Discrete Space-time as an Alternate Model for Dimensions in the Physical World

AN
8/24/2007

Quantum mechanics and general relativity have scored spectacular successes, the former in explaining the three subatomic forces and the latter in explaining gravitational force on the cosmic scale¹ . However, in certain analyses these two theories have been plagued with infinities using Cartesian mathematics based on dimensionless points^{1, 2} . In this paper, it is proposed that there exist calculable limits in space-time dimensions below which there would be no physical meaning due to the discrete nature of space-time. As proof of concept, two examples of these limits are approximated using vector field analysis. The physical world can be modeled as a discrete four-dimensional manifold.

General relativity are based on a model of continuum space-time² . In this model, the position of a space-time point is uniquely defined by a set of four real numbers, three for space and one for time, specified as coordinates in an established coordinate system. Let us examine the components of the field equation of general relativity²²

G_{μν +} Λ g_μν = 8 π G T_μν (1)

Where G_μν: curvature tensor

Λ: cosmological constant

g_μν : space-time metric tensor

T_μν: stress-energy tensor

G_μν is a function of g_μν and its first and second derivatives, in a precise mathematical technique by Newton and Leibniz. This model, using a set of real numbers from integers and fractions, is an abstract mathematical process, and there is no reason to think that it reflects a true model of the real world. In quantum mechanics, the model of continuum space-time seems even more inappropriate ^{2, 3, 4} . According to Heisenberg uncertainly principle, an infinite amount of energy would be required to localize a particle to a true (dimensionless) point in space-time. Despite the indication that space-time cannot be subdivided without limit, quantum theory (either in the form of Schrödinger wave equation or Heisenberg’s matrix mechanics) continues to use the mathematics of continuity ⁵ . All the calculations are performed over the fields of real numbers, i.e. fields with coordinates of the continuum. It is ironic that modern quantum field theory still utilizes the concept of dimensionless points in quantum fields. It has been suggested that the problematic infinities and the associated need for renormalization arise from such inconsistency. The infinities arise from integrals of the general form ² :

∫ (1/xⁿ)dx (2)

With n≥ 1 , and x= 0 ->R (where R is a real number)

The lower limit of zero is the source of infinity and it is crucial to know that the variable x is a coordinate in space-time with continuum property. From this mathematical expression, it can be seen that the infinity arises as quantum field theory is probing down to zero distance. It has been suggested that this infinity may be alleviated by introducing some non-zero limit on space-time coordinates ² . In renormalization process, one positive infinity can be used to cancel out another negative infinity leaving a finite result if one does not carry out the integration with the variable x preceding all the way to an actual zero value (see equation (2)). One problem remains is that there is no established method to determine the lower limit for this variable.

The preceding argument raises the possibility that dimensions of space-time may set certain limits below which there would be no physical meaning and conventional mathematical analyses would not apply ⁶. In continuum space, a dimensional measurement (length, area, or volume) can be any real number and can be as close to zero as one wants to. On the other hand, if the geometry is discrete or granular, then the measurement results can not be smaller than a certain limit, i.e. a minimum possible limit length. Since nothing can travel faster than light ⁷, the minimum possible time limit corresponding to each minimum possible length limit is defined as the ratio of the minimum possible length limit to the speed of light ⁸.

It has been argued that a radical overhaul of the space-time concept is needed to move towards a more discrete topological character, a quantum theory of space-time ⁹ . For space dimensions, we call these limits “quantum length limit” in lack of a more suitable term. For time dimension, the limit is called “quantum time limit”. These quantum limits are different for different masses in different fields. They are dependent on the mass of the object and the associated dominant force in consideration. This article proposes the discrete space-time model as an alternate model for dimensions in the physical world and outlines the approximating procedures using classical electrodynamics and mechanics for calculating the quantum limits. The numerical results obtained from approximating methods should be considered as qualitative, and not quantitative. They are strictly used for demonstrating proof of concept. Two examples using this model illustrate the use of quantum limits to electric field and gravity field to resolve the infinity results.

Infinite Mass of an Electron

This section will analyze the difficulty in calculating the mass of an electron using a continuum model and suggest the use of a quantum length limit to circumvent the infinity result.

Figure 1: An electron in a hydrogen atom

In the following calculations, classical electrodynamics will be used to get approximate results. The most dominant energy associated with an electron is due to electromagnetic force. Gravitational force’s contribution is considered negligible in any particle analysis ¹⁰ .

