exponent137
Oct18-06, 10:12 AM
Abstract
Firstly, the article shows that the gravitational constant G inside of a black hole is rapidly changing. It then shows how this variable quantity allows us to understand why elementary particles are formed by gravity. The analysis of dimensionless constants, formed from gravitational constant, Planck’s constant, the speed of light and the elementary particle masses offers additional arguments for the theory that elementary particles are formed by gravity.
Comment
This manuscript is part of article, published in www.philica.com as number 17 [1a]. This manuscript can be found in article [1a] in sections 1, 2, 3 and in appendix A.
1. Introduction
This article presents the problem of merging general relativity theory (GR) and quantum mechanics (QM) from an alternative point of view.
As first in section 2 there are shown four possible approaches to theory that gravitational constant is rapidly variable. Importance of rapid variability is that it enables possibility that elementary particles are black holes. Because G is variable, Planck's mass is also variable. This means that it can sometimes be smaller than the masses of the elementary particles, so it follows that elementary particles can be formed by gravity.
Dimensionless constants b_i are formed from G, masses of elementary particles, m_i, Planck's constant, h, and speed of light, c. It is shown in section 3 that the existence of the dimensionless constants b_i is an additional argument that elementary particles are formed by gravity. A further argument that elementary particles are formed by gravity is the theory of Mark Hadley [1..6], which explains quantum logic with only four-dimensional time-symmetric geons. The significance of the constants b_i for Hadley’s theory is also shown in section 3.
There are some new (or unnoticed) things in this article, regarding other theories of quantum gravity:
1. Principle of equivalence in quantum regime is changed
2. Gravitational constant is rapidly variable
3. Black holes can be lighter than Planck’s mass
4. Importance of dimensionless gravitational constants of particles is analysed
5. Different view on relativistic mass
My theory persists on importance of equivalence principle and it does not demand change of other basic principles as Lorentz principle, the same G at small distances, etc.. Therefore all measurements, which try to refute these principles, but do not succeeded, are important for me. Planck mass physics can be tested by high energy cosmic particles and this is important for testing of all quantum gravity theories.
2. The rapid variability of the gravitational constant G
Various independent arguments that G is rapidly variable inside of BHs are given below.
2.1 Approach to G variability from Hadley’s equation of quantum gravity
Mark Hadley [1] shows the connection between GR and QM by the example of Einstein's equation of GR.
G_{uv} = C T_{uv} (2.1)
The quantity C is a proportionality constant, and tensor T_uv is the energy-momentum part of the equation, which forms the curvature of space-time described by tensor G_uv. It is known that the energy-momentum part T_uv is quantized. Therefore, Page and Geilker [7] assumed that the quantum version of equation (2.1) equals:
G_{uv} = C \langle \Psi|T_{uv}|\Psi \rangle (2.2)
This equation was obtained by replacing the energy-momentum tensor in (2.1) by the expectation value of the energy-momentum operator with respect to a quantum state PSI. Hadley wrote that the left side of equation is incorrect, because it is in contradiction with QM - some physical quantities can be precisely determined in this way, which is in contradiction to QM. The expectation value can also be equal to a value that does not exist. For example, the mean spin of an electron is zero, but electron spin can only be equal to h/(4pi) or –h/(4pi). He corrected equation (2.2) and obtained:
\langle G_{uv}\rangle = C \langle \Psi|T_{uv}|\Psi\rangle (2.3)
Assuming that equation (2.2) is incorrect and equation (2.3) is correct, G_uv is shown to be rapidly variable . One possible explanation for this is that the gravitational constant G is rapidly variable. Therefore, formula (2.3) is an assumption for preliminary conclusion that G is rapidly variable.
2.2 Approach to G variability from Hawking radiation
Let us consider more precisely how the derivations of Unruh radiation and Hawking BH radiation imply a variable G.
If we are moving under acceleration, we feel Unruh radiation. If we are moving under constant acceleration, a, Unruh radiation is the thermical radiation where the temperature, T, of this radiation equals [8] :
T = \frac {h a}{4\pi^2 c k} (2.4)
where k is Boltzmann's constant.
In other words, when classical acceleration and quantum field are connected, it appears radiation with temperature, T, which is proportional to acceleration, a.
