General Relativity in the Undergraduate Physics Curriculum

[ZapperZ]

This is an interesting paper. In arguing on why GR can be a part of an undergraduate curriculum via a "physics first" approach, James Hartle has made a rather "understandable" description of GR. I think this might be a terrific introduction to GR to a lot of physics students, whether they intend to go into this area or not.

http://arxiv.org/abs/gr-qc/0506075

Zz.

 

[robphy]

Gravity: An Introduction to Einstein's General Relativity
James B. Hartle
ISBN: 0805386629
Publisher: Addison Wesley; 1st edition (December 26, 2002)



http://www.aw-bc.com/catalog/academic/product/0,1144,0805386629,00.html
http://wps.aw.com/aw_hartle_gravity_1 (Companion Website... has sample chapters)

http://online.itp.ucsb.edu/online/plecture/hartle/

 

[James R]

Sounds like a good approach to me. Many undergraduate students are interested in general relativity. The main problem is their lack of mathematical background.

A "physics first" approach skirts around that problem, while giving students a bit of a feel for the subject.

 

[neurocomp2003]

my GR teacher was decent as an undergrad...i learned more from the problmes he gave...I think they need to strengthen the programming side.

 

[ohwilleke]

I agree with the premise of the article. Mastering differential geometry and tensor mathematics is a non-trivial exercise, yet the vast majority of the epxeriment and observational (and even theoretical) work in the field pretty much limits itself to a handful of greatly simplified cases where the full mathematical suite of tools necessary fully follow GR are not required.

 

[pervect]

This is an interesting paper. In arguing on why GR can be a part of an undergraduate curriculum via a "physics first" approach, James Hartle has made a rather "understandable" description of GR. I think this might be a terrific introduction to GR to a lot of physics students, whether they intend to go into this area or not.

http://arxiv.org/abs/gr-qc/0506075

Zz.

I gather Hartle has a textbook he wrote using this approach. I was hoping that it would cover the topic of accelerated observers, but nearly as I can tell it does not conver this topic. This is based on a review where the table of contents is pasted, the "peek into book" function isn't implemented for the book I looked at :-(.

I was looking at some of the other introduction to relativity books via Amazon, and they don't seem to cover the accelerated observer either! (Schutz, Rindler).

Is there anyone who has an introductory level book that does cover this important topic? Or is the treatment in "Gravitation" the simplest one avaliable? (it's a nice treatment, but it does use tensor notation). It seems like the topic of the accelerated observer has come up a lot recently (aside from several recent posts to the relativity forum, it's popped up again in the college homework forum).

 

[robphy]

Try
Leslie Marder, Time and the space-traveller
http://www.worldcatlibraries.org/wcpa/top3mset/36df9466cefce179.html
and
James J. Callahan, The Geometry of Spacetime

http://www.worldcatlibraries.org/wcpa/top3mset/5d5c8379d3e93584a19afeb4da09e526.html

In addition, these may be useful (although I haven't looked at these in detail yet)
http://www.vallis.org/research/res_foundations.html
M. Vallisneri, Relativity and Acceleration, thesis, doctorate in physics (University of Parma, Italy, 2000), http://www.vallis.org/publications/tesidott.pdf
Massimo Pauri, Michele Vallisneri,
Marzke-Wheeler coordinates for accelerated observers in special relativity
http://xxx.arxiv.org/abs/gr-qc/0006095

 

[pmb_phy]

There are several textbooks out there on this topic. One is called Gravity from the Ground up," by Schutz. Another one is Feyman who connects gravity to acceration. Peacock also includes this in a recent article. There are many more that I mentioned that I have been guided to. Someday I'll get around to that point but is rather tricky,

Pete

 

[CarlB]

I'm very much in favor of teaching GR at the undergraduate level. Heck, I think it would be better if there was more concentration on the practical uses of the various black hole metrics at the graduate level.

Carl

 

[pmb_phy]

I'm very much in favor of teaching GR at the undergraduate level. Heck, I think it would be better if there was more concentration on the practical uses of the various black hole metrics at the graduate level.

