should be advisable. Disagree with that you if you like, but know that thousands of undergraduates at many very good universities learn GR precisely in that way every year. If one's view of pedagogy is so narrow that they can't accept that those students come out of it with a solid foundational grasp of the material (upon which deeper study can build), then I would recommend the underlying "the only effective teaching style is mine" arrogant attitude should be reexamined.

 

[WannabeNewton]

If you think the differential geometry presented in Carroll's text will be enough to actually understand the GR literature once you are done with the book, then you are mistaken. The differential geometry in that text won't even be enough to do the exercises in Do Carmo's Riemannian Geometry text and this text is seen as one of the easier differential geometry texts out there. Take a look at Hawking and Ellis' GR text to see the kind of math one would need to know beforehand. I refer you again to the thread "topology of space-times", which you can find by doing a quick search, to see the kind of misconceptions that come up when someone tries to "learn" math from a physics textbook. This is about as inefficient as "learning" functional analysis from a QM book like Ballentine. Books like Wald and Ballentine etc. do a great job with the physics and, as Nugatory said, do a great job with showing you how to use the math but this hardly counts as learning it. In the end, it comes down to how much you actually want to know and how much mainstream literature you actually want to read.

 

[micromass]

Whether we're all talking about the same thing depends on whether or not you missed the fact that I said, "Any standard intro GR text, of which [yenchin] mentioned a couple, will teach all the necessary differential geometry in the first few chapters." A book like Carroll's does an entirely competent job of teaching the math and teaching how to use the math in close succession. Differential geometry is, of course, more broadly useful in physics and so there is nothing wrong with studying it more in depth from dedicated textbooks. Such study would also be expected to bring an enhanced understanding of some concepts in relativity. My point was simply that it is not necessary to do so before studying GR since modern textbooks almost invariably present the subject in a mathematically self-contained way. In my experience, this is how a considerable majority of physics students—excluding those doing a double major with mathematics—encounter the machinery of differential geometry for the first time.

I have found this to be an effective approach. However, as I have acknowledged, that is a matter of taste. What is not a matter of taste is the claim that it is necessary to study differential geometry before a book like Carroll, Schutz, or Wald is accessible. That is not an opinion; it is just factually wrong. Going beyond that, my opinion is that it is not only unnecessary, there's no particular reason why it should be advisable. Disagree with that you if you like, but know that thousands of undergraduates at many very good universities learn GR precisely in that way every year. If one's view of pedagogy is so narrow that they can't accept that those students come out of it with a solid foundational grasp of the material (upon which deeper study can build), then I would recommend the underlying "the only effective teaching style is mine" arrogant attitude should be reexamined.

I'm sorry, but if students learn differential geometry from a GR text, then they won't have a solid foundational grasp of the material. They can't. Those books aren't made to give you a solid foundational grasp.

The GR books teach differential geometry in a way that the students are able to use the math. And that works rather well, I think. The books teach enough math in order to be able to grasp the physics and do the physical problems. They don't teach enough math in order to understand differential geometry. And there's nothing wrong with that. The books aren't made to teach differential geometry, the books are made to teach physics. And that's what they do. I'm not saying that you need to learn differential geometry before GR (it would be better though), you'll do fine without it. It's just wrong to think that you'll have a foundational grasp of the mathematics, because that is false.

In my opinion, every professional researches in theoretical fields of GR should absolutely complete a actual differential geometry book and not just rely on the physics texts. It is very beneficial to go through books such as Lee. Students who first encounter GR will be fine without the differential geometry (although it helps a lot).

 

[VantagePoint72]

If you think the differential geometry presented in Carroll's text will be enough to actually understand the GR literature once you are done with the book, then you are mistaken.

You are misrepresenting what I am saying, and when I have very explicitly said things like:

Differential geometry is, of course, more broadly useful in physics and so there is nothing wrong with studying it more in depth from dedicated textbooks. Such study would also be expected to bring an enhanced understanding of some concepts in relativity.

and

solid foundational grasp of the material (upon which deeper study can build)

I am beginning to wonder if it is intentional. I have no where said that the introduction to differential geometry that comes from a GR text should be the end of one's journey into differential geometry. I have said that is an acceptable start; that is, it is enough to begin digging into the mechanics of GR are start developing some physical context to motivate deeper study. I have made very clear that my point is: it is not necessary (and, in my own experience, not even especially helpful) to independently study differential geometry before beginning to explore GR. You are twisting this into, 'It is not necessary to study differential geometry'. Not only have I not said that, I've said the exact opposite.

