The Planck length is about 10−20 times the diameter of a proton, and thus is exceedingly small.
Simple dimensional analysis shows that the measurement of the position of physical objects with precision to the Planck length is problematic. Indeed, we will discuss the following thought experiment. Suppose we want to determine the position of an object using electromagnetic radiation, i.e., photons. The greater the energy of photons, the shorter their wavelength and the more accurate the measurement. If the photon has enough energy to measure objects the size of the Planck length, it would collapse into a black hole and the measurement would be impossible. Thus, the Planck length sets the fundamental limits on the accuracy of length measurement.
Suppose we have a generator of photons of different energies. The question is whether it is possible to increase the energy of the photons to infinity? Let us analyze this situation. According to general relativity, any form of energy, including massless photons, should generate a gravitational field. The higher the energy of the photon, the more powerful gravitational field is generated. We know that the photon has a kinetic energy , where is the photon momentum, аnd its speed. This energy is positive. But the photon has, according to general relativity, gravitational (potential) energy. This energy is negative. Typically, the potential energy of the photon is simply ignored. We find its formula from the analogy with the potential energy of massive particles. For a homogeneous sphere of radius and mass , its gravitational energy has the form
where is the gravitational constant, is the mass of the ball, and its radius. But a photon has no mass . Therefore is replaced by the , where is the photon momentum and is the speed of light in a vacuum. Then the gravitational energy of the self-interacting photon has the form
where is necessary to compare with the photon's wavelength . The total energy of the photon is the sum of kinetic and potential energies and has the following form
(here photon spin is not considered, but it is not essential).
Consider this equation from the quantum point of view. We use the Heisenberg uncertainty principle between the momentum of the photon and its coordinates. We assume that , where is the Dirac constant. Using this relation (substituting ), we find the function from the last equation:
where is the fundamental Planck length, which appears here automatically.
Graph of the function
When we construct a graph of the function , we can see that as the photon wavelength decreases, its energy increases. The maximum total energy is approximately equal to the Planck energy, where the photon wavelength is approximately equal to the Planck length. However, if the photon momentum continues to increase, its total energy begins to decrease due to the increase of the gravitational energy of the photon. When the wavelength of the photon is equal to the Planck length, its total energy zero; The photon collapses and turns into a microscopic black hole, the hypothetical Planck particle.
with metric coefficients , taken from Schwarzschild solution; where is the action and is the particle mass. It is a generalization of the equation between energy and momentum in special relativity . This equation is generally covariant (physical content of equations does not depend on the choice of coordinate system). This Hamilton-Jacobi equation has the form
It can be rewritten as follows:
where is the angular momentum of a particle and is the gravitational radius of the central attracting body.
The following assumptions are necessary for the approach above: 1) The mass of the particle is zero, 2) Angular momentum (spin of the photon) can be neglected, 3) The Heisenberg uncertainty principle is simplified to . We then obtain an approximate equation for the total energy of the photon
where is the wavelength of a photon and is the gravitational radius. For self-interacting photons, mass should be replaced by ; where = is the momentum of a photon. The resulting equation agrees with the equation for the total energy of the photon to within a factor of .
To account for the angular momentum of the photon in the above equation it is necessary to substitute with ; where is the quantum number of the total angular momentum of the photon. The angular momentum of a photon leads to the formation of internal event horizon in Planck black hole.
Analysis of the Hamilton-Jacobi equation for the self-interacting photon in spaces of different dimensions indicates a preference (energy gain) for three-dimensional space for the emergence of the Planck black holes - both real and virtual (quantum foam), see Figure.
Graphs of the collapse of the photon in the spaces of different dimensions.
Indeed, according to Ehrenfest, expressions for the potential energy in spaces of various dimensions are of the form
where - the interaction constant in -dimensional space. With the usual Newton's constant it is linked through cross-linking potentials for 3-dimensional space and the corresponding -dimensional space.
For the potential energy of the photon, these equations have the form (given that )
Then the total energy of the photon is approximately equal to
where on the space dimension is independent.
Graphics functions are shown in Figure (here ).