The distribution of the energy of the photons dp/dW equals:
\frac {\mathrm{d}p}{\mathrm{d}W} = \frac {1}{kT (\exp{\frac{W}{kT}}-1)} (2.5)
Let us assume that acceleration of an accelerated body can only be measured by Unruh radiation, and let us consider a single moment in time when a single photon is perceived. The precise temperature of this radiation cannot be derived from the energy of one photon, because the measurement of temperature means to measure the mean of the energies of photons. Constant temperature means only the constant mean of the energy of radiated photons. The energy of a photon is triggered by an acceleration a. Thus, it cannot be assumed that acceleration a can be constant in small time intervals. QM states that if something cannot be noticed in any way, it does not exist. Therefore the uniform acceleration of a body is only an approximation, but in fact acceleration can change thermically, according to the formula:
a = 4 \pi^2 \nu \;c \;k (2.6)
where the energy of the photon, W = hv, is changing thermically and v is the frequency of this photon. Of course the assumption of Unruh radiation as the only possibility of measurement is real only for very small accelerated bodies.
This example can be compared to the radiation of a BH. Acceleration a means gravitational acceleration, g, at the horizon of a BH. The BH emits radiation at temperature, T. But, if this BH is very small (close to Planck's mass), the temperature of the radiation may be the only possibility for measuring gravitational acceleration, g. I cannot imagine another means of measuring this acceleration. So, the constant acceleration at a BH horizon does not exist and the acceleration at BH horizon changes thermically.
If g changes rapidly, then G changes rapidly.
Hawking radiation is only a semi-classical approximation. Therefore, the assumption above is also only a semi-classical approximation. But, this is also useful. Hawking radiation is an assumption without experimental verification. Therefore, there are many possibilities for modification.
The assumption that an acceleration can be measured only by Hawking radiation gives also variable G. Variable G enables BHs that are smaller than m_pl, as will be seen in section 2.5. The acceleration g at the horizon of such BHs can only be measured by Hawking radiation. Therefore, this is an assumption which becomes fulfilling.
2.3 Approach to G variability from different quantum principle of equivalence
A variable G can be obtained in still another way: Constant acceleration cannot be achieved without gravity, because the energy of a body can be entered in “steps” with photons or any other particles. (Gravity must not be used as a source of acceleration, because Unruh acceleration is an analogy for gravitational acceleration.) The principle of equivalence says that it is not possible to differentiate if we are in a homogeneous gravitational field or in an elevator with uniform acceleration. But I would add here: “If an elevator with constant acceleration does not exist, it is not necessary that a homogeneous gravitational field exists.” If homogeneous gravitational field does not exist, G as constant in small time intervals does not exist.
The most uniform acceleration in quantum regime can be achieved by many photons of very small energies. But, equal energies of large number of these photons cannot be achieved. They demand too much information for their creation. Including the second law of thermodynamics, entropy of these photons should be maximal. This means that their energy should be distributed thermically. So the most possible uniform acceleration in quantum regime can be achieved with many low-energy photons, where their energy is thermically distributed and so acceleration is thermically distributed. Due to the equivalence principle, acceleration in gravitational field cannot be uniform, but it is rapidly thermically changing. This can be generalized that G of the black hole is rapidly thermically changing.
2.4 Approach to G variability from discussing gravity of an alone elementary particle
Variable G above was derived theoretically, but it can also be derived semi-empirically. If constant G is supposed, gravitational field of a single elementary particle (for instance neutron) cannot be detected, because change of a momentum (of any body) due to neutron’s gravitational field is smaller than the uncertainty of momentum of measured body. (Alternatively, shift of trajectory is smaller than l_pl.) The inability of existence of black holes, smaller than m_pl also gives inability of measurement of gravitational field of a single elementary particle. If the neutron’s gravitational field cannot be detected, the gravitational field of many particles also cannot be detected; otherwise this is in contradiction with principle of uncertainty. A possible and simple solution is that G is rapidly variable. I suppose possibility that G equals zero sometimes, that gravitational field is not negative (positive energy theorem), and we should suppose that uncertainty of G is maximally possible. This gives the same solution as with Hawking radiation above and with “acceleration with small photons” example. At the same time, this is one sort of test of Hawking radiation.
2.5 Consequences of variable G
The radius r of a BH equals:
r = \frac{2Gm}{c^2} (2.7)
where m represents the mass of a BH. Because G is changing rapidly, it is expected that r is also changing rapidly. But, this is not the case. The radius of a BH, r, approximately represents the distance which must be traversed by a photon to reach the horizon where it returns inside of the horizon. If the BH is not extremely small, the average G in this journey is not very different from the ideal, mean G. This means r is not changing as fast as g and G, therefore G is changing approximately thermically.
