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If we consider a particle to be point-like, wouldn't it produce a Schwarzschild spacetime in it's vicinity?

What does spacetime look-like in the vicinity of point particles? What experiments have been done in regards to detailing spacetime geometry in the vicinity of point particles?

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There are two important (intrinsic) length scales which can be associated to a point-like concentration of mass. The first is its Compton wavelength, and the second is its Schwarzschild radius: $$\lambda_c = \frac{2\pi \hbar}{mc}\qquad R_s = \frac{2Gm}{c^2}$$

The Compton wavelength sets the length scale at which quantum field theory becomes necessary. Interactions which probe such distances require energies comparable to the rest mass of the particle, and therefore are coupled to relativistic effects (e.g. pair production, relativistic kinematics, etc). On the other hand, the Schwarzschild radius sets the length scale at which spacetime curvature becomes important, and gravitational effects (e.g. the existence of an event horizon) become important.

The key thing to note is that the Compton wavelength and the Schwarzschild radius depend on the mass of the particle in opposite ways. For an electron, the Compton wavelength is approximately $10^{-12}$ m, while the Schwarzschild radius is on the order of $10^{-57}$ meters. As a result, at distances within the realm of experimental accessibility, the dynamics of the electron are completely dominated by QFT effects.

Contrast this with a stellar mass black hole, with $M\sim 10^{30}$ kg. The Compton wavelength is $10^{-72}$ m, while the Schwarzschild radius is about $1.5$ km. In constrast to the case of the electron, QFT effects are completely swamped by GR effects - at least until one probes (ludicrously) deep within the event horizon.

In between these two extremes lies the mass range $m \sim 10\ \mu$g - the Planck mass scale. This is the mass regime in which the Schwarzschild radius and the Compton wavelength are on the same order of magnitude. For such objects, relativistic quantum effects are of equal importance to spacetime curvature effects. The dynamics of such a particle cannot be adequately described by one without the other, and so this is the scale at which a coherent theory of quantum gravity would be needed.

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At first glance, considering spherical symmetry and so on, not necessarily the truth for a particle (if it were charged then the metric would correspond to the Reissner–Nordström spacetime), the mass is too low to produce anything in terms of a curvature or any other notable effect. To get a feeling, plug in the mass of a subatomic particle into the Schwarzschild radius formula (see https://en.wikipedia.org/wiki/Schwarzschild_radius) and you would see that the radius is well below the Planck length and thus beyond of our present understanding and description of physics.

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The question assumes that it is possible to think classically, in terms of a point particle in classical spacetime. However, quantum mechanics tells us that we cannot, in the general case, describe a particle as having exact position, meaning that our classical description of spacetime has already broken down. It is not simply that the particle has unknown position. The property of position for a particle does not generally exist except in measurement of position, and then it only exists to the accuracy of measurement. This is a distance scale many orders of magnitude greater than the Schwarzschild radius of an elementary particle.

One way I have thought about this is to recognise that position exists as a result of electromagnetic interactions between photons and charged particles. This was proposed in Bondi's Relativity and Commonsense, and it can be fleshed out by using Feynman's description of QED based on Feynman diagrams. Mathematically, Feynman diagrams are graphs. The lines and vertices have meaning, the paper on which the graph is drawn does not. From this perspective spacetime does not appear in the fundamental description of matter, or in the vicinity of point particles, but is an emergent property from the interactions of matter. I have explored the relevance to general relativity in the final chapters of my second and third books.

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