General Relativity for Undergraduates

Ideas, Approaches, Experiences, Insights -- Articles by Speakers and Participants
(click below to view)

Relativistic Effects in the Global Positioning System
by Neil Ashby

Some Thoughts on Involving Undergraduate Students in GR-Related Research 
by Thomas Baumgarte


Light Cones in the Schwarzschild Geometry
by Jeff Bowen

Spinning Charged Bodies and the Linearized Kerr Metric (PDF)
by Joel Franklin

Quantum gravity with undergraduates
by Seth Major

General Relativity in theUndergraduate Physics Curriculum
by James Hartle

Pedagogical Strategy
by James Hartle

Tips on Teaching GR (with Tensors) to Undergraduates
by Tom Moore

Acceleration of Light at Earth’s Surface
by Richard Mould

A One-Term (1 Quarter) Undergraduate Course on General Relativity with Applications
by George W. Rainey

Teaching General Relativity: A Seven-Layer Cake
by Ian H. Redmount

The challenge of changing deeply held student beliefs about the relativityof simultaneity
by Stamatis Vokos

Student understanding of time in special relativity: Simultaneity and reference frames
by Stamatis Vokos

About Teaching General Relativity:
history, motivation, experiment

by Rainer Weiss
(4.7 Mb PDF)






Light Cones in the Schwarzschild Geometry

by Jeff Bowen[*]

Undergraduates studying general relativity need guidance to develop useful skills.  A step-by-step worksheet for plotting the Schwarzschild light cones is presented and described.  First, the students seek to characterize the null rays by setting the metric ds2 to zero. For Schwarzschild in Eddington-Finkelstein coordinates, there are then three different types of solutions.  These are expressed in terms of , and conditions for the slope are found.  Finally, a table of r and light cone slope values makes it easy to draw typical light cones on a plot. 

I have taught a general relativity course at Bucknell many times over the last twenty years.  I used Misner, Thorne, and Wheeler the first few times, and then, when it became available in 2003, Jim Hartle's Gravity the last two times.  One tool that I find particularly useful to help students visualize the structure of curved spacetime is a plot of the light cones at various places.  It is especially effective when the students create the plot themselves.

 Analysis of Worksheet
To that end, I have developed a worksheet on constructing light cone plots for use in class.  (A copy of the worksheet is attached as the last page.)  Before using this worksheet, students have seen the static spherical solution in Schwarzschild coordinates, and have gone through the algebra to transform to Eddington-Finkelstein coordinates.  Now with the worksheet, it's time to get a feel for the coordinates and the nature of the horizon by comparing the light cone structure of Schwarzschild with that of flat Minkowski spacetime.

 In class, I have my students pair up at their tables, and each student has a separate worksheet.  The idea is that they complete the sheet in collaboration with their partner, and I interact with the class as a whole as we go through each part. 

 For the first part, I ask things like, "Why the zero on the left?" or "What happened to the dq part?".  I may also do some individual coaching; for instance, "What simple functional form could v take for the first type?" (Answer: any constant).

 For part II of the worksheet, I encourage students to use the results from part I to change to the new time-like variable, and to check that their answer is the same as my given answer.

 For part III, the students should use the results of I.2), II.1) and II.3) to construct line segments through my given points on the    plot.  For the part II.3) result, they should first fill out the short table of slope values for representative locations.  Most students correctly interpret the 2M+ and 2M- notation to mean approaching 2M from just above and just below, so answers like "big positive" or "big negative" are sufficient.  What's important is that they can translate their entries to the plot.

The last thing to put in the plots is to indicate which way the light cones open.  Students can show this by drawing little ellipses onto the X's they've made at the various locations on the plot.  The ellipse should connect two adjacent rays that form the X, and also cover the time-like region between the rays.

 For example, look at the r = 3M point, and consider the line r = 3M (with q and f constant).  Is this time-like?  By putting these values into the original metric, students can see that


 so yes, the path is time-like.  This means the ellipses are drawn horizontally so that the forward light cone opens upward on the plot.  On the other hand, the line at r = M is spacelike, so the forward cones open upward and to the left.

When all the plots are completed, we conclude the exercise with a discussion of what the plot can tell you: how the light cones tip, and what consequences that tipping might have for particle motion in Schwarzschild.  The nature of a horizon is revealed!




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