Hawking radiation is also a semi-classical approximation, because the radius of a BH cannot be precisely determined and the back-reaction of a gravitational field when a photon flies away is not respected. Nevertheless, it is most likely that higher approximations would also give thermical radiation for a BH, because if a multitude of small BHs can be in thermical equilibrium, the radiation must be thermical. (The third possible instance of a higher approximation is the above example, where G is changing approximately thermically.)
Probably, G is changing thermically, approximately thermically or in some other way that it can sometimes reach arbitrarily high value. However, if G is changing, then Planck's mass is also changing. Planck's mass m_pl equals:
m_{pl} = \left(\frac{hc}{2\pi G}\right)^{1/2} (2.8)
If G can maybe reach an arbitrarily high value, then it follows that m_pl can reach arbitrarily small values. Therefore, the value of this m_pl can sometimes be smaller than the masses of elementary particles. This offers the possibility that elementary particles can be BHs.
A problem of physics that it is not yet explained is how elementary particles can be BHs. One possibility is that G is very large at very small distances. The question is, why it is larger than normal. The possibility of a variable G is better supported by explanation than the possibility of an enlarged G at small distances.
Important elements of quantum gravity (QG) theories are also BHs remnants with mass m_pl. But no one has ever seen one of these objects. I think that objects with this mass do not exist, but elementary particles, as BHs, exist and variable G supports them.
(If elementary particles are BHs, which radiate, then the decay of a particle is a BH radiation. It follows that electrons and protons are BH remnants.)
A variable m_pl also gives a variable Planck’s distance, l_pl, and a variable Planck’s time, t_pl. The quantities l_pl and t_pl are defined as:
l_{pl} = \left (\frac{h G}{2\pi c^3}\right)^{1/2} (2.9)
t_{pl} = \left (\frac{h G}{2\pi c^5}\right)^{1/2} (2.10)
So space-time is not grained, but it can achieve arbitrarily small distances and times. The limitation for perception at very small scales is only the probability of perception, which is very small as distances are very small.
Firstly, the article shows that the gravitational constant G inside of a black hole is rapidly changing. It then shows how this variable quantity allows us to understand why elementary particles are formed by gravity. The analysis of dimensionless constants, formed from gravitational constant, Planck’s constant, the speed of light and the elementary particle masses offers additional arguments for the theory that elementary particles are formed by gravity.
Comment
This manuscript is part of article, published in www.philica.com as number 17 [1a]. This manuscript can be found in article [1a] in sections 1, 2, 3 and in appendix A.
1. Introduction
This article presents the problem of merging general relativity theory (GR) and quantum mechanics (QM) from an alternative point of view.
As first in section 2 there are shown four possible approaches to theory that gravitational constant is rapidly variable. Importance of rapid variability is that it enables possibility that elementary particles are black holes. Because G is variable, Planck's mass is also variable. This means that it can sometimes be smaller than the masses of the elementary particles, so it follows that elementary particles can be formed by gravity.
Dimensionless constants b_i are formed from G, masses of elementary particles, m_i, Planck's constant, h, and speed of light, c. It is shown in section 3 that the existence of the dimensionless constants b_i is an additional argument that elementary particles are formed by gravity. A further argument that elementary particles are formed by gravity is the theory of Mark Hadley [1..6], which explains quantum logic with only four-dimensional time-symmetric geons. The significance of the constants b_i for Hadley’s theory is also shown in section 3.
There are some new (or unnoticed) things in this article, regarding other theories of quantum gravity:
1. Principle of equivalence in quantum regime is changed
2. Gravitational constant is rapidly variable
3. Black holes can be lighter than Planck’s mass
4. Importance of dimensionless gravitational constants of particles is analysed
5. Different view on relativistic mass
My theory persists on importance of equivalence principle and it does not demand change of other basic principles as Lorentz principle, the same G at small distances, etc.. Therefore all measurements, which try to refute these principles, but do not succeeded, are important for me. Planck mass physics can be tested by high energy cosmic particles and this is important for testing of all quantum gravity theories.
2. The rapid variability of the gravitational constant G
Various independent arguments that G is rapidly variable inside of BHs are given below.