From Coulomb’s law, the force between the electron and the proton in the hydrogen atom (see Fig. 2) is ^{11, 12} :

F = ke²/ r² (3)

Where k : the constant in Coulomb’s law

e : the charge of the electron (and the proton with opposite sign)

r : the distance between the center points of the electron and the proton

The potential energy of the electron is ^{11, 12} :

W = ke² ∫ (1/r²)dr where r= 0 ->∞

= -ke² [(1/∞) - (1/0)]

The energy of the electron goes to infinity as r approaches zero, and thus the mass of the electron approaches infinity, a nonsensible result¹⁰ . This is illustrated in Figure 2.

Figure 2: The energy of the electron (W) goes to infinity as r approaches zero

However, since the energy of the electron is known from experiment data, the lower limit for r can be determined to avert this infinity result. The process of obtaining the lower limit for r (L₀) with prior data of energy (W) is demonstrated in Figure 3.

Figure 3: The finite energy (W) of the electron sets a limit on r (L₀)

Electron in atoms (such as hydrogen atom) typically travel at speeds one hundred times less than the speed of light. Therefore, the change in electron mass according to special relativity is negligible (0.01%) and the rest mass, instead of the relativistic mass, can be used in calculations ¹ .

If one sets a lower limit of L₀ for r, the energy of the electron becomes:

W = -ke² [(1/∞) - (1/L₀)]

= ke² /L₀

This is not an infinite value. The rest mass associated with this energy is ⁷

m_e= W/c² = ke² /(L₀ c²)

With known values of m_e, k, e, and c; L₀ can be calculated [data from Notes]

L_o = ke² /(m_e c²) = 2.8 x 10^-15 m

For an electron with its associated electric field, this value of L₀ (quantum length limit) is the minimum space measurement limit at this point charge (r = 0). Below this limit, there is no meaningful physical property in this discrete space model. According to the uncertainty principle, to localize a point particle in space-time would require an infinite amount of energy. The use of quantum length limit would alleviate this difficulty. Note that from the results of scattering experiments at high energy, an atom is an object with spatial extent of approximately 10^-10 m ^{10, 13}. The atomic spatial extent is therefore about 10⁵ times the size of the quantum length limit L_o.

In quantum field theory, interaction terms in Feynman diagrams become infinite as the distance between point particles become vanishingly small ⁵ . With the proposed quantum length limit being the limiting factor, infinities from calculations may potentially be averted. This approach may be compared to the concept of electromagnetic quanta in solving the ultraviolet catastrophe predicted by black body radiation ¹ .

The quantum time limit for the electron can be determined:

T₀ = L₀/c = 0.9 x 10^-23 sec

Correlation with the uncertainty principle

Heisenberg’s uncertainty principle indicates that point particles behave in some respects like waves. They do not have a definite position but one “smeared out” with a certain probability distribution which is proportional to the square of the wave amplitude ¹⁴ as shown in Schrödinger wave equation⁶

d² ψ/ dx² + (8 π² m/h²) (E - V) ψ = 0 (4)

where

ψ : wave function

x: position

m: mass

h: Planck constant

E: energy

V: potential energy

The proposed existence of quantum length limit is likely to be associated with such particle behavior subjected to measurement in a continuum model.

Correlation with the infinities in quantum field theory

Quantum field theory describes interaction between particles (for example, two electrons) by taking account a series of processes, from the simplest one to evermore complex ones. The simplest one, known as lowest order process, is the exchange of a single virtual photon. Higher order terms involve an emitted photon that creates, while on transit, an electron-positron pair. This pair quickly annihilates to produce a photon again that may be absorbed by the second electron. If we start to probe down to smaller distances we find more and more possible processes that may occur. The possibility of more complex processes becomes less likely but cannot be ignored in quantum field theory calculations since these processes lead to infinity answers. These infinities were finally resolved with a mathematical tricks called “renormalization” has been considered unsatisfactory by many¹⁶. This the renormalization technique does not even work for gravitational force in quantum field theory ¹⁹. According to the uncertainty principle, even empty space is filled with infinite number of particle-antiparticle pairs and the resulting infinite mass would curve up space to an infinitely small size ¹⁴ . Taking the lesson of Plank more than 100 years ago, one has to consider the discrete nature of space-time and not probing down beyond the limits of space-time. Only then the infinity answers given by quantum field theory can be averted without using any mathematical tricks.

Singularity of Black Hole

This section will discuss the finding of singularity of black hole based on a continuum model and suggest the use of a quantum length limit to circumvent the infinity result.