 

[VantagePoint72]

I'm sorry, but if students learn differential geometry from a GR text, then they won't have a solid foundational grasp of the material. They can't. Those books aren't made to give you a solid foundational grasp.

Foundational in the sense of contextualizing further study. I really don't disagree with anything you've said, I'm struggling to understand why it is you think you disagree with me. Perhaps you have been mislead by WannabeNewton's extreme misrepresentation of what I am arguing. As you said:

The books teach enough math in order to be able to grasp the physics and do the physical problems. They don't teach enough math in order to understand differential geometry. And there's nothing wrong with that. The books aren't made to teach differential geometry, the books are made to teach physics. And that's what they do.

And that is precisely why I am arguing that for someone interested in the physics, they are the correct starting point. The only thing I have disputed is the recommendation to study differential geometry independently before beginning a study of GR. I agree that it is indispensable tool for physicists, and I have said as much already. However, when someone has the goal in mind of understanding specific theories in physics, I think there is much to recommend helping them get to the heart of what interests them as quickly as good pedagogy allows. That will, as I've said, contextualize the further study.

 

[micromass]

Foundational in the sense of contextualizing further study. I really don't disagree with anything you've said, I'm struggling to understand why it is you think you disagree with me. Perhaps you have been mislead by WannabeNewton's extreme misrepresentation of what I am arguing. As you said:



And that is precisely why I am arguing that for someone interested in the physics, they are the correct starting point. The only thing I have disputed is the recommendation to study differential geometry independently before beginning a study of GR. I agree that it is indispensable tool for physicists, and I have said as much already. However, when someone has the goal in mind of understanding specific theories in physics, I think there is much to recommend helping them get to the heart of what interests them as quickly as good pedagogy allows. That will, as I've said, contextualize the further study.

Well, we agree then. I responded because you said this:

Any standard intro GR text, of which you mentioned a couple, will teach all the necessary differential geometry in the first few chapters.

To me, it looked like you were saying that the GR texts teach all the math and that you don't need to learn any diff geo afterwards. But I misinterpreted your statement then.

 

[WannabeNewton]

I am beginning to wonder if it is intentional. I have no where said that the introduction to differential geometry that comes from a GR text should be the end of one's journey into differential geometry. I have said that is an acceptable start; that is, it is enough to begin digging into the mechanics of GR are start developing some physical context to motivate deeper study. I have made very clear that my point is: it is not necessary (and, in my own experience, not even especially helpful) to independently study differential geometry before beginning to explore GR. You are twisting this into, 'It is not necessary to study differential geometry'. Not only have I not said that, I've said the exact opposite.

I know what you said and I'm saying that if you go into a physics book without knowing the mathematics beforehand from an actual math text, then you will not have an adequate understanding of the math when you finish, as far as the mathematics is concerned. You will only be able to apply it to the level of physics problems presented in that specific text. Schutz was my first GR text, then I used Carroll and Wald in conjunction and I didn't learn differential geometry beforehand so I'm speaking from experience. I had to go back and learn the math properly from a math text in order to clear up the horribly hand-wavy presentations of the mathematics presented in books like Carroll and Schutz (Wald is quite an exception in the realm of undergrad / first year grad physics texts since he is actually rigorous with the mathematics). There is absolutely no harm in learning the math properly from a math text before delving into the physics if it means you will have a much better understanding of the math and can focus on the actual physics instead of worrying about the hand-wavy arguments given regarding the differential geometry.

 

[VantagePoint72]

To me, it looked like you were saying that the GR texts teach all the math and that you don't need to learn any diff geo afterwards. But I misinterpreted your statement then.

Fair enough. By 'necessary differential geometry' I meant 'necessary to understand the rest of the textbook', not 'necessary for your future life as a physicist'.

 

[VantagePoint72]

I know what you said and I'm saying that if you go into a physics book without knowing the mathematics beforehand from an actual math text, then you will not have an adequate understanding of the math as far as the mathematics is concerned. You will only be able to apply it to the level of physics problems presented in that specific text. Schutz was my first GR text, then I used Carroll and Wald in conjunction and I didn't learn differential geometry beforehand so I'm speaking from experience. I had to go back and learn the math properly from a math text in order to clear up the horribly hand-wavy presentations of the mathematics presented in books like Carroll and Schutz (Wald is quite an exception in the realm of undergrad / first year grad physics texts since he is actually rigorous with the mathematics).

