It follows from equations of quantum mechanics for discrete energy levels of the hydrogen atom and its size that the energy of the photon which the atom emits is inversely proportional to the size of the atom. This conclusion is consistent with the quantum theory of gravity and contradicts the general theory of relativity.
One can better understand any theory by some example. If we have two different theories, then to better understand their difference, we should take a simple example, which explain these two theories in different ways. We immediately “catch two birds with one stone.” First, we will better understand the essence of each theory, and secondly, clearly understand their difference.
Now we will compare two theories of gravity: the general theory of relativity and the quantum theory of gravitation. A simple example is the ordinary hydrogen atom, which consists of a proton and an electron.
Here is the formula for the Bohr radius of the hydrogen atom in the CGS system (the simplest form):
Here e is the electron charge (which equals the proton charge), m is an electron’s mass, and ћ is Planck’s constant.
Wikipedia says: “They often use the Bohr radius in atomic physics as an atomic unit of length.” Simply put, this is the natural standard of length. If we consider that the space-time scale changes in the gravitational field, then we should understand this change by the example of the simplest atom.
In the quantum theory of gravitation and in the general relativity, all standards (scales) of length vary in a gravitational field in the same proportion. If, for example, the meter standard is reduced by 1 percent, then any other standard, including the sizes of all atoms, should also decrease. So, we choose the Bohr radius (1) as the standard of length. This is the simplest and most understandable standard.
It is important that we see its connection with other fundamental constants. Therefore, if the Bohr radius, for example, decreases by 1 percent in a gravitational field, then the electron charge, the electron mass and Planck’s constant change so that the calculated Bohr radius decrease by formula (1) also equals 1 percent.
It is worth emphasizing that the electron charge in both gravity theories (the quantum theory of gravitation and the general relativity) does not change in a gravitational field.
Express the electron charge using the Bohr radius from Eq. (1):
All the fundamental quantities in the right-hand side of equation (2) can somehow change or not change in a gravitational field. And they change the way that the electron charge remains constant.
The energy levels En in a hydrogen atom have a discrete spectrum of values and are determined by the Bohr formula:
Здесь mP – масса протона. При переходе электрона с уровня Еn на уровень Еk (n > k) излучается фотон с энергией:
ε = ћω = Еn – Еk и частотой: ω = (Еn– Еk)/ћ. Введём новую величину Z:
The value Z depends only on the electron charge and dimensionless constants. So, it does not depend on the gravitational potential. The photon energy is:
ε = Zm/ћ2 (5)
Its frequency is:
ω = Zm/ћ3 (6)
Rewrite equation (5):
Z = ε ћ2/m (7)
Divide equation (7) into equation (2) and obtain:
Z/ e2= ε a (8)
ε a = const (9)
We got a simple but interesting result. The energy of processes occurring in atom, including the emission of photons, is uniquely related to its size. The smaller the size of atom, the greater the energy of emitted photon. And vice versa. This is clear even from the most general considerations. If the system of charges is squeezed in 2 times, then the field will increase 4 times, the energy density of the field will increase, respectively, 16 times. And the volume will decrease 8 times. Therefore, the total energy will increase 2 times, that is, inversely proportional to the size.
Equation (9) is derived from the basic formulas of quantum mechanics. Thus, quantum mechanics imposes a rigid restriction on any theory of gravity. The space-time scale in modern physics is connected with an atom and with processes taking place in it, including radiation. Therefore, the space-time scale changes in a gravitational field so as to satisfy equations (9) and (2).
Let’s see how the space-time scale changes in the quantum theory of gravitation and the general relativity. We will consider a static and weak gravitational field. This is enough to understand the difference between two theories. A weak field is, in which the objects are accelerated to speeds much less than the speed of light.
We have two hydrogen atoms. The first places on the Earth’s surface, and the other is on height H. Question: How will the properties of the upper atom change with respect to the lower one? And vice versa.
1. Atomic size
According to the fundamental formulas of the quantum theory of gravitation, in a gravitational field the speed of light increases, Planck’s constant decreases and the electron mass decreases. Therefore, in accordance with equation (1), size of an atom also decreases in a gravitational field. In particular, size of the lower atom will be smaller by a relative value of gH/c2. If size of the lower atom is а0, and the upper аН, then:
If we introduce the notation for changing the gravitational potential Δφ = – gH, then:
So, according to the quantum theory of gravitation, the hydrogen atom immersed in the gravitational field will be smaller.
What about a hydrogen atom in the general relativity?
According to the general relativity, the speed of light in a weak gravitational field (|Δφ| « с2) decreases in proportion to (1 + 2Δφ/с2), and the duration of the second increases proportionally (1 – Δφ/с2), Δφ < 0. Therefore, the size of meter and, hence, the size of atom will change just like in the quantum theory of gravitation:
Here is how Professor M. Bowler comments in his book Gravitation and Relativity, Moscow: Mir, 1979. This is a textbook on general relativity, written on the basis of lectures that M. Bowler read to students at Oxford University. Here is the beginning of Chapter 6, “Deformation of Reference Systems,” pages 89 and 90:
Fig. 1. M. Bowler “Gravitation and Relativity”, p. 89 and 90
So, according to the quantum theory of gravitation and the general relativity, the size of atom decreases in a gravitational field. In a weak field, this decrease is described by the identical equations (10) and (13).
2. Photon’s energy
If we assume that the electron charge does not change in a gravitational field (this is assumed both in the quantum theory of gravitation and the general relativity), then the energy of photon emitted by a hydrogen atom is determined by equation (9). That is, photon’s energy is inversely proportional to the size of atom. Since the lower hydrogen atom has a smaller size, the photon emitted by it must have a greater energy than the photon emitted by the upper atom. This agrees with the quantum theory of gravitation and contradicts the general relativity since it is assumed in the general relativity that the energy (and frequency) of photon emitted by the lower atom is smaller.