A point mass is a geometric zero-dimensional point that is assigned a finite mass ¹⁶. The density of a point mass is infinite since a point has no volume. A point mass may not exist in reality. It is only a useful mathematical tool to solve analytical problems involving bodies of which dimensions are much less than the distance between them. A black hole is one of the two objects in the physical world that has been thought to have infinite density according to general relativity ^{17, 18}. The other entity with infinite density is the space-time point of the big bang at the birth of the universe ¹⁴.

Previous analyses unequivocally showed that stars with mass more than twice that of the Sun will implode inward beyond the thresholds for white-dwarves and neutron stars to become black holes ¹⁷. The special relativity theory limits the maximum difference in the velocity of the matter particles in the star to the speed of light. Consequently, when the star density gets sufficiently high, the repulsion caused by the Pauli exclusion principle would be less than the gravity attraction ¹⁴. Gravity, a dominant force in a black hole, overwhelms both the degeneracy pressure of electrons and the nuclear force between neutrons. The black hole forms an event horizon, a boundary of a region that prevents light from escaping ¹⁹. According to general relativity, the materials that form black hole reach a singularity with infinite density. This represents a point at which the laws of physics break down. The prediction of singularities by general relativity suggests that this theory is not a complete one ²⁰. Since the singular points have to be taken out of the space-time manifold, the field equations are not defined at these points and one cannot predict what will come out of these singularities.

Now let us consider two black holes of minimum required mass M (1.4 times the mass of the Sun) which is known as the Chandrasekhar limit ¹⁴. The two black holes move towards each other and merging under gravitational attraction (see Fig. 4). In this case, we assume that the black holes are non-rotating (no angular momentum) and the effect of other forces is negligible compared to that of gravity. As the black holes move toward each other, their kinetic energy increases and their potential energy decreases according to the law of conservation of energy ¹⁰. Some of the kinetic energy may be transformed into heat through collision with particles and radiated away as x-ray or other forms of electromagnetic radiation. However, most of the kinetic energy would be captured by the merged black hole ²¹. Black holes are thought to emit radiation (Hawking radiation) due to interaction of virtual particle-antiparticle pairs appearing close to the event horizon ^{17, 14}. The loss of energy due to this black body radiation is likely to be small in the duration of this merging process ¹⁴. The total energy of the black holes is approximately equal to the gravitational energy provided that the two black holes are merged without losing any mass in the process. In the following calculations, classical mechanics with gravity being treated as a vector field will be used to get approximate results instead of using the more sophisticated tensor analysis.

Figure 4: Two colliding black holes

The distance between two black holes is r. The gravitational force between them is ²²

GM²/r² (5)

where G : the gravitational constant

r : the distance between the center points of the two black holes.

The total potential energy of the two black holes is ²²

U = 2 ∫ (GM²/r²)dr where r= 0 -> ∞

= - 2GM²[(1/∞) - (1/0)]

As the two black holes approach each other, r approaches zero and the energy of the system (the merged black hole) becomes infinite, forming a singularity.

Since the combined mass (hence energy) of the black holes is known, a lower limit for r can be determined to avert this singularity.

If one sets a lower limit of L₀ for r, the energy of the merged black hole becomes:

U = 2GM²/ L₀ , a finite value

The energy associated with the combined mass (2M) of the two black holes⁷ is 2Mc²

2GM²/ L_{0 =} 2Mc²

With known values of G, M, and c; L₀ can be calculated (Data from Notes)

L_{0 =} GM/(c²) = 2 x 10³ m

For a black hole with 2.8 the mass of the Sun, this value for L₀ (quantum length limit) is the minimum space measurement limit at the position of the black hole, below which there is no meaningful physical property in this discrete space model.

This proposed quantum length limit for the black hole indicates the discrete nature of space-time where the gravitational effect is extreme with large space-time curvature. Beyond such situation, the flat space-time can be approximated as a continuous model as seen in general relativity ¹⁹.

When two black holes collide and merge together to form a single black hole, the area of event horizon is equal to at least the sum of two original event horizon areas ^{14, 20}. The area of the event horizon for this single black hole is at least ¹⁹

A≥ A₁ + A_2, where A₁ and A₂ are event horizon area of each black hole before merging

Since the two black holes are of the same size in this case,

A≥ 2A₁

A≥ 2(4π R² ) , where R is the radius of event horizon of each black hole¹⁹

With R = 2GM/c²

A≥ 8π (2GM/c²)²

If R_f is the radius of the merged black hole’s event horizon,

4 π R_f ² ≥ 8π (2GM/c²)²

R_f ≥ 2^1/2 (2GM/c²)

With known values of G, M, and c; R_f can be calculated (data from Notes)