Carl

I'm sure that you're students would enjoy learning how the knowledge of relativity alowed physics to design the GPS system/

Pete

 

[metrictensor]

I see no reason why GR can't be introduced in the undergrad curriculum. I am reading a very good book that is a mix of a popular and mathematical treatment of GR. Relativity Theory by Amos Harpaz. Very good on the conceptual stuff although it does leave some "examples" unjustified or at least hastly so. But it is the only book I have read where the ideas begin to make sense while not being watered down to the popular level. Popular books are useless anyway if you are serious. In many cases after I understand something as it should the impression I got from the popular books was wrong.

For example, is spacetime really curved in the same way as a road is? I think a better explanation is that the equations of motion are constrained in such a way. To think there is something actually there that is being bent seems wrong.

 

[pervect]

In what way is a road curved? This is a vague enough statement that I'm not sure what you mean by it.

GR has an important entity called the "Riemann curvature tensor" which is nonzero (has nonzero elements) near large masses. This tensor measures something that is called "intrinsic curvature".

http://mathworld.wolfram.com/IntrinsicCurvature.html

Intrinsic Curvature


A curvature such as Gaussian curvature which is detectable to the "inhabitants" of a surface and not just outside observers. An extrinsic curvature, on the other hand, is not detectable to someone who can't study the three-dimensional space surrounding the surface on which he resides.

The 2-dimensional version of this tensor has the property that the surface of a sphere is curved (has non-zero components in the tensor) while the surface of a plane is flat.

The tensor doesn't apply to a line, you need at least 2 dimensions to have intrinsic curvature as per the quote.

 

[metrictensor]

In what way is a road curved? This is a vague enough statement that I'm not sure what you mean by it.

It is very clear. Curvature is a measure of somethings deviation from flat space.

GR has an important entity called the "Riemann curvature tensor" which is nonzero (has nonzero elements) near large masses. This tensor measures something that is called "intrinsic curvature".

http://mathworld.wolfram.com/IntrinsicCurvature.html



The 2-dimensional version of this tensor has the property that the surface of a sphere is curved (has non-zero components in the tensor) while the surface of a plane is flat.

The tensor doesn't apply to a line, you need at least 2 dimensions to have intrinsic curvature as per the quote.

I understand the math but you are making the assumption that the Riemann tensor is valid. I am interested in the idea of whether spacetime being curved is a model or there is actually something like a road or train tracks that is curved.

If we look at the Earth orbiting the sun we can model it as two balanced forces, i.e. the centripetal and centrefugal. We can also look at it as moving in such a way because there are constraints on the equations of motion.

 

[ZapperZ]

If we look at the Earth orbiting the sun we can model it as two balanced forces, i.e. the centripetal and centrefugal. We can also look at it as moving in such a way because there are constraints on the equations of motion.

Er... hang on a minute. You see nothing wrong with saying that?

If you have "two balanced forces", then what is causing the net central acceleration of the orbiting Earth around the sun? Or are you also disputing Newton's First Law?

Zz.

 

[metrictensor]

Er... hang on a minute. You see nothing wrong with saying that?

If you have "two balanced forces", then what is causing the net central acceleration of the orbiting Earth around the sun? Or are you also disputing Newton's First Law?

Zz.

Maybe that wasn't a good example but that is all it was. Consider a wire loop with bead on it. The bead is constrained to moving only in a circle no matter what forces are acting on the bead. We want to look at the motion of the bead and determine the forces acting on it.

The only force that affects the bead's motion is tangent to the circle, i.ae. it pushed the bead around the circle. Any force acting away from the center of the bead will be balanced by a reactionary force supplied by the bead and acting towards the center.

Now we can look at this in two ways. First, the motion of the bead is constrained by the loop and the only force acting on it is the one that is tangent to the circle. We could alternatively say that there are 3 forces acting on the bead, the one above and an action and reaction force normal to the surface of the loop. Since these two forces are an action-reaction pair they will always be equal and opposite and therefore are independent of the mass of the object.

 

[ZapperZ]

Maybe that wasn't a good example but that is all it was. Consider a wire loop with bead on it. The bead is constrained to moving only in a circle no matter what forces are acting on the bead. We want to look at the motion of the bead and determine the forces acting on it.

The only force that affects the bead's motion is tangent to the circle, i.ae. it pushed the bead around the circle. Any force acting away from the center of the bead will be balanced by a reactionary force supplied by the bead and acting towards the center.

