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4.4: Experimental Tests

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    Universality of Tensor Behavior

    The techniques developed in this chapter allow us to make a variety of new predictions that can be tested by experiment. In general, the mathematical treatment of all observables in relativity as tensors means that all observables must obey the same transformation laws. This is an extremely strict statement, because it requires that a wide variety of physical systems show identical behavior. For example, we already mentioned in section 2.5 the 2007 Gravity Probe B experiment (discussed in detail on in sections 5.5 and 6.2), in which four gyroscopes aboard a satellite were observed to precess due to special- and general-relativistic effects. The gyroscopes were complicated electromechanical systems, but the predicted precession was entirely independent of these complications. We argued that if two different types of gyroscopes displayed different behaviors, then the resulting discrepancy would allow us to map out some mysterious vector field. This field would be a built-in characteristic of spacetime (not produced by any physical objects nearby), and since all observables in general relativity are supposed to be tensors, the field would have to transform as a tensor. Let’s say that this tensor was of rank 1. Since the tensor transformation law is linear, a nonzero tensor can never be transformed into a vanishing tensor in another coordinate system. But by the equivalence principle, any special, local property of spacetime can be made to vanish by transforming into a free-falling frame of reference, in which the spacetime is has a generic Lorentzian geometry. The mysterious new field should therefore vanish in such a frame. This is a contradiction, so we conclude that different types of gyroscopes cannot differ in their behavior.

    This is an example of a new way of stating the equivalence principle: there is no way to associate a preferred tensor field with spacetime.12

    12Note

    This statement of the equivalence principle, along with the others we have encountered, is summarized in Appendix C.

    Speed of Light Differing from c

    In a Lorentz invariant theory, we interpret c as a property of the underlying spacetime, not of the particles that inhabit it. One way in which Lorentz invariance could be violated would be if different types of particles had different maximum velocities. In 1997, Coleman and Glashow suggested a sensitive test for such an effect.13

    Assuming Lorentz invariance, a photon cannot decay into an electron and a positron (example 12):

    \[ \gamma → e^{+} + e^{−}\]

    Suppose, however, that material particles have a maximum speed cm = 1, while photons have a maximum speed cp > 1. Then the photon’s momentum four-vector, (E, \(\frac{E}{c_{p}}\)) is timelike, so a frame does exist in which its three-momentum is zero. The detection of cosmicray gammas from distant sources with energies on the order of 10 TeV puts an upper limit on the decay rate, implying

    \[c_p  −1 \lesssim 10^{−15}.\]

    An even more stringent limit can be put on the possibility of cp < 1. When a charged particle moves through a medium at a speed higher than the speed of light in the medium, Cerenkov radiation results. If \(c_p\) is less than 1, then Cerenkov radiation could be emitted by high-energy charged particles in a vacuum, and the particles would rapidly lose energy. The observation of cosmic-ray protons with energies ∼ 108 TeV requires

    \[c_p − 1 \lesssim −10^{−23}.\]

    Degenerate Matter

    The straightforward properties of the momentum four-vector have surprisingly far-reaching implications for matter subject to extreme pressure, as in a star that uses up all its fuel for nuclear fusion and collapses. These implications were initially considered too exotic to be taken seriously by astronomers. For historical perspective, consider that in 1916, when Einstein published the theory of general relativity, the Milky Way was believed to constitute the entire universe; the “spiral nebulae” were believed to be inside it, rather than being similar objects exterior to it. The only types of stars whose structure was understood even vaguely were those that were roughly analogous to our own sun. (It was not known that nuclear fusion was their source of energy.) The term “white dwarf” had not been invented, and neutron stars were unknown.

    An ordinary, smallish star such as our own sun has enough hydrogen to sustain fusion reactions for billions of years, maintaining an equilibrium between its gravity and the pressure of its gases. When the hydrogen is used up, it has to begin fusing heavier elements. This leads to a period of relatively rapid fluctuations in structure. Nuclear fusion proceeds up until the formation of elements as heavy as oxygen (Z = 8), but the temperatures are not high enough to overcome the strong electrical repulsion of these nuclei to create even heavier ones. Some matter is blown off, but finally nuclear reactions cease and the star collapses under the pull of its own gravity.

    To understand what happens in such a collapse, we have to understand the behavior of gases under very high pressures. In general, a surface area A within a gas is subject to collisions in a time t from the n particles occupying the volume V = Avt, where v is the typical velocity of the particles. The resulting pressure is given by P ∼ \(\frac{npv}{V}\), where p is the typical momentum.

