SCEST21: Schrodinger's Cat, and Einstein's Space-time, in the 21st Century
A blogspot for discussing the connection between quantum foundations and quantum gravity
Managed by: Tejinder Pal Singh, Physicist, Tata Institute of Fundamental Research, Mumbai
If you are a professional researcher / student researching on these topics, and would like to post an article here with you as author, you are welcome to do so. Please e-mail your write-up to tpsingh@tifr.res.in and it will be uploaded here.
Keywords: Quantum foundations; Quantum gravity; Schrodinger's cat; Spontaneous collapse theory;
Trace dynamics; Non-commutative geometry; Spontaneous quantum gravity; Classical general relativity
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January 11, 2020
Why does a charged rotating black hole have the same gyromagnetic ratio as an electron?!
We know that a charge moving in a magnetic field experiences a force. A moving charge is of course a current. So, a current carrying wire experiences a force in a magnetic field. The same would be true if the wire forms a closed loop, say for simplicity a circular loop with some radius, in which a charge is going around with some angular velocity. The response of such a charge to an external magnetic field depends on the value of the charge, the angular speed, and the area of the loop. These quantities combine to define the magnetic moment of the current loop, which is simply current times the area. The response of a current in a loop depends on the magnetic moment, and is essentially magnetic moment times the external magnetic field.
It is easy to show that the magnetic moment is proportional to the orbital angular momentum of the charge in the loop. The ratio of the magnetic moment to the angular momentum is a constant q/2m, independent of the parameters of the orbit, where q is the magnitude of the charge, and m the mass of the particle. This ratio, q/2m, known as the gyromagnetic ratio, is the object of our interest in today’s post.
So if we have an electron with a charge e and mass m going around in a loop, it will have a g-ratio of e/2m. Now, an electron also has a spin, I.e. an intrinsic angular momentum, distinct from orbital angular momentum. It is a purely quantum mechanical effect, arising out of the Dirac equation, and the spin of an electron is one-half the Planck constant. The electron also has an intrinsic magnetic moment because it is a charge with angular momentum. Again, this magnetic moment is proportional to intrinsic angular momentum, but the g-ratio is now e/m, not e/2m. This can be proved from the Dirac equation. The g-ratio of a Dirac fermion is twice the classical value - this can be proved from the Dirac equation. Nothing terribly surprising if the quantum value is twice the classical value. The surprise comes now, when we talk of black holes.
Black holes are solutions of Einstein equations, and their properties are determined by the laws of classical general relativity. These laws are *classical*, not quantum, and black holes are classical objects. A black hole is described by at most three parameters - its mass, charge, and angular momentum, let’s label them M, Q and J respectively. The composition of this mass is not relevant for describing a black hole. Already we see something striking - an electron is also described by its mass, charge, and (spin) angular momentum! A black hole with no charge or angular momentum is described only by its mass M, and is called the Schwarzschild black hole. A Reissner-Nordstrom black hole has mass and charge, but no angular momentum. The most general black hole is the Kerr-Newman black hole, where M, Q and J are all non-zero. This is the charged rotating black hole. It has a magnetic moment, and a g-ratio, which can be calculated from Einstein equations. The g-ratio turns out to be Q/M, not Q/2M, agreeing with the Dirac quantum value for the electron, and disagreeing with the classical value! Even though black holes are classical objects.
This should set alarm bells ringing. In my view, this is a great hint that, despite appearances. Einstein equations have something to do with the Dirac equation, and black holes have something to do with Dirac fermions. Thus, since classical general relativity (with matter) is an approximation to a quantum gravity theory (along with matter), the latter theory must explain why a black hole has a g-ratio twice the classical value. This should be a requirement for a quantum gravity theory to be viable.
There is already considerable evidence in known physics that Einstein equations are deeply connected with the Dirac equation. Firstly, if we ask how to describe the dynamics of a relativistic particle of some mass m, the answer is ambiguous. We could describe it by the Dirac equation, which claims to hold for arbitrary values of mass; or we could describe it by Einstein equations, which also claim to hold for arbitrary values of the mass. Obviously, both descriptions cannot be correct. From experience we know that Dirac equation works for small particles, which are quantum in nature, and Einstein equations work for large objects, which are classical in nature. But how small is small, and how large is large? Neither the Dirac equation nor Einstein equations have a mass scale. Obviously then, there ought to exist an underlying theory with a mass scale (or equivalently a length scale), such that for masses much smaller than this scale, the theory reduces to Dirac equation for quantum systems. And for masses much larger than this scale, the theory reduces to Einstein equations for classical systems.
The next big hint that Einstein equations are connected with Dirac equations, comes from geometry - something we discussed in an earlier post. Given a Riemannian manifold, and the standard Dirac operator on it, the sum of the eigenvalues of the square of the Dirac operator is proportional to the Einstein-Hilbert action on this manifold. This remarkable result again suggests an Einstein-Dirac connection, provided we figure out a way to include Dirac fermions in the theory.
This is precisely what has been done in our recently proposed theory of spontaneous quantum gravity. An atom of space-time-matter has an associated length scale L, which is interpreted as its Compton wave length. If L is much larger than Planck length the theory reduces to quantum theory and the Dirac equation for a fermion. If L is much smaller than Planck length, the equations describing the STM atoms reduce to Einstein equations of general relativity, and the collection of STM atoms can be shown to be a classical black hole! One way to understand this is to note that given L, one can define another length from it, namely the square of Planck length divided by L. This is a quantity equal to the Schwarzschild radius, and it exceeds Planck length when L goes below Planck length. Thus when L crosses from a value larger than Planck length to a value smaller than Planck length, there is a cross-over from the Dirac fermion phase to the Einstein black hole phase. The net mass in the system crosses over from less than Planck mass to greater than Planck mass. A collection of entangled STM atoms behaves quantum mechanically, obeying the Dirac equation, if the total mass is less than Planck mass. The collection behaves like a black hole if he total mass exceeds Planck mass.
There in fact is a duality which maps a Dirac fermion to a black hole. It can be shown that if a solution describes an STM atom with Compton wavelength L, the adjoint of this solution describes a black hole with Schwarzschild radius L and Compton length L’=L_P^2/L. Now, if we associate an electric charge e with the Dirac fermion, we can associate a dipole moment eL with the STM atom. There is strong evidence that this dipole moment remains unchanged under the said duality map, which maps the Dirac fermion to a black hole with charge Q and mass M=1/L’ in such a way that eL = QL’. But this product is nothing but the gyromagnetic ratio. Hence this duality between Dirac fermions and black holes explains why a charged rotating black hole has the same gyromagnetic ratio as the electron. Black holes and electrons are simply two different states of atoms of space-time-matter. We consider this to be compelling evidence that spontaneous quantum gravity is a viable theory of quantum gravity.
