Discreteness of Curved Spacetime from GUP

Abstract

Diverse theories of quantum gravity expect modifications of the Heisenberg's uncertainty principle near the Planck scale to a so-called Generalized uncertainty principle (GUP). It was shown by some authors that the GUP gives rise to corrections to the Schrodinger , Klein-Gordon, and Dirac equations. By solving the GUP corrected equations, the authors arrived at quantization not only of energy but also of box length, area, and volume. In this paper, we extend the above results to the case of curved spacetime (Schwarzschild metric). We showed that we arrived at the quantization of space by solving Dirac equation with GUP in this metric.

1. Introduction

Diverse approaches to quantum gravity expect a minimum measurable length and a modification of the Heisenberg uncertainty principle to a so-called generalized uncertainty principle or GUP. This implies a modification of the commutation relations between position coordinates and momentum. In [1], the following proposed GUP is consistent with Doubly special relativity or DSR theories and black hole physics which ensure that [𝑥_𝑖,𝑥_𝑗]=0=[𝑝_𝑖,𝑝_𝑗]: 𝑥_𝑖⋅𝑝_𝑗=𝑖ℎ𝛿_𝑖𝑗−𝑎𝑝𝛿_𝑖𝑗+𝑝_𝑖𝑝_𝑗𝑝+𝑎²𝑝²𝛿_𝑖𝑗+3𝑝_𝑖𝑝_𝑗,ℎΔ𝑥Δ𝑝≥21−2𝑎𝑝+4𝑎²𝑝²≥ℎ2𝑎1+𝑝²+4𝑎²Δ𝑝²+4𝑎²𝑝²−2𝑎𝑝², (1) where 𝑎=𝑎₀/𝑀_pl𝑐=𝑎₀𝐿_pl/ℎ, 𝑀_pl = Planck mass, 𝐿_pl≃10⁻³⁵ m = Planck length, and 𝑀_pl𝑐² = Planck energy ≈10¹⁹ GeV. GUP-induced terms become important near the Planck scale (for earlier version of GUP motivated by string theory, black hole physics, and DSR, see [2–14], and for some phenomenological implications, see [1, 15, 16]).

Equation (1) implies the following minimum measurable length and maximum measurable momentum [1, 17]: Δ𝑥≥(Δ𝑥)_min≈𝑎₀𝐿_pl,Δ𝑝≤Δ𝑝_max≈𝑀_pl𝑐𝑎₀. (2) It is natural to take 𝑎₀=1; for more details see [17].

The following definitions are proposed in [1] and used in [1, 17]:𝑥_𝑖=𝑥_0𝑖,𝑝_𝑖=𝑝_0𝑖1−𝑎𝑝₀+2𝑎²𝑝²₀ (3) with 𝑥_0𝑖 and 𝑝_0𝑗 satisfying the canonical commutation relations [𝑥_0𝑖,𝑝_0𝑗]=𝑖ℎ𝛿_𝑖𝑗, such that 𝑝_0𝑖=−𝑖ℎ(𝜕/𝜕𝑥_0𝑖), 𝑝²₀=∑³_𝑗=1𝑝_0𝑗𝑝_0𝑗.

In [1], it was shown that any nonrelativistic Hamiltonian of the form 𝐻=𝑝²/2𝑚+𝑉(⇀−𝑟) can be written as 𝐻=𝑝²₀/2𝑚−(𝑎/𝑚)𝑝³₀+𝑉(⇀−𝑟)+𝑂(𝑎²) using (3). This corrected Hamiltonian implies not only the usual quantization of energy, but also that the box length is quantized. In [17], the above results were extended to a relativistic particle in two and three dimensions. In this paper we study Dirac equation in Schwarzschild metric using GUP and show that we arrive at the quantization of space.