    Definition

    • Nondegenerate gas: In an ordinary gas such as air, the particles are nonrelativistic, so \(v = \frac{p}{m}\), and the thermal energy per particle is \(\frac{p^{2}}{2m} ∼ kT\), so the pressure is \(P ∼ \frac{nkT}{V}\).
    • Nonrelativistic, degenerate gas: When a fermionic gas is subject to extreme pressure, the dominant effects creating pressure are quantum-mechanical. Because of the Pauli exclusion principle, the volume available to each particle is \(∼ \frac{V}{n}\), so its wavelength is no more than \(∼ (\frac{V}{n})^{1/3}\), leading to \(p = \frac{h}{\lambda} ∼ h(\frac{n}{V})^{1/3}\). If the speeds of the particles are still nonrelativistic, then \(v = \frac{p}{m}\) still holds, so the pressure becomes \(P ∼ (\frac{h^{2}}{m})(\frac{n}{V})^{5/3}\).
    • Relativistic, degenerate gas: If the compression is strong enough to cause highly relativistic motion for the particles, then v ≈ c, and the result is \(P ∼ hc(\frac{n}{V})^{4/3}\).

    As a star with the mass of our sun collapses, it reaches a point at which the electrons begin to behave as a degenerate gas, and the collapse stops. The resulting object is called a white dwarf. A white dwarf should be an extremely compact body, about the size of the Earth. Because of its small surface area, it should emit very little light. In 1910, before the theoretical predictions had been made, Russell, Pickering, and Fleming discovered that 40 Eridani B had these characteristics. Russell recalled: “I knew enough about it, even in these paleozoic days, to realize at once that there was an extreme inconsistency between what we would then have called ‘possible’ values of the surface brightness and density. I must have shown that I was not only puzzled but crestfallen, at this exception to what looked like a very pretty rule of stellar characteristics; but Pickering smiled upon me, and said: ‘It is just these exceptions that lead to an advance in our knowledge,’ and so the white dwarfs entered the realm of study!”

    S. Chandrasekhar showed in that 1930’s that there was an upper limit to the mass of a white dwarf. We will recapitulate his calculation briefly in condensed order-of-magnitude form. The pressure at the core of the star is P ∼ ρgr ∼ GM2/r4 , where M is the total mass of the star. The star contains roughly equal numbers of neutrons, protons, and electrons, so M = Knm, where m is the mass of the electron, n is the number of electrons, and K ≈ 4000. For stars near the limit, the electrons are relativistic. Setting the pressure at the core equal to the degeneracy pressure of a relativistic gas, we find that the Chandrasekhar limit is \(∼ (\frac{hc}{G})^{3/2} (Km)^{−2} = 6M_{\odot}\). A less sloppy calculation gives something more like 1.4M. The selfconsistency of this solution is investigated in homework problem 15.

    Figure 4.4.1.png

    Figure \(\PageIndex{1}\) - Subrahmanyan Chandrasekhar (1910-1995)

    What happens to a star whose mass is above the Chandrasekhar limit? As nuclear fusion reactions flicker out, the core of the star becomes a white dwarf, but once fusion ceases completely this cannot be an equilibrium state. Now consider the nuclear reactions

    $$n \rightarrow p + e^{-} + \bar{\nu}$$

    $$p + e^{-} \rightarrow n + \nu,$$

    which happen due to the weak nuclear force. The first of these releases 0.8 MeV, and has a half-life of 14 minutes. This explains why free neutrons are not observed in significant numbers in our universe, e.g., in cosmic rays. The second reaction requires an input of 0.8 MeV of energy, so a free hydrogen atom is stable. The white dwarf contains fairly heavy nuclei, not individual protons, but similar considerations would seem to apply. A nucleus can absorb an electron and convert a proton into a neutron, and in this context the process is called electron capture. Ordinarily this process will only occur if the nucleus is neutron-deficient; once it reaches a neutronto-proton ratio that optimizes its binding energy, neutron capture cannot proceed without a source of energy to make the reaction go. In the environment of a white dwarf, however, there is such a source. The annihilation of an electron opens up a hole in the “Fermi sea.” There is now an state into which another electron is allowed to drop without violating the exclusion principle, and the effect cascades upward. In a star with a mass above the Chandrasekhar limit, this process runs to completion, with every proton being converted into a neutron. The result is a neutron star, which is essentially an atomic nucleus (with Z = 0) with the mass of a star!

