In this paper, we study the relativistic energy spectrum for Dirac fermions under rainbow gravity effects in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(3+1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dimensional Bonnor–Melvin–Lambda spacetime, where we work with the curved Dirac equation in cylindrical coordinates. Using the tetrad formalism of General Relativity and considering a first-order approximation for the trigonometric functions, we obtain a Bessel equation. To solve this differential equation, we also consider a region where a hard-wall confining potential is present (i.e., a finite distance where the radial wave function is null). In other words, we define a second boundary condition (Dirichlet boundary condition) to achieve the quantization of the energy. Consequently, we obtain the spectrum for a fermion/antifermion, which is quantized in terms of quantum numbers n, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$m_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$m_s$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation>, where n is the radial quantum number, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$m_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> is the total magnetic quantum number (or angular quantum number), <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$m_s$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation> is the spin magnetic quantum number, and explicitly depends on the rainbow functions <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$F(\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$G(\xi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, curvature parameter <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, cosmological constant <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Lambda$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>, fixed radius <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$r_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>r</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>, rest energy <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$m_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>, and on the z-momentum <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$p_z$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mi>z</mi> </msub> </math></EquationSource> </InlineEquation>. So, analyzing this spectrum according to the values of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$m_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$m_s$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation>, we see that for <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$m_j>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mi>j</mi> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$m_s=-1/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mi>s</mi> </msub> <mo>=</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> (positive angular momentum and spin down), and for <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$m_j<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mi>j</mi> </msub> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$m_s=+1/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mi>s</mi> </msub> <mo>=</mo> <mo>+</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> (negative angular momentum and spin up), the spectrum is the same (i.e., there are two different cases for the same spectrum). Besides, we graphically analyze the behavior of the spectrum for the three scenarios of rainbow gravity as a function of <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\Lambda$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$r_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>r</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> for three different values of n.

Dirac fermions under rainbow gravity effects in the Bonnor–Melvin–Lambda spacetime

期刊论文
发表日期：

2026年3月9日

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R. R. S. Oliveira

摘要

In this paper, we study the relativistic energy spectrum for Dirac fermions under rainbow gravity effects in the $(3+1)$ $(3 + 1)$ -dimensional Bonnor–Melvin–Lambda spacetime, where we work with the curved Dirac equation in cylindrical coordinates. Using the tetrad formalism of General Relativity and considering a first-order approximation for the trigonometric functions, we obtain a Bessel equation. To solve this differential equation, we also consider a region where a hard-wall confining potential is present (i.e., a finite distance where the radial wave function is null). In other words, we define a second boundary condition (Dirichlet boundary condition) to achieve the quantization of the energy. Consequently, we obtain the spectrum for a fermion/antifermion, which is quantized in terms of quantum numbers n, $m_j$ $m_{j}$ and $m_s$ $m_{s}$ , where n is the radial quantum number, $m_j$ $m_{j}$ is the total magnetic quantum number (or angular quantum number), $m_s$ $m_{s}$ is the spin magnetic quantum number, and explicitly depends on the rainbow functions $F(\xi )$ $F (ξ)$ and $G(\xi )$ $G (ξ)$ , curvature parameter $\alpha$ $α$ , cosmological constant $\Lambda$ $Λ$ , fixed radius $r_0$ $r_{0}$ , rest energy $m_0$ $m_{0}$ , and on the z-momentum $p_z$ $p_{z}$ . So, analyzing this spectrum according to the values of $m_j$ $m_{j}$ and $m_s$ $m_{s}$ , we see that for $m_j>0$ $m_{j} > 0$ with $m_s=-1/2$ $m_{s} = - 1 / 2$ (positive angular momentum and spin down), and for $m_j<0$ $m_{j} < 0$ with $m_s=+1/2$ $m_{s} = + 1 / 2$ (negative angular momentum and spin up), the spectrum is the same (i.e., there are two different cases for the same spectrum). Besides, we graphically analyze the behavior of the spectrum for the three scenarios of rainbow gravity as a function of $\Lambda$ $Λ$ , $r_0$ $r_{0}$ , and $\alpha$ $α$ for three different values of n.