Well, this is a happy coincidence: I followed literally the exact same sequence when I first began studying GR. Clearly you and I hold this approach in different regard. For me, I appreciated the broad (but increasingly more detailed with each of those books) understanding of GR that came from viewing geometry through the lens of physics. When I went on to study differential geometry on its own, I found it much more satisfying to be able to see immediately how most of the things I was studying enhanced my understanding of relativity. My physical knowledge made it clear exactly which bits were most relevant and merited the deepest study. In the end, studying differential geometry not only helped me better understand the GR I already knew; already knowing a fair amount of GR helped me better understand the differential geometry I was studying. Which has, of course, been useful well beyond just GR.

I am happy that that is the path through GR and geometry that my undergraduate and graduate education lead me on. Apparently you are not. That's fine: as I've said from the very beginning, there is a significant element of personal taste here. However, surely you can't sustain the position that the sequence of GR and then more differential geometry that I'm advocating invariably leads to inadequate understanding. I would presume you think you turned out fine.

 

[WannabeNewton]

Well, this is a happy coincidence: I followed literally the exact same sequence when I first began studying GR. Clearly you and I hold this approach in different regard. For me, I appreciated the broad (but increasingly more detailed with each of those books) understanding of GR that came from viewing geometry through the lens of physics. When I went on to study differential geometry on its own, I found it much more satisfying to be able to see immediately how most of the things I was studying enhanced my understanding of relativity. My physical knowledge made it clear exactly which bits were most relevant and merited the deepest study. In the end, studying differential geometry not only helped me better understand the GR I already knew; already knowing a fair amount of GR helped me better understand the differential geometry I was studying. Which has, of course, been useful well beyond just GR.

I certainly agree that viewing the geometry through the lens of physics is quite important and very satisfying (this is possibly one of the bigger reasons I got into physics). I would never want a fleshed out exposition of differential geometry in Schutz for example because he provides such an awesome intuition for the physics that I certainly think would be ruined by adding too much math. I don't think it would be good to add too much math to the physics book itself, I'm saying it might help to learn it before using the book. I can't say that learning GR has helped me personally understand differential geometry better but I can say that it has helped me appreciate it more, that's for sure.

I am happy that that is the path through GR and geometry that my undergraduate and graduate education lead me on. Apparently you are not. That's fine: as I've said from the very beginning, there is a significant element of personal taste here. However, surely you can't sustain the position that the sequence of GR and then more differential geometry that I'm advocating invariably leads to inadequate understanding. I would presume you think you turned out fine.

I am not unhappy as much as I am confused when I read math in physics texts where the mathematical facts are not proven / exposed rigorously. This, I agree, is a matter of purely personal taste. I certainly don't disagree with your advice that learning GR then learning differential geometry is one advantageous route to take; I'm just saying that there can be people out there who are not satisfied with the exposition of the math in the GR texts to begin with (as I was). Looking back, I do agree that there is quite a bit of personal taste involved in this. However you can agree that just learning the relevant geometry through physics texts cannot be the end all be all of the education.

 

[VantagePoint72]

I think perhaps an analogy is in order:

Calculus is very much the language of mechanics. However, students (in the educational systems with which I'm familiar) don't typically begin working with it until the final year before they begin university, and really get to delve into until first year uni. Nonetheless, we do not wait until then to begin educating people in physics. We present an algebraic version of it throughout high school to begin fostering physical intuition and motivating further study. Indeed, my first year physics class as undergraduate even included precisely the sort of 'on-the-fly' instruction in calculus that I am describing with differential geometry. Obviously that introduction was not the end of calculus education at my undergrad institution. However, it made it possible to start working with the fundamental machinery of classical mechanics. The same thing happened with my first dedicated course in electromagnetism: a vector calculus preview was necessary since the multivariable course was taught concurrently. For those students interested in mathematics primarily for its use as the language of physics, the physical knowledge provides excellent context for the deeper mathematical study. More importantly, it makes it possible to start understanding the physics as quickly as possible.

I think a dogmatic insistence on a rigorous study of differential geometry prior to embarking on a study of GR [note: this is not directed at anyone in particular, I'm just emphasizing my original point] is precisely as wrong-headed as insisting high school students not be taught a lick of physics, nor undergraduates until they've finished with real analysis and partial differential equations. I think it fundamentally confuses logical dependence with pedagogical dependence. There will, of course, be students interested in the mathematics for its own sake. These are students who will pursue a dedicated education in mathematics alongside their physics education. However, for the students primarily interested in the application of mathematics to physics, I do not believe any favours are done by placing as much abstract mathematics as possible between them and their exploration of physical concepts.

 

[VantagePoint72]

However you can agree that just learning the relevant geometry through physics texts cannot be the end all be all of the education.

I do (and have already said) that I readily agree with that.