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# 8: Sources

• 8.1: Sources in General Relativity (Part 1)
The Schrödinger equation and Maxwell’s equations treat spacetime as a stage on which particles and fields act out their roles. General relativity, however, is essentially a theory of spacetime itself. The role played by atoms or rays of light is so peripheral that by the time Einstein had derived an approximate version of the Schwarzschild metric, and used it to find the precession of Mercury’s perihelion, he still had only vague ideas of how light and matter would fit into the picture.
• 8.2: Sources in General Relativity (Part 2)
Summarizing the story of the Kreuzer and Bartlett-van Buren results, we find that observations verify to high precision one of the defining properties of general relativity, which is that all forms of energy are equivalent to mass. That is, Einstein’s famous E = mc² can be extended to gravitational effects, with the proviso that the source of gravitational fields is not really a scalar m but the stress-energy tensor T.
• 8.4: Cosmological Solutions (Part I)
We are thus led to pose two interrelated questions. First, what can empirical observations about the universe tell us about the laws of physics, such as the zero or nonzero value of the cosmological constant? Second, what can the laws of physics, combined with observation, tell us about the large-scale structure of the universe, its origin, and its fate?
• 8.5: Cosmological Solutions (Part 2)
The Friedmann equations only allow a constant a in the case where Λ is perfectly tuned relative to the other parameters, and even this artificially fine-tuned equilibrium turns out to be unstable. These considerations make a static cosmology implausible on theoretical grounds, and they are also consistent with the observed Hubble expansion.
• 8.5: Cosmological Solutions (Part 3)
With hindsight, we can see that in a quantum-mechanical context, it is natural to expect that fluctuations of the vacuum, required by the Heisenberg uncertainty principle, would contribute to the cosmological constant.
• 8.6: Sources in General Relativity (Part 3)
“Does a small enough physical object always have a world-line that is approximately a geodesic?” In other words, do Eötvös experiments give null results when carried out in laboratories using real-world apparatus of small enough size? We would like something of this type to be true, since general relativity is based on the equivalence principle, and the equivalence principle is motivated by the null results of Eötvös experiments.
• 8.7: Cosmological Solutions (Part 4)
In this section we discuss the predictions of general relativity concerning the effect of cosmological expansion on small, gravitationally bound systems such as the solar system or clusters of galaxies. The short answer is that in most realistic cosmologies (but not necessarily in “Big Rip” scenarios) the effect of expansion is not zero, but is many orders of magnitude too small to measure. Many readers will probably be willing to accept these assertions and skip the following demonstrations.
• 8.8: Mach's Principle Revisited
Robert Dicke and his student Carl Brans came up with a theory of gravity that made testable predictions, and that was specifically designed to be more Machian than general relativity.
• 8.9: Historical Note - The Steady-state Model
The steady-state model, originated by the British trio of Fred Hoyle, Hermann Bondi, and Thomas Gold. who imagined that the universe could, although expanding, remain locally in the same state at all times. If this were to happen, the empty space being opened up between the galaxies would have to be filled back in by the spontaneous creation of matter.  The model holds a strong philosophical appeal because it applies not just to conditions everywhere in space but also at all times.
• 8.E: Sources in General Relativity (Exercise)