Home Page for Physics 171 (General Relativity) for the 2015 Fall Quarter

This page contains copies of the class handouts, and other items of interest to the Physics 171 class. This course is being offered during the 2015 fall quarter at the University of California, Santa Cruz.

ANNOUNCEMENTS

Have a happy and restful holiday season! Feel free to stop by my office before the Christmas break or any time during the winter quarter (and retrieve your graded homework and exam). If you wish to stop by before Christmas, the best time to meet would be Tuesday December 15--Friday December 18 in the afternoon.

This website will be kept live for the remainder of the academic year 2015--2016.

Solutions to an alternative problem set 3 have been posted to Section IV of this website.

Solutions to the final exam have been posted to Section IV of this website.

Celebrate the 100th birthday of general relativity with a short 3 minute video narrated by David Tennant (who played the tenth incarnation of Dr. Who). Check it out at this link. And don't forget the cake.

Check out http://sci.esa.int/lisa-pathfinder/, which reported the launching of Lisa Pathfinder. This is a mission to test the technology which will be used in the future by eLISA to search fro gravitational waves in the sky.

[ I. General Information and Syllabus | II. Links to the Web Site for the Textbook | III. Problem Sets and Exams | IV. Solutions to Problem Sets and Exams | V. Other Class Handouts | VI. A Free Book on Tensor Calculus | VII. Free textbooks and lecture notes on general relativity and cosmology| VIII. Related Web Pages of Interest]

The General Information and Syllabus handout is available in either PDF or Postscript format     [PDF | Postscript]
Some of the information in this handout is reproduced here.

General Information
Instructor	Howard Haber
Office	ISB 326
Phone	459-4228
Office Hours	Mondays and Tuesdays, 2--3 pm
e-mail	haber@scipp.ucsc.edu
webpage	scipp.ucsc.edu/~haber/

Teaching Assistant

Class Hours

Lectures: Tuesdays and Thursdays, 12--1:45 pm, Nat. Sci. Annex 102
Discussion Section: Wednesdays, 5--6:10 pm, Thimann 101

Required Textbook

Relativity, Gravitation and Cosmology: A Basic Introduction, second edition, by Ta-Pei Cheng (Oxford University Press, 2010).

The most recent compilation of the errata to the 2014 reprinting of the second edition of the textbook can be found here:   [PDF]

If you have the first printing of the second edition, please consult the webpage of Cheng's textbook for the relevant errata.

Course Grading and Requirements

40% regular problem sets
25% Midterm Exam (take-home exam: pick up evening of November 6, 2015; return morning of November 9, 2015)
35% Final Exam (Tuesday December 8, 2015, 8--11 am)

Problem sets will be handed out on a regular basis. The homework problem sets are not optional. You are encouraged to discuss the class material and homework problems with your classmates and to work in groups, but all submitted problems should represent your own work and understanding.

The final exam will be held in Nat. Sci. Annex 102 and will cover the complete course material. You must take the final exam to pass the course.

Course Syllabus

The course outline is available in either PDF or Postscript format     [PDF | Postscript]

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• Webpage of Cheng's textbook
• Please make sure you have the second printing of the second edition of the textbook, since many significant typographical errors have been corrected. If you have the first printing of the second edition, please consult the webpage of Cheng's textbook for the relevant errata.
• The most recent compilation of the errata to the 2014 reprinting of the second edition of the textbook has also been posted to the webpage of Cheng's textbook. For your convenience, this most recent compilation of errata can also be found here:   [PDF]

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Problem sets and exams are available in either PDF or Postscript formats

• Problem Set #1     [PDF | Postscript]
• Problem Set #2     [PDF | Postscript]
• Problem Set #3     [PDF | Postscript]
• Alternate Problem Set #3     [PDF | Postscript]
• Midterm Take-home Exam     [PDF | Postscript]
• Problem Set #4     [PDF | Postscript]
• Problem Set #5     [PDF | Postscript]
• Final Exam         [PDF | Postscript]

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The problem set solutions are available in either PDF or Postscript formats.