2. GUP Dirac Equations in Schwarzschild Metric

Dirac equation in Schwarzschild metric without (GUP) can be written as follows [18]: 𝑐⇀−𝛼⋅𝑝⇀−+𝛽𝑚𝑐²𝐸𝜓=𝜁𝜓, (4) where 𝑚 is the rest mass of the particle, 𝜓 is the Dirac spinor, ⇀−𝛼 and 𝛽 are Dirac matrices, ⇀−𝑝≡(𝑝_𝑟,𝑝_𝜃,𝑝_𝜑) are momentum operators, 𝜁=1−𝑟_𝑠/𝑟, 𝑟_𝑠 is the Schwarzschild radius of massive body, related to its mass 𝑀 by 𝑟_𝑠=2𝐺𝑀/𝐶², 𝐺 is the gravitational constant, and 𝑐 is the speed of light in free space. Using 𝜓=^𝜒1_𝜒2, (4) can be written as 𝑐⇀−𝛼⋅𝑝𝜒⇀−₂+𝑚𝑐²𝜒₁=𝐸𝜁𝜒₁,𝑐⇀−𝛼⋅𝑝𝜒⇀−₁−𝑚𝑐²𝜒₂=𝐸𝜁𝜒₂. (5) Now, using GUP correction (3) and (5) take the form 𝑐⇀−𝛼⋅𝑝−⇀₀𝜒₂+𝑚𝑐²𝜒₁−𝑐𝑎𝑝²₀𝜒₁=𝐸𝜁𝜒₁,𝑐⇀−𝛼⋅𝑝−⇀₀𝜒₁−𝑚𝑐²𝜒₂−𝑐𝑎𝑝²₀𝜒₂=𝐸𝜁𝜒₂, (6) where 𝑝²₀=−ℎ²𝜁𝑟²𝜕𝑟𝜕𝑟²𝜁𝜕+1𝜕𝑟𝑟²×𝜕sin𝜃𝜕𝜕𝜃sin𝜃+1𝜕𝜃𝑟²sin²𝜃𝜕²𝜕𝜑². (7) We study Dirac equations in (6) in Schwarzschild metric with spherical cavity with radius 𝑅 defined by the potential 𝑈(𝑟)𝑈(𝑟)=0,𝑟≤𝑅,=𝑈₀⟶∞,𝑟>𝑅, (8) so, we can write the corrected GUP Dirac equations with spherical cavity defined by (8) in Schwarzschild metric as 𝑐⇀−𝛼⋅𝑝−⇀₀𝜒₂+𝑚𝑐²+𝑈𝜒₁−𝑐𝑎𝑝²₀𝜒₁=𝐸𝜁𝜒₁,𝑐⇀−𝛼⋅𝑝−⇀₀𝜒₁−𝑚𝑐²+𝑈𝜒₂−𝑐𝑎𝑝²₀𝜒₂=𝐸𝜁𝜒₂. (9) Notice that, when 𝑎=0, 𝜁=1, equations in (9) are usual Dirac equations in flat spacetime. When 𝑎≠0, 𝜁=1, (9) are Dirac equations with GUP in flat spacetime proposed in [17]. When 𝑎=0, 𝜁≠1, equations in (9) are Dirac equations in Schwarzschild metric without GUP defined in [18]. We follow the analysis of [17, 19] and related references [20, 21].