Now we can look at this in two ways. First, the motion of the bead is constrained by the loop and the only force acting on it is the one that is tangent to the circle. We could alternatively say that there are 3 forces acting on the bead, the one above and an action and reaction force normal to the surface of the loop. Since these two forces are an action-reaction pair they will always be equal and opposite and therefore are independent of the mass of the object.

The bead example is even worse, because the bead is confined to move in a circle NO MATTER what the nature and magnitude of the force are. I don't need a central force to have the bead move in a circular motion.

However, this is NOT the same as "earth orbiting the sun" that you brought up. There are no "balanced forces" here. There's only ONE net central force that is responsible for the observed motion of earth. There is no "wire" to constrain the motion.

Zz.

 

[pmb_phy]

It is very clear. Curvature is a measure of somethings deviation from flat space.

If we're talking about spacetime curvature then you've come to the wrong conclusion. If it was I who came to the conclusion then I blame it on the meds I'm on.

To give an answer which might give you a better visual understanding think of the three models of the universe. There the is open universe, closed universe and the flat universe. In this example the "flat" does not refer to the spacetime, it referres to spatial flatness. So here we have an example of a spatially flat universe whose spacetime is curved.

Pete

 

[metrictensor]

The bead example is even worse, because the bead is confined to move in a circle NO MATTER what the nature and magnitude of the force are. I don't need a central force to have the bead move in a circular motion.

However, this is NOT the same as "earth orbiting the sun" that you brought up. There are no "balanced forces" here. There's only ONE net central force that is responsible for the observed motion of earth. There is no "wire" to constrain the motion.

Zz.

The point is that you can look at the problem in two ways. Either a constraint on the motion of the object caused by the bead or due to the balance of two forces.

The whole point of GR is that there is a wire. That Newton's explanation does not hold up to the principles of relativity.

 

[metrictensor]

If we're talking about spacetime curvature then you've come to the wrong conclusion. If it was I who came to the conclusion then I blame it on the meds I'm on. Pete

I am talking about curvature and how it is defined.

 

[ZapperZ]

The point is that you can look at the problem in two ways. Either a constraint on the motion of the object caused by the bead or due to the balance of two forces.

The whole point of GR is that there is a wire. That Newton's explanation does not hold up to the principles of relativity.

No you can't. Those two are not the same. I have explained why they are not the same.

In plane polar coordinates, if I apply an F_{\theta} to your bead, the bead will simply move faster but keeps the identical path. If I do that to the earth, the trajectory will change, and can even change severely. So how can you think these two are the same?

Secondly, look at how you would solve this problem using lagrangian mechanics. Where and what exactly is the nature of the "constraint" here both BOTH situations. You will see that they are clearly not identical.

Zz.

 

[pervect]

It is very clear. Curvature is a measure of somethings deviation from flat space.

That's not a very clear statement unless you describe how you intend to measure the "deviation from flat space".

I understand the math but you are making the assumption that the Riemann tensor is valid.

Whatever you're trying to say is even less clear now that you are suggesting that you are not utiliziting the standard way of measuring curvature in GR, the Riemann tensor.

Some of your other remarks suggest that you may be talking about geodesic motion, rather than curvature.

 

[metrictensor]

That's not a very clear statement unless you describe how you intend to measure the "deviation from flat space".

If you are on the top of a hill you can imagine a line going straght off the top parallel to the base of the mountain. As you jump you deviate from this line and your movement is a curved path.

If you like Wikipedia check this out http://en.wikipedia.org/wiki/Curvature

 

[pervect]

OK, thanks for the quote - it clarified what you were talking about considerably.

You appear to be talking and thinking about extrinsic curvature. The sense in which space-time is curved is not that it has extrinsic curvature, but rather intrinsic curvature. From the Wikipedia article you quote:

The Gaussian curvature, named after Carl Friedrich Gauss, is equal to the product of the principal curvatures, k1*k2. It has the dimension of 1/length2 and is positive for spheres, negative for one sheet hyperboloids and zero for planes. It determines whether a surface is locally convex (when it is positive) or locally saddle (when it is negative).

The above definition of Gaussian curvature is extrinsic in that it uses the surface's embedding in R3, normal vectors, external planes etc. Gaussian curvature is however in fact an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. Formally, Gaussian curvature only depends on the Riemannian metric of the surface. This is Gauss' celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking.