R_f ≥ 5.87 x 10³ m

If D_f is the diameter of the merged black hole’s event horizon,

D_f ≥ 2 x 5.87 x 10³ m = 11.74 x 10³ m

The diameter of this event horizon is therefore at least more than 6 times the size of the quantum length limit L₀

The quantum time limit for the black hole can be determined:

T₀ = L₀/c = 7 x 10^-6 sec

Implications for large scale measurements

Theoretical research done on black holes has indicated that their entropy is equal to ¼ of the event horizon area and this area represents the amount of information trapped beyond the event horizon ^{6, 19, 20}. If the space inside the black hole event horizon is continuous, then it would contain an infinite amount of information exceeding the holding capacity of the event horizon area ¹⁹. This argument further supports the discrete space model for black hole geometry.

On the large scale of the physical world, the use of quantum length limit due to gravity is also expected to alleviate uncertainty in measurement using conventional mathematics. This would be similar to the use of quantum length limit in measurements involving electromagnetic force at very small scale.

Quantum length limit over distance from a black hole as a point mass

The calculation of quantum length limit for the black hole shown above is intended for a point mass in space at its location. The quantum length limit is expected to decrease progressively over distance from this point. As the distance approaches infinity, the quantum length limit will approach the Planck length, L_p, the smallest length allowed by quantum mechanics¹. Note that L_p has a value of 10^-35 m, which is much smaller than the quantum length limit as calculated for the black hole in this study. The precise distribution of quantum length limit is unknown but can be tentatively assumed to be inversely proportional to square of the distance from the point mass. This is reasonable with the gravitational force having this same distribution. The most simple equation of this distribution is of the following form:

L_r = 1/(r² + A) + B (6)

Where:

L_r : quantum length limit as a function of r

r : distance from the point mass

A, B : coefficients to be determined

From the preceding calculation, for a black hole with mass M, the quantum length limit at the point mass is

L_{(r =0) =} GM/(c²)

As the distance approaches infinity, the ratio = 1/(r² + A) approaches zero and the quantum length limit approaches the Planck length, L_p

L_{(r =∞) =} L_p

These two boundary conditions yield

A = c² / (GM - L_p c²)

B = L_p

The distribution of quantum length limit over distance from the black hole as a point mass is shown in Figure 5. Quantum length limit has significant effect on space-time at cosmic level with extreme mass density (black holes). At increasing distances from the black hole, the value of quantum length limit vanishes progressively and approaching quantum length value, L_p . Conventional mathematics based on calculus with real numbers can be fully utilized in the latter.

Figure 5: The distribution of quantum length limit over distance from the black hole

Quantum length limit over distance from an electron as a point charge

For an electron in a hydrogen atom, similar calculations can be performed to show that

the distribution of the quantum length limit is:

L_r = 1/(r² + A) + B

Where:

L_r : quantum length limit as a function of r

r : distance from the electron as a point charge

A = (m_e c²)/(ke² - m_e L_p c²)

B = L_p

The distribution of quantum length limit is shown graphically in Figure 6. Similarly, quantum length limit has significant effect on space-time at subatomic level (electrons). At increasing the distances from the electron, the value of quantum length limit vanishes progressively and approaching quantum length value, L_p . Conventional mathematics based on calculus with real numbers can be fully utilized in the latter.

Figure 6: The distribution of quantum length limit over distance from the electron

Discussion

Calculations using quantum field theory have lead to infinite results when smaller and smaller distances between particles are approached ¹⁵. The problem of infinity in fact had actually been around long before the birth of quantum mechanics. In the 18th century, physicists had no answer for the effect of an electric field on the source charge ^{1, 15}. An electric charge will generate an electric field around it. According to Coulomb law, the force on a charged particle in this field would be inversely proportional to the square of its distance from the source charge, known as the “inverse square law” (see equation 3). But how do we work out the effect of this field on the source charge itself? Here we would get an infinite result due to division of a number by zero. In quantum field theory, the renormalization technique has been used to cancel out the infinity seen in quantum electrodynamics, also for quantum chromodynamics and radioactive decay ¹⁹. Despite very accurate results obtained with renormalization, as confirmed by experimental data, many physicists think that this technique is awkward, should not be necessary and hold out for something more fundamental ¹⁵. Similar infinities arise with the effect of gravitational field on the source mass itself in Newtonian mechanics (see equation 5). Here the renormalization technique does not even work for gravitational force in quantum field theory ¹⁹. According to the uncertainty principle, even empty space is filled with infinite number of particle-antiparticle pairs and the resulting infinite mass would curve up space to an infinitely small size ¹⁴. It has been stated that quantum physicists in the 1940s rediscovered the infinity problems, associated with point particles, already identified by the 18th century physicists ¹.