2.1 Approach to G variability from Hadley’s equation of quantum gravity
Mark Hadley [1] shows the connection between GR and QM by the example of Einstein's equation of GR.
G_{uv} = C T_{uv} (2.1)
The quantity C is a proportionality constant, and tensor T_uv is the energy-momentum part of the equation, which forms the curvature of space-time described by tensor G_uv. It is known that the energy-momentum part T_uv is quantized. Therefore, Page and Geilker [7] assumed that the quantum version of equation (2.1) equals:
G_{uv} = C \langle \Psi|T_{uv}|\Psi \rangle (2.2)
This equation was obtained by replacing the energy-momentum tensor in (2.1) by the expectation value of the energy-momentum operator with respect to a quantum state PSI. Hadley wrote that the left side of equation is incorrect, because it is in contradiction with QM - some physical quantities can be precisely determined in this way, which is in contradiction to QM. The expectation value can also be equal to a value that does not exist. For example, the mean spin of an electron is zero, but electron spin can only be equal to h/(4pi) or –h/(4pi). He corrected equation (2.2) and obtained:
\langle G_{uv}\rangle = C \langle \Psi|T_{uv}|\Psi\rangle (2.3)
Assuming that equation (2.2) is incorrect and equation (2.3) is correct, G_uv is shown to be rapidly variable . One possible explanation for this is that the gravitational constant G is rapidly variable. Therefore, formula (2.3) is an assumption for preliminary conclusion that G is rapidly variable.
2.2 Approach to G variability from Hawking radiation
Let us consider more precisely how the derivations of Unruh radiation and Hawking BH radiation imply a variable G.
If we are moving under acceleration, we feel Unruh radiation. If we are moving under constant acceleration, a, Unruh radiation is the thermical radiation where the temperature, T, of this radiation equals [8] :
T = \frac {h a}{4\pi^2 c k} (2.4)
where k is Boltzmann's constant.
In other words, when classical acceleration and quantum field are connected, it appears radiation with temperature, T, which is proportional to acceleration, a.
The distribution of the energy of the photons dp/dW equals:
\frac {\mathrm{d}p}{\mathrm{d}W} = \frac {1}{kT (\exp{\frac{W}{kT}}-1)} (2.5)
Let us assume that acceleration of an accelerated body can only be measured by Unruh radiation, and let us consider a single moment in time when a single photon is perceived. The precise temperature of this radiation cannot be derived from the energy of one photon, because the measurement of temperature means to measure the mean of the energies of photons. Constant temperature means only the constant mean of the energy of radiated photons. The energy of a photon is triggered by an acceleration a. Thus, it cannot be assumed that acceleration a can be constant in small time intervals. QM states that if something cannot be noticed in any way, it does not exist. Therefore the uniform acceleration of a body is only an approximation, but in fact acceleration can change thermically, according to the formula:
a = 4 \pi^2 \nu \;c \;k (2.6)
where the energy of the photon, W = hv, is changing thermically and v is the frequency of this photon. Of course the assumption of Unruh radiation as the only possibility of measurement is real only for very small accelerated bodies.
This example can be compared to the radiation of a BH. Acceleration a means gravitational acceleration, g, at the horizon of a BH. The BH emits radiation at temperature, T. But, if this BH is very small (close to Planck's mass), the temperature of the radiation may be the only possibility for measuring gravitational acceleration, g. I cannot imagine another means of measuring this acceleration. So, the constant acceleration at a BH horizon does not exist and the acceleration at BH horizon changes thermically.
If g changes rapidly, then G changes rapidly.
Hawking radiation is only a semi-classical approximation. Therefore, the assumption above is also only a semi-classical approximation. But, this is also useful. Hawking radiation is an assumption without experimental verification. Therefore, there are many possibilities for modification.
The assumption that an acceleration can be measured only by Hawking radiation gives also variable G. Variable G enables BHs that are smaller than m_pl, as will be seen in section 2.5. The acceleration g at the horizon of such BHs can only be measured by Hawking radiation. Therefore, this is an assumption which becomes fulfilling.
2.3 Approach to G variability from different quantum principle of equivalence
A variable G can be obtained in still another way: Constant acceleration cannot be achieved without gravity, because the energy of a body can be entered in “steps” with photons or any other particles. (Gravity must not be used as a source of acceleration, because Unruh acceleration is an analogy for gravitational acceleration.) The principle of equivalence says that it is not possible to differentiate if we are in a homogeneous gravitational field or in an elevator with uniform acceleration. But I would add here: “If an elevator with constant acceleration does not exist, it is not necessary that a homogeneous gravitational field exists.” If homogeneous gravitational field does not exist, G as constant in small time intervals does not exist.