An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. She runs around P while the thread is completely stretched and measures the length C(r) of one complete trip around P. If the surface were flat, she would find C(r) = 2πr. On curved surfaces, the formula for C(r) will be different, and the Gaussian curvature K at the point P can be computed as
<br /> K = \lim_{r \rarr 0} (2 \pi r - \mbox{C}(r)) \cdot \frac{3}{\pi r^3}.<br />

The reason that we study the intrinsic curvature of space-time and not the extrinsic curvature is that we live in space-time and cannot step outside it. Because of this there is no way we can determine the extrinsic curvature of space-time. The concept of the extrinsic curvature of space-time would have an operational meaning only to some entity that could stand outside of space-time, which we cannot do.

However, like the ant on the sphere in the above example from the Wikipedia, we can (and do) study the intrinsic curvature of space-time.

 

[metrictensor]

OK, thanks for the quote - it clarified what you were talking about considerably.

You appear to be talking and thinking about extrinsic curvature. The sense in which space-time is curved is not that it has extrinsic curvature, but rather intrinsic curvature. From the Wikipedia article you quote:



The reason that we study the intrinsic curvature of space-time and not the extrinsic curvature is that we live in space-time and cannot step outside it. Because of this there is no way we can determine the extrinsic curvature of space-time. The concept of the extrinsic curvature of space-time would have an operational meaning only to some entity that could stand outside of space-time, which we cannot do.

However, like the ant on the sphere in the above example from the Wikipedia, we can (and do) study the intrinsic curvature of space-time.

I understand the difference but they are both curvatures. It is only a matter of where you can detect them from that is cause for the difference. Whether intrinsic or extrinsic there is still a curvature. I understand the mathematical difference in terms of embedding and not embedding.

It makes sense that spacetime is intrinsic because you can't step out of it. But is spacetime curved or are the equations of motion simply such that they follow curved paths. The question boils down to "is there something that actually bends?" All our examples from eveyday life are based on extrinsic curvatue. That is, we look at what is being curved from an perspective outside the object.

 

[pervect]

I understand the difference but they are both curvatures. It is only a matter of where you can detect them from that is cause for the difference. Whether intrinsic or extrinsic there is still a curvature. I understand the mathematical difference in terms of embedding and not embedding.

It makes sense that spacetime is intrinsic because you can't step out of it. But is spacetime curved or are the equations of motion simply such that they follow curved paths. The question boils down to "is there something that actually bends?" All our examples from eveyday life are based on extrinsic curvatue. That is, we look at what is being curved from an perspective outside the object.

Given that we have a defintion of distance, we can calculate the curvature of space (assuming we can find a suitable notion of space as a specific surface of spacetime, something that we can in fact accomplish) from the distance measurements. This is done by applying the formula for intrinsic curvature such as the one I quoted from the Wikipedia. The ability to measure distances allows us to caculate whether or not a manifold does or does not have intrinsic curvature.

The results we get for GR is that space around a massive object is curved very slightly. For Newtonian gravity, space is not curved around a massive object. Experimental results so far favor GR. The change in the length of the meter predicted by GR illustrates the curvature of space in GR, this is a consequence of the constancy of the speed of light, and gravitational time dilation.

Moving on to the curvature of space-time (rather than the curvature of space), we use for our distance measure the Lorentz interval, rather than the usual notion of distance that we use in space. Again, we find in GR that space-time is curved as well as space is curved.

If we restrict ourselves to Newtonian gravity rather than General relativity, we wind up having a choice. We can interpret gravity as a force, as Newton does, our we can interpret it as a curvature of space-time. This is done in various relativity books. We have this choice because Newtonian gravity doesn't have a pre-defined notion of "distance" in space-time. When we define how we measure "distances" in space-time, we also define whether or not it's curved - Newtonian theory treats space and time as separate. In Newtonian theory, as I mentioned before, space itself is not curved, unlike in GR.

In GR, though, we don't have this choice, unless we totally abandon our standard notions of distance (in space), and of the Lorentz interval (in space-time). With the standard notions of distance, space and space-time must be curved in GR.

 

[George Jones]

I gather Hartle has a textbook he wrote using this approach. I was hoping that it would cover the topic of accelerated observers, but nearly as I can tell it does not conver this topic. This is based on a review where the table of contents is pasted, the "peek into book" function isn't implemented for the book I looked at :-(.

I was looking at some of the other introduction to relativity books via Amazon, and they don't seem to cover the accelerated observer either! (Schutz, Rindler).