Thus gain in energy, apparently, predetermined three-dimensionality of the observed space, given that the Planck virtual black holes form the so-called spacetime quantum foam, which is the foundation of the "fabric" of the Universe.
There is currently no proven physical significance of the Planck length; it is, however, a topic of theoretical research. Since the Planck length is so many orders of magnitude smaller than any current instrument could possibly measure, there is no way of examining it directly. According to the generalized uncertainty principle (a concept from speculative models of quantum gravity), the Planck length is, in principle, within a factor of order unity, the shortest measurable length – and no improvement in measurement instruments could change that.
Physical meaning of the Planck length can be determined as follows:
In the derivation of his equations, Einstein suggested that physical spacetime is Riemannian, ie curved. A small domain of it is approximately flat spacetime.
For any tensor field value we may call a tensor density, where is the determinant of the metric tensor . The integral is a tensor if the domain of integration is small. It is not a tensor if the domain of integration is not small, because it then consists of a sum of tensors located at different points and it does not transform in any simple way under a transformation of coordinates. Here we consider only small domains. This is also true for the integration over the three-dimensional hypersurface .
The resulting tensor equation can be rewritten in another form. Since then
where is the Schwarzschild radius, is the 4-speed, is the gravitational mass. This record reveals the physical meaning of . There is a similarity between the obtained tensor equation and the expression for the gravitational radius of the body (the Schwarzschild radius). Indeed, for static spherically symmetric field and static distribution of matter have . In this case we obtain
In a small area of spacetime is almost flat and this equation can be written in the operator form
It is also seen that the spacetime metric is always fluctuates even in the absence of an external gravitational field. This gives rise to the so-called quantum foam, consisting of virtual Planckian black holes. But these fluctuations in the macrocosm and in the world of atoms are very small compared to and become noticeable only at the Planck scale. Fluctuations need to be considered when using the Minkowski metric of special relativity for very small regions of space and large momenta.
Prescribed above uncertainty relation valid for strong gravitational fields, as in any sufficiently small domain of a strong field spacetime is essentially flat.
This implies that the Planck scale is the limit below which the very notions of space and length cease to exist. Any attempt to investigate the possible existence of shorter distances (less than 1.6 ×10−35 m), by performing higher-energy collisions, would inevitably result in black hole production. Higher-energy collisions, rather than splitting matter into finer pieces, would simply produce bigger black holes. Reduction of the Compton wavelength of the particle increases the Schwarzschild radius. The resulting uncertainty relation generates at the Planck scale virtual black holes.
In some forms of quantum gravity, the Planck length is the length scale at which the structure of spacetime becomes dominated by quantum effects, and it is impossible to determine the difference between two locations less than one Planck length apart. The precise effects of quantum gravity are unknown; it is often guessed that spacetime might have a discrete or foamy structure at a Planck length scale.
If large extra dimensions exist, the measured strength of gravity may be much smaller than its true (small-scale) value. In this case the Planck length would have no fundamental physical significance, and quantum gravitational effects would appear at other scales.
In string theory, the Planck length is the order of magnitude of the oscillating strings that form elementary particles, and shorter lengths do not make physical sense. The string scale is related to the Planck scale by , where is the string coupling constant. Contrary to what the name suggests, the string coupling constant is not constant, but depends on the value of a scalar field known as the dilaton.
In loop quantum gravity, area is quantized, and the Planck area is, within a factor of order unity, the smallest possible area value.
The size of the Planck length can be visualized as follows: if a particle or dot about 0.1mm in size (which is at or near the smallest the unaided human eye can see) were magnified in size to be as large as the observable universe, then inside that universe-sized "dot", the Planck length would be roughly the size of an actual 0.1mm dot. In other words, the diameter of the observable universe is to within less than an order of magnitude, larger than a 0.1 millimeter object, roughly at or near the limits of the unaided human eye, by about the same factor (1031) as that 0.1mm object or dot is larger than the Planck length. More simply – on a logarithmic scale, a dot is halfway between the Planck length and the size of the observable universe.