Questions have been raised regarding the validity of space-time continuity, based on real numbers, that is almost universally assumed in physical theories ⁹. Several approaches have been proposed to take us away from a continuous space-time toward something of a more discrete topological character. They include quantum loop theory, spin network, periodic lattice space-time, and string theory ⁹ . String theory^{7, 11} is currently the most well known theory that attempts to unify gravity with other three forces in nature (strong, weak and electromagnetic) by postulating that all forces and particles arise from different vibration modes of one-dimensional strings with exceeding small length, near the Planck length (10^-35 m). String theory is successful in eliminating infinities arising from the continuous space-time model. However, it requires supersymmetry and 10 dimensions for space-time for string theory to work. Supersymmetric partners of known particles have not been observed. The extra 6 space dimensions are assumed to be curled up (hidden) in Calabi-Yau manifolds and they are too small to be ever confirmed by experiments. Furthermore, various configurations of the extra dimensions (called topologies) can exist and string theory predicts the presence of multiple universes (bubbles), up to an unimaginable number of 10⁵⁰⁰, each with its own set of physical laws. In each bubble, an observer conducting experiments at low energy (as we do) will only see a specific universe with its own laws of physics. Information from other bubbles cannot reach this observer due to the vast separating space which expands too rapidly for light to overrun. String theory also has no explicit and precise formulation, only approximate equations. The exact equations for string theory are unclear, and important physical concepts remain to be discovered⁵. The difficulties in string theory have brought up questions about whether string theory would be a viable theory that could be verified or falsified by experimental confirmation²⁴.

Quantum mechanics and general relativity have scored spectacular successes, the former in explaining the three subatomic forces and the latter in explaining gravitational force on the cosmic scale of stars and galaxies ¹. However, in certain analyses these two theories have been plagued with infinities using Cartesian mathematics based on dimensionless points and real numbers ^{1, 2}. This difficulty is most likely due to the quantum fluctuation in space-time geometry. According to general relativity, space-time is dynamical. It affects, and in turn is affected by, the particles that constitute space-time itself. Quantum mechanics dictate uncertainty in position and momentum of the particles in this space-time with consequent subjection of geometry to quantum fluctuation.

In this article, it is proposed that there exist calculable limits in space-time dimensions below which there would be no physical meaning due to the discrete nature of space-time. Space-time would seem smooth and continuous when viewed on scales much larger than these limits. As proof of concept, two examples of these limits are approximated using vector field analysis. These limits are different for different points in space-time and they are determined by the mass of the object and the associated dominant force under consideration. The physical world can be modeled as a discrete four-dimensional manifold. Conventional mathematics, based on real numbers and formulated in the language of differential equations, can continue to be used in analysis involving quantum mechanics and general relativity given this constraint. Conceptually, it may be proposed that space-time has more than 4 dimensions to account for quantum length limits and quantum time limit: 4 conventional dimensions and 4 associated quantum limit dimensions (see Fig. 7).

Figure 7: Conventional dimensions and associated quantum length limit dimensions (time dimension not shown)

At point A (electron or black hole): 3 conventional space dimensions and 3 associated quantum length limit dimensions

At point B (at significant distance from point A): 3 conventional space dimensions with associated quantum length limit dimension L_p (Planck length)

Furthermore, the two examples in this article using discrete space-time model show that: (1) space-time can be dynamical at subatomic level, and (2) quantum effect can be present at the cosmic scale. Existing theories using continuous space-time model have limited dynamical space-time to the cosmic scale, and quantum effect to the subatomic one.

Quantum length limits have significant effect on space-time at subatomic level (electrons) and cosmic level with extreme mass density (black holes) represented as point A in Fig. 7. At increasing distances from point A, such as point B in Fig. 7, the quantum length limits decrease progressively to the value of Planck length and conventional mathematics with calculus based on real numbers can be utilized upon the 4 conventional dimensions, still as approximation methods but with excellent accuracy.

Notes ^{1, 12, 13, 22, 23}

Rest mass of electron me = 9.11 x 10^-31 kg

Electron charge e = 1.602 x 10^-19 C

Speed of light c = 3 x 10⁸ m/s

Constant k in Coulomb’s law (for vacuum) k=8.987 x 10⁹ Nm²/c²

1.4 mass of the Sun M= 1.4 x (2 x 10³⁰ kg) = 2.8 x 10³⁰ kg

Gravitational constant G = 6.67 x 10^-11 Nm²Kg^-2

Planck length L_p = 10^-35 m

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