The most uniform acceleration in quantum regime can be achieved by many photons of very small energies. But, equal energies of large number of these photons cannot be achieved. They demand too much information for their creation. Including the second law of thermodynamics, entropy of these photons should be maximal. This means that their energy should be distributed thermically. So the most possible uniform acceleration in quantum regime can be achieved with many low-energy photons, where their energy is thermically distributed and so acceleration is thermically distributed. Due to the equivalence principle, acceleration in gravitational field cannot be uniform, but it is rapidly thermically changing. This can be generalized that G of the black hole is rapidly thermically changing.
2.4 Approach to G variability from discussing gravity of an alone elementary particle
Variable G above was derived theoretically, but it can also be derived semi-empirically. If constant G is supposed, gravitational field of a single elementary particle (for instance neutron) cannot be detected, because change of a momentum (of any body) due to neutron’s gravitational field is smaller than the uncertainty of momentum of measured body. (Alternatively, shift of trajectory is smaller than l_pl.) The inability of existence of black holes, smaller than m_pl also gives inability of measurement of gravitational field of a single elementary particle. If the neutron’s gravitational field cannot be detected, the gravitational field of many particles also cannot be detected; otherwise this is in contradiction with principle of uncertainty. A possible and simple solution is that G is rapidly variable. I suppose possibility that G equals zero sometimes, that gravitational field is not negative (positive energy theorem), and we should suppose that uncertainty of G is maximally possible. This gives the same solution as with Hawking radiation above and with “acceleration with small photons” example. At the same time, this is one sort of test of Hawking radiation.
2.5 Consequences of variable G
The radius r of a BH equals:
r = \frac{2Gm}{c^2} (2.7)
where m represents the mass of a BH. Because G is changing rapidly, it is expected that r is also changing rapidly. But, this is not the case. The radius of a BH, r, approximately represents the distance which must be traversed by a photon to reach the horizon where it returns inside of the horizon. If the BH is not extremely small, the average G in this journey is not very different from the ideal, mean G. This means r is not changing as fast as g and G, therefore G is changing approximately thermically.
Hawking radiation is also a semi-classical approximation, because the radius of a BH cannot be precisely determined and the back-reaction of a gravitational field when a photon flies away is not respected. Nevertheless, it is most likely that higher approximations would also give thermical radiation for a BH, because if a multitude of small BHs can be in thermical equilibrium, the radiation must be thermical. (The third possible instance of a higher approximation is the above example, where G is changing approximately thermically.)
Probably, G is changing thermically, approximately thermically or in some other way that it can sometimes reach arbitrarily high value. However, if G is changing, then Planck's mass is also changing. Planck's mass m_pl equals:
m_{pl} = \left(\frac{hc}{2\pi G}\right)^{1/2} (2.8)
If G can maybe reach an arbitrarily high value, then it follows that m_pl can reach arbitrarily small values. Therefore, the value of this m_pl can sometimes be smaller than the masses of elementary particles. This offers the possibility that elementary particles can be BHs.
A problem of physics that it is not yet explained is how elementary particles can be BHs. One possibility is that G is very large at very small distances. The question is, why it is larger than normal. The possibility of a variable G is better supported by explanation than the possibility of an enlarged G at small distances.
Important elements of quantum gravity (QG) theories are also BHs remnants with mass m_pl. But no one has ever seen one of these objects. I think that objects with this mass do not exist, but elementary particles, as BHs, exist and variable G supports them.
(If elementary particles are BHs, which radiate, then the decay of a particle is a BH radiation. It follows that electrons and protons are BH remnants.)
A variable m_pl also gives a variable Planck’s distance, l_pl, and a variable Planck’s time, t_pl. The quantities l_pl and t_pl are defined as:
l_{pl} = \left (\frac{h G}{2\pi c^3}\right)^{1/2} (2.9)
t_{pl} = \left (\frac{h G}{2\pi c^5}\right)^{1/2} (2.10)
So space-time is not grained, but it can achieve arbitrarily small distances and times. The limitation for perception at very small scales is only the probability of perception, which is very small as distances are very small.