We assume the form of Dirac spinor as 𝜒𝜓=¹𝜒₂=𝑔_𝑘(𝑟)𝛾^𝑗3_𝑗𝑙(𝑟)̂𝑖𝑓_𝑘(𝑟)𝛾^𝑗3_𝑗𝑙−(𝑟)̂𝛾_{𝑗3𝑗𝑙},(𝑟)̂=𝑙12𝑗₃−1212∣𝑗𝑗₃×𝑌^𝑗3_𝑙^−(1/2)(𝑟)̂10+𝑙12𝑗₃+12−12∣𝑗𝑗₃×𝑌^𝑗3_𝑙^+(1/2)(𝑟)̂01, (10)(11) where 𝑌^𝑗3_𝑙^−(1/2)(𝑟)̂ and 𝑌^𝑗3_𝑙^+(1/2)(𝑟)̂ are spherical harmonics and 𝑗₁𝑗₂𝑚₁𝑚₂∣𝑗𝑗₃ are Clebsh-Gordon coefficients, 𝜒₁, 𝜒₂ are eigenstates of 𝐿² (𝐿⇀−is the angular momentum operator) with eigenvalues ℎ²𝑙(𝑙+1) and ℎ²𝑙⁻(𝑙⁻+1), respectively, such that the following hold: if 1𝑘=𝑗+2>0, (12) then 1𝑙=𝑘=𝑗+2,𝑙⁻1=𝑘−1=𝑗−2, (13) and if 1𝑘=−𝑗+2<0, (14) then ()1𝑙=−𝑘+1=𝑗−2,𝑙⁻1=−𝑘=𝑗+2. (15)We use (⇀−𝛼⋅⇀−𝐴)(⇀−𝛼⋅⇀−𝐵)=⇀−𝐴⋅⇀−𝐵+𝑖⇀−𝛼⋅(⇀−𝐴×⇀−𝐵) and the related identity (⇀−𝛼⋅⇀−𝑟)(⇀−𝛼⋅⇀−𝑟)=𝑟²; so we have ⇀−𝛼⋅𝑝−⇀₀=⇀−𝛼⋅⇀−𝑟⇀−𝛼⋅⇀−𝑟⇀−𝛼⋅𝑝−⇀₀𝑟²=⇀−𝛼⋅⇀−𝑟𝑟²⇀−𝑟⋅𝑝−⇀₀+𝑖⇀−𝛼⋅⇀−𝑟×𝑝−⇀₀=⇀−𝛼⋅⇀−𝑟𝑟²⇀−𝑟⋅𝑝−⇀₀+𝑖⇀−𝛼⋅𝐿.⇀− (16) But from the definition of the momentum operators in Schwarzschild metric [18], we can write (16) as ⇀−𝛼⋅𝑝−⇀₀=⇀−𝛼⋅𝑟̂−𝑖ℎ𝜁𝜕+𝑖𝜕𝑟𝑟⇀−𝛼⋅𝐿⇀−. (17) Also, we have ⇀−𝛼⋅𝜒⇀−𝐿+1_1,2=∓𝜒_1,2,⇀−𝛼⋅𝑟̂𝛾^𝑗3_𝑗𝑙(𝑟)̂=−𝛾^𝑗3_𝑗𝑙−(𝑟)̂,⇀−𝛼⋅𝑟̂𝛾^𝑗3_𝑗𝑙−(𝑟)̂=−𝛾^𝑗3_𝑗𝑙(𝑟)̂. (18) Next, from the definition of 𝑝²₀ in Schwarzschild metric [18] we can write 𝑝²₀𝐹(𝑟)𝑌^𝑚_𝑙=ℎ²−𝜁𝑟²𝑑𝑟𝑑𝑟²𝜁𝜕+𝑙()𝜕𝑟𝑙+1𝑟²𝐹(𝑟)𝑌^𝑚_𝑙. (19) So, using (17), (18), and (19), we can obtain from (9) the following equations: −𝑐ℎ𝜁𝑑𝑓_𝑘+𝑐()𝑑𝑟𝑘−1𝑟𝑓_𝑘+𝑚𝑐²+𝑈𝑔_𝑘+𝑐𝑎ℎ²𝜁𝑟²𝑑𝑑𝑟𝜁𝑟²𝑑𝑔_𝑘−𝑙()𝑑𝑟𝑙+1𝑟²𝑔_𝑘=𝐸𝜁𝑔_𝑘,𝑐ℎ𝜁𝑑𝑔_𝑘+𝑐()𝑑𝑟𝑘+1𝑟𝑔_𝑘−𝑚𝑐²+𝑈𝑓_𝑘+𝑐𝑎ℎ²𝜁𝑟²𝑑𝑑𝑟𝜁𝑟²𝑑𝑓_𝑘−𝑙𝑑𝑟⁻(𝑙⁻+1)𝑟²𝑓_𝑘=𝐸𝜁𝑓_𝑘. (20) It can be shown that MIT bag boundary condition (at 𝑟=𝑅) is equivalent to [19, 20] 𝜓𝜓=0. (21) As in [17], we can expect new nonperturbative solutions of the forms 𝑓_𝑘=𝐹_𝐾(𝑟)𝑒^{𝑖𝜖𝑟/𝑎ℎ} and 𝑔_𝑘=𝐺_𝐾(𝑟)𝑒^{𝑖𝜖𝑟/𝑎ℎ} (where 𝜖=𝑂(1)) for which (20) simplifies to 𝑑𝑎ℎ²𝑔_𝑘𝑑𝑟²=𝜁𝑑𝑓_𝑘,𝑑𝑑𝑟𝑎ℎ²𝑓_𝑘𝑑𝑟²=−𝜁𝑑𝑔_𝑘,𝑑𝑟 (22) where we have dropped terms which are ignorable for small 𝑎.

When 𝜁=1, equations in (22) are identical to the (60)-(61) in reference [17], and in this case we have the following solutions: 𝑓^𝑁_𝑘=𝑖𝑁𝑒^{𝑖𝑟/𝑎ℎ}, 𝑔^𝑁_𝑘=𝑁𝑒^{𝑖𝑟/𝑎ℎ}, 𝑁 is constant; so we can assume the solutions of (22) as 𝑓^𝑁_𝑘=𝑖𝑁𝑒^{𝑖𝑟/𝑎ℎ},𝑔^𝑁_𝑘=𝑁𝐵(𝑟)𝑒^{𝑖𝑟/𝑎ℎ}. (23) By applying (23) on (22), we find that 𝑎²ℎ²𝑑²𝐵(𝑟)𝑑𝑟²+𝐵(𝑟)+2𝜁−𝜁=0. (24) Consider that 𝑟_𝑠 is very small, so we can approximate (24) to 𝑎²ℎ²𝑑²𝐵(𝑟)𝑑𝑟²+𝐵(𝑟)−1−3𝑟_𝑠2𝑟=0. (25) The solution of (25) takes the form 𝐵(𝑟)=𝑐₁𝑟sin𝑎ℎ+𝑐₂𝑟cos+𝑎ℎ1.5𝑟_𝑠𝐶𝑎ℎ^𝑖𝑟𝑟𝑎ℎsin𝑎ℎ−𝑆_𝑖𝑟𝑟𝑎ℎcos𝑎ℎ+1, (26) where 𝑐₁ and 𝑐₂ are constants, 𝑆_𝑖(𝑟/𝑎ℎ) and 𝐶_𝑖(𝑟/𝑎ℎ) are the sine integral function and cosine integral function defined as 𝑆_𝑖(𝑦)=∫^𝑦₀(sin𝑡/𝑡)𝑑𝑡 and 𝐶_𝑖(𝑦)=−∫^∞_𝑦(cos𝑡/𝑡)𝑑𝑡. For more details about this functions see [22, 23].