• Solution Set #1     [PDF | Postscript]
• Solution Set #2     [PDF | Postscript]
• Solution Set #3     [PDF | Postscript]
• Alternate Solution Set #3     [PDF | Postscript]
• Midterm Exam Solutions     [PDF | Postscript]
• Solution Set #4     [PDF | Postscript]
• Solution Set #5     [PDF | Postscript]
• Final Exam Solutions     [PDF | Postscript]

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1. THE CONFRONTATION BETWEEN GENERAL RELATIVITY AND EXPERIMENT.
By Clifford M. Will (Florida U.). Mar 28, 2014. 113pp.
Published in Living Rev.Rel. 17 (2014) 4
DOI: 10.12942/lrr-2014-4
e-Print: arXiv:1403.7377 [gr-qc] | PDF (arXiv version)| PDF (Journal version)

References | BibTeX | LaTeX(US) | LaTeX(EU) | Harvmac | EndNote
ADS Abstract Service; Link to Fulltext

2. TESTING GENERAL RELATIVITY WITH PRESENT AND FUTURE ASTROPHYSICAL OBSERVATIONS.
By Emanuele Berti (Mississippi U.) et al.. Jan 28, 2015. 186 pp.
Published in Class.Quant.Grav. 32 (2015) 243001
DOI: 10.1088/0264-9381/32/24/243001
e-Print: arXiv:1501.07274 [gr-qc] | PDF (arXiv version)| PDF (Journal version)

References | BibTeX | LaTeX(US) | LaTeX(EU) | Harvmac | EndNote
ADS Abstract Service; Link to Fulltext

3. A handout entitled The velocity and momentum four-vectors examines the properties of the velocity and momentum four-vectors of special relativity, and provides a careful derivation of the relativistic law of addition of velocities. In addition, an appendix provides a derivation of the most general Lorentz boost matrix.   [PDF | Postscript].

4. A handout entitled How do the connection coefficients transform under a general coordinate transformation? derives the transformation law for the connection coefficients with respect to general coordinate transformations. The transformation law contains an inhomogeneous piece, which implies that the connection coefficients are not the components of a tensor. Nevertheless, this inhomogeneous piece is critical in establishing two results. First, the connection coefficients vanish in the local inertial frame. Second, presence of the connection in the definition of the covariant derivative is precisely what is needed so that the covariant derivative of a tensor is also a tensor.   [PDF | Postscript].

5. A handout entitled Parallel Transport and Curvature presents the details of the calculation that demonstrates the connection (no pun intended!) between the change in a vector parallel transported around an infinitesimal closed loop and the Riemann curvature tensor. The derivation presented is slightly more general than the one given in Box 13.2 on pp. 311--312 of our textbook.   [PDF | Postscript].

6. A handout entitled The Cosmological Constant Problem discusses some of the implications of adding a cosmological constant to Einstein's field equations of general relativity. In particular, the handout describes the worst prediction in the history of physics. The predicted value of the vacuum energy density (due to quantum fluctuations) is a factor of 10 larger than the observed value. Moreover, if an additional contribution to the vacuum energy exists to cancel off the contribution of quantum fluctuations, the cancellation must be unimaginably precise (accurate to 123 significant figures). This is one of the greatest unsolved problems of fundamental physics.   [PDF | Postscript].

7. ASTROPHYSICS AND COSMOLOGY REVIEWS from the Particle Data Group.

Astrophysical constants and parameters	PDF	(2 pages)
Experimental tests of gravitational theory	PDF	(13 pages)
Big Bang cosmology	PDF	(33 pages)
Cosmological parameters	PDF	(21 pages)
Dark matter	PDF	(21 pages)
Dark energy	PDF	(19 pages)
Cosmic background radiation	PDF	(24 pages)

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A free textbook entitled Introduction to Tensor Calculus and Continuum Mechanics by John H. Heinbockel is available via the links below. Check it out if you would like more practice in using tensors and manipulating indices.

• bookpdf.zip     PDF format file for downloading (file size approximately 4.3 MB).
• bookps.zip     Postscript format file for downloading (file size approximately 2.6 MB).