    Observational evidence for the existence of neutron stars came in 1967 with the detection by Bell and Hewish at Cambridge of a mysterious radio signal with a period of 1.3373011 seconds. The signal’s observability was synchronized with the rotation of the earth relative to the stars, rather than with legal clock time or the earth’s rotation relative to the sun. This led to the conclusion that its origin was in space rather than on earth, and Bell and Hewish originally dubbed it LGM-1 for “little green men.” The discovery of a second signal, from a different direction in the sky, convinced them that it was not actually an artificial signal being generated by aliens. Bell published the observation as an appendix to her PhD thesis, and it was soon interpreted as a signal from a neutron star. Neutron stars can be highly magnetized, and because of this magnetization they may emit a directional beam of electromagnetic radiation that sweeps across the sky once per rotational period — the “lighthouse effect.” If the earth lies in the plane of the beam, a periodic signal can be detected, and the star is referred to as a pulsar. It is fairly easy to see that the short period of rotation makes it difficult to explain a pulsar as any kind of less exotic rotating object. In the approximation of Newtonian mechanics, a spherical body of density \(\rho\), rotating with a period \(T = \sqrt{\frac{3 \pi}{G \rho}}\), has zero apparent gravity at its equator, since gravity is just strong enough to accelerate an object so that it follows a circular trajectory above a fixed point on the surface (problem 14). In reality, astronomical bodies of planetary size and greater are held together by their own gravity, so we have \(T \gtrsim \frac{1}{\sqrt{G \rho}}\) for any body that does not fly apart spontaneously due to its own rotation. In the case of the Bell-Hewish pulsar, this implies \(\rho \gtrsim 10^{10}\) kg/m3, which is far larger than the density of normal matter, and also 10-100 times greater than the typical density of a white dwarf near the Chandrasekhar limit.

    An upper limit on the mass of a neutron star can be found in a manner entirely analogous to the calculation of the Chandrasekhar limit. The only difference is that the mass of a neutron is much greater than the mass of an electron, and the neutrons are the only particles present, so there is no factor of K. Assuming the more precise result of 1.4M\(\odot\) for the Chandrasekhar limit rather than our sloppy one, and ignoring the interaction of the neutrons via the strong nuclear force, we can infer an upper limit on the mass of a neutron star:

    $$1.4 M_{\odot} \left(\dfrac{Km_{e}}{m_{n}}\right)^{2} \approx 5 M_{\odot}$$

    The theoretical uncertainties in such an estimate are fairly large. Tolman, Oppenheimer, and Volkoff originally estimated it in 1939 as 0.7M\(\odot\), whereas modern estimates are more in the range of 1.5 to 3M\(\odot\). These are significantly lower than our crude estimate of 5M\(\odot\), mainly because the attractive nature of the strong nuclear force tends to pull the star toward collapse. Unambiguous results are presently impossible because of uncertainties in extrapolating the behavior of the strong force from the regime of ordinary nuclei, where it has been relatively well parametrized, into the exotic environment of a neutron star, where the density is significantly different and no protons are present. There are a variety of effects that may be difficult to anticipate or to calculate. For example, Brown and Bethe found in 199414 that it might be possible for the mass limit to be drastically revised because of the process e → K +\(\nu_{e}\), which is impossible in free space due to conservation of energy, but might be possible in a neutron star. Observationally, nearly all neutron stars seem to lie in a surprisingly small range of mass, between 1.3 and 1.45M\(\odot\), but in 2010 a neutron star with a mass of 1.97 ± .04 M\(\odot\) was discovered, ruling out most neutron-star models that included exotic matter.15

    For stars with masses above the Tolman-Oppenheimer-Volkoff limit, theoretical predictions become even more speculative. A variety of bizarre objects has been proposed, including black stars, gravastars, quark stars, boson stars, Q-balls, and electroweak stars. 

    It seems likely, however, both on theoretical and observational grounds, that objects with masses of about 3 to 20 solar masses end up as black holes; see section 6.3.

    References

    13 arxiv.org/abs/hep-ph/9703240

    14 H.A. Bethe and G.E. Brown, “Observational constraints on the maximum neutron star mass,” Astrophys. J. 445 (1995) L129. G.E. Brown and H.A. Bethe, “A Scenario for a Large Number of Low-Mass Black Holes in the Galaxy,” Astrophys. J. 423 (1994) 659. Both papers are available at adsabs.harvard.edu.

    15 Demorest et al., arxiv.org/abs/1010.5788v1.

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