Therefore, the solutions of (22) take the form 𝑓^𝑁_𝑘=𝑖𝑁𝑒^{𝑖𝑟/𝑎ℎ},𝑔^𝑁_𝑘=𝑁[1+𝜎(𝑟)]𝑒^{𝑖𝑟/𝑎ℎ}, (27) where 𝜎(𝑟)=𝑐₁𝑟sin𝑎ℎ+𝑐₂𝑟cos+𝑎ℎ1.5𝑟_𝑠𝐶𝑎ℎ^𝑖𝑟𝑟𝑎ℎsin𝑎ℎ−𝑆_𝑖𝑟𝑟𝑎ℎcos.𝑎ℎ (28) Here, one must have lim_{𝑟𝑠→𝑜}𝑐₁=lim_{𝑟𝑠→𝑜}𝑐₂=0, and, in this case (𝜎(𝑟)=0, 𝑟_𝑠=0), the results are the same of [17].

Now, the boundary condition (21) gives |𝑔_𝑘(𝑅)+𝑔^𝑁_𝑘(𝑅)|²=|𝑓_𝑘(𝑅)+𝑓^𝑁_𝑘(𝑅)|², (29) which to 𝑂(𝑎) translates to 𝑔²_𝑘−𝑓²_𝑘+2𝑁𝑔_𝑘((𝑅𝑅1+𝜎))cos𝑎ℎ−𝑓_𝑘𝑅sin𝑎ℎ+𝑁²(1+𝜎(𝑅))²−1=0. (30) From (30), we have 𝑓_𝑘=𝑔_𝑘,𝑅tan(𝑅),𝜎(𝑅)(𝑅)𝑎ℎ=1+𝜎=0,or𝜎=−2. (31)(32) Observing (28), we can write 𝜎(𝑅)cos(𝑅/𝑎ℎ)=𝑐₁𝑅tan𝑎ℎ+𝑐₂+1.5𝑟_𝑠𝐶𝑎ℎ^𝑖𝑅𝑅𝑎ℎtan𝑎ℎ−𝑆_𝑖𝑅.𝑎ℎ (33)

From (32) we have (𝑅)𝑅if𝜎=0thentan(𝑅)𝑅𝑎ℎ=1,if𝜎=−2thentan𝑎ℎ=−1. (34)

For the case of 𝜎(𝑅)=0 and 𝑟_𝑠=0 we have the same result of discreteness of space in flat spacetime [17].

For the second case 𝜎(𝑅)=−2, we have, from (33), 4𝑎ℎ3𝑟_𝑠()+cos𝑅/𝑎ℎ2𝐶𝑎ℎ3𝑟_𝑠=𝐶_𝑖𝑅𝑎ℎ+𝑆_𝑖𝑅,𝑎ℎ𝐶isconstant. (35) If we take the constant 𝐶=0, then we have 4𝑎ℎ3𝑟_𝑠cos(𝑅/𝑎ℎ)=𝐶_𝑖𝑅𝑎ℎ+𝑆_𝑖𝑅.𝑎ℎ (36)

Suppose that we choose 𝑟_𝑠=(4/3)𝑎ℎ (very small as assumption); then we have 1cos(𝑅/𝑎ℎ)=𝐶_𝑖𝑅𝑎ℎ+𝑆_𝑖𝑅.𝑎ℎ (37)

Equation (37) has infinite number of solutions; we write some numerical values of it, (𝑅/𝑎ℎ)=5,565, (𝑅/𝑎ℎ)=7.159, (𝑅/𝑎ℎ)=11, and 755, (𝑅/𝑎ℎ)=13.448. So, the radius of cavity 𝑅 has been quantized in terms of 𝑎ℎ and we again arrived at the quantization of space in Schwarzschild-like metric.

3. Conclusion

Dirac equations with GUP in Schwarzschild metric have been studied. We showed that the assumption of existence of a minimum measurable length and a corresponding modification of uncertainty principle yields discreteness of space in this metric. But the question now arises, what is the guarantee that this result will continue to hold for more generic curved spacetimes? We expect that this discreteness will always appear provided that generalized uncertainty principle enters into the theory, but in fact we have no mathematical proof of existence of such discreteness of space if we work on the general metric of the general relativity theory.