Is there anyone who has an introductory level book that does cover this important topic? Or is the treatment in "Gravitation" the simplest one avaliable? (it's a nice treatment, but it does use tensor notation). It seems like the topic of the accelerated observer has come up a lot recently (aside from several recent posts to the relativity forum, it's popped up again in the college homework forum).

Hartle does cover the topic of accelerated observers, albeit briefly. Do you mean general accelerated motion, or do you mean uniform acceleration? Section 5.5 of Hartle, Observers and Observations, is about general motion, though the examples given deal with uniform acceleration.

Which book by Schutz? Which book by Rindler? I would be completely astonished to find that Rindler, of Rindler spacetime fame, doesn't treat acceleration!

Regards,
George

 

[pmb_phy]

I am talking about curvature and how it is defined.

You used the term "deviation" - Please define "deviation" without using the definition of curvature.

Pete

 

[metrictensor]

You used the term "deviation" - Please define "deviation" without using the definition of curvature.

Pete

In this case it means changing direction.

 

[pervect]

Hartle does cover the topic of accelerated observers, albeit briefly. Do you mean general accelerated motion, or do you mean uniform acceleration? Section 5.5 of Hartle, Observers and Observations, is about general motion, though the examples given deal with uniform acceleration.

Which book by Schutz? Which book by Rindler? I would be completely astonished to find that Rindler, of Rindler spacetime fame, doesn't treat acceleration!

Regards,
George

I don't have/own all the books to look at (I own MTW & Wald), I was basically browsing the table of contents on amazon.com trying to find a good book to recommend.

Basically I want something that covers the deriviation of the relativistic rocket equation (a simple constant proper acceleration problem), and which also tells people about the limitations on the coordinate system of an accelerated observer and the Rindler horizon. A little bit about Thomas precession would be a plus.

I also was surprised not to find anything in Rinder. The book I looked at was "Relativity: Special, General, and Cosmological".

Taking a second look, I do see it being covered briefly there, but only for a very few pages (70-73).

https://www.amazon.com/gp/product/0198508360/?tag=pfamazon01-20

 

[George Jones]

I don't have/own all the books to look at (I own MTW & Wald), I was basically browsing the table of contents on amazon.com trying to find a good book to recommend.

Basically I want something that covers the deriviation of the relativistic rocket equation (a simple constant proper acceleration problem), and which also tells people about the limitations on the coordinate system of an accelerated observer and the Rindler horizon. A little bit about Thomas precession would be a plus.

I also was surprised not to find anything in Rinder. The book I looked at was "Relativity: Special, General, and Cosmological".

Taking a second look, I do see it being covered briefly there, but only for a very few pages (70-73).

https://www.amazon.com/gp/product/0198508360/?tag=pfamazon01-20

Years ago, I read parts of the second edition of Rindler's Essential Relativity, and I think it had a little about horizons and acceleration, though probably not more than the book you mention. Also, I haven't looked at this book in quite some time, so I could be wrong.

I can't think of any books pitched at a level lower than Gravitatation that cover all the (interesting) topics that you mention. Spacetime Physics by Taylor and Wheeler does treat, either in exercises or examples, both the constant acceleration rocket and Thomas precession.

A few years ago on sci.physics, I had a go at using proper time clocks and light signals to derive the coordinate system given in Chapter 6 of Gravitation. I don't think my derivation was very clear, but I'll give a link anyway.

http://groups.google.ca/group/sci.physics/msg/91d8704c115d8899?dmode=source&hl=en

These corrdinates and horizons are related to the Unruh effect, which in turn is related to Hawking radiation - accelerated observers play the role of hovering Schwarzschild observers. For my money, the best quantitative description available of this is in Chaper 9, Quantum Field Theory in Curved Spacetime, of Carroll's (Who, despite having published at least a dozen papers in Physical Review during his time there, was recently denied tenure by the University of Chicago!) superb grad text, Spacetime Geometry: An Introduction to General Relativity.

Regards,
George

 

[robphy]

Sorry to resurrect this thread... but I stumbled upon the more-appropriate URL (from ucsb) that I intended to list in my post, #2, above.

http://online.itp.ucsb.edu/online/pcgm17/hartle/
"How to Teach Relativity to Undergraduates"
Jim Hartle, UCSB (Mar 10, 2001)