The above files are zip files that should be unzipped on a Windows based PC. You should be warned that I have not succeeded in printing out any of the above files obtained after unzipping (although they can be viewed successfully with acrobat reader or ghostview). For your convenience, each chapter of the book appears separately as a pdf and a postscript file below. I made the pdf files from the postscript (rather than use bookpdf.zip) and I was able to print out the resulting pdf files.

Part 1 contains the book cover, preface and a table of contents. Parts 2--5 cover topics of tensor algebra and calculus and Part 6 introduces some differential geometry and applies it to general relativity. Parts 7--12 cover topics of continuum mechanics. Part 13 is the bibliography and three appendices and Part 14 is the index.

WARNING! You may receive a printer error if you try to print the postscript files above. To obtain a hard copy of these chapters, I recommend printing the pdf files.

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• Benjamin Crowell has written a textbook entitled General Relativity which is roughly at the same level as our textbook. The book is meant to be especially well adapted for self-study, and answers are given in the back of the book for almost all the problems. The author has generously made the book available as a free download.   [PDF]


• Alan Macdonald has provided with a minimum of mathematical machinery and in the fewest possible pages, a clear and careful explanation of the physical principles and applications of classical general relativity. Check out his free textbook entitled General Relativity in a Nutshell.   [PDF]


• Sean M. Carroll has provided a copy of a set of lecture notes for a course in general relativity, which has since been expanded into a textbook, Spacetime and Geometry: An Introduction to General Relativity. You can find the lecture notes, along with some useful general relativity links, at the following web page.


• Andrew Hamilton has provided a draft of a textbook entitled General Relativity, Black Holes, and Cosmology. Although this book contains a significant amount of graduate school level material, Part I of this book, entitled "Fundamentals", provides a readable treatment of general relativity that can clarify and supplement the material that is being presented in the class lectures.     [PDF]


• Daniel Baumann, an assistant professor at the University of Cambridge in England, has recently taught a course on cosmology. He has written up a very nice set of lecture notes. An outline to the notes can be found here. Chapters 1 and 3 of these notes are the most relevant to our class. If you want to get a nice introduction to Cosmic Inflation, check out Chapter 2. The full pdf file of the lecture notes are available for downloading.     [PDF]

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• A complete online course on relativity is available, entitled Reflections on Relativity by Kevin Brown.


• A web-based textbook on relativity by David White, entitled Modern Relativity, is available at this link.


• Sean Carroll provides a webpage with many links to resources to aid in the learning of general relativity. Click here. Unfortunately, some of the links provided are broken.


• The Original Usenet Physics FAQ, currently edited by Don Koks, contains a number of interesting FAQs on relativity and cosmology.


• The arXiv.org e-print archive provides a warehouse for research papers in many areas of physics and mathematics. Check out the archives for General Relativity and Quantum Cosmology and the archives for Astrophysics. One can easily search for papers in the e-print archives as well as older papers (published before arXiv.org was founded in the early 1990s) using the INSPIRE database.


• Is energy conserved in general relativity? The answer to this question is not straightforward. Check out what Sean Carroll has to say in the following blog entry. A more detailed discussion is given by Michael Weiss and John Baez here. Finally, check out the blog of Lubos Motl; a link is provided here.


• In the November 2015 issue of Physics Today, there is a very interesting article by Michel Janssen and Jürgen Renn entitled How Einstein found his field equations. In his later years, Einstein often claimed that he had obtained the field equations of general relativity by choosing the mathematically most natural candidate. His writings during the period in which he developed general relativity tell a different story.     [PDF]


• Nathan Reed's blog entitled Algorithms and Time Warps has a number of entries on cosmology. Check out his blog from August 3, 2011 entitled Numerically modeling the expansion of the universe where he displays some interesting plots that exhibit the expanding universe. He even provides a link to his Python code that was used to generate the data for the plots.


• Celebrate the 100th birthday of general relativity with a short 3 minute video narrated by David Tennant (who played the tenth incarnation of Dr. Who). Check it out at this link. And don't forget the cake.
  
  
  

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