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Paulo Custodio

FMU, Computer Science, Faculty Member

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Professor of the FMU Laureate - Brazil, Computer Science and Software Development.
phD in Astronomy by USP at 2001.
Research: Black Holes, Inflation and Cosmology.
Numerical algorithms for prime numbers.

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Papers by Paulo Custodio

Cosmological Quintessence Accretion Onto Primordial Black Holes: Conditions for Their Growth to the Supermassive Scale

International Journal of Modern Physics D, 2005

In this work we revisit the growth of small primordial black holes (PBHs) immersed in a quintesse... more In this work we revisit the growth of small primordial black holes (PBHs) immersed in a quintessential field and/or radiation to the supermassive black hole (SMBHs) scale. We show the difficulties of scenarios in which such huge growth is possible. For that purpose we evaluated analytical solutions of the differential equations (describing mass evolution) and point out the strong fine tuning for the conclusions. The timescale for growth in a model with a constant quintessence flux is calculated and we show that it is much bigger than the Hubble time. The fractional gain of the mass is further evaluated in other forms, including quintessence and/or radiation. We calculate the cosmological density Ω due to quintessence necessary to grow BHs to the supermassive range and show it to be much bigger than one. We also describe the set of complete equations analyzing the evolution of the BH + quintessence universe, showing some interesting effects such the quenching of the BH mass growth du...

Thermodynamics of Black Holes in Finite Boxes, 2010

We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black ho... more We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black hole mass evolution is analyzed in an initially empty box. Using the conservation of the energy and the Hawking evaporation flux, we deduce a minimal volume above which one black hole can loss all of its mass to the box, a result which agrees with the previous analysis made by Page. We then obtain analogous results using a box initially containing radiation, allowed to be absorbed by the black hole. The equilibrium times and masses are evaluated and their behavior discussed to highlight some interesting features arising. These results are generalized to N black holes + thermal radiation. Using physically simple arguments, we prove that these black holes achieve the same equilibrium masses (even that the initial masses were different). The entropy of the system is used to obtain the dependence of the equilibrium mass on the box volume, number of black holes and the initial radiation. The equilibrium mass is shown to be proportional to a positive power law of the effective volume (contrary to naive expectations), a result explained in terms of the detailed features of the system. The effect of the reflection of the radiation on the box walls which comes back into the black hole is explicitly considered. All these results (some of them counter-intuitive) may be useful to formulate alternative problems in thermodynamic courses for graduate and advanced undergraduate students. A handful of them are suggested in the Appendix.

This article describes the Robertson-Walker equations from the Cartan's structure equations.

Primordial black holes in the very early Universe

Physical Review D, 1989

ABSTRACT

Efeitos astrofísicos da matéria escura: evolução e sobrevivência de buracos negros

A inflação e fósseis primordiais: pertubações de densidade e buracos negros primordialmente formados pelo inflaton

Survival of small PBHs in the Very Early Universe

viXra, 2017

The formation of Primordial Black Holes is a robust prediction of several gravitational theories.... more The formation of Primordial Black Holes is a robust prediction of several gravitational theories. Whereas the creation of PBHs was very active in the remote past, such process seem to be very negligible at the present epoch. In this work, we estimate the effects from the radiation surrounding PBHs due to the absorption term in the equations that describe how their masses depend on time. The Hawking radiation contributes with mass loss and the absorption term contributes with gain, but a interesting competition between these terms is analysed. These effects are included in the equations describing PBHs and its mass density as the universe evolves in time and the model is able to describes the evolution of the numerical density of PBHs and the mass evolution and comparisons with cosmological constraints set upper limits in their abundances. We evaluate the effect of this accretion onto PBHs and we get some corrections for the initial masses that indicates some deviations from de- faul...

Supermassive Black Holes May Be Limited by the Holographic Bound

General Relativity and Gravitation, 2003

The Evolution of Primordial Black Hole Masses in the Radiation-Dominated Era

General Relativity and Gravitation, 2002

We discuss in this work the behaviour of primordial black holes (PBHs) in the radiative era. Taki... more We discuss in this work the behaviour of primordial black holes (PBHs) in the radiative era. Taking into account the Hawking evaporation and the absorption of energy we revisit the complete differential equation for the evolution of the mass of a PBH. We show that the mass can grow in this cosmological phase in a very slow fashion (even when considering the very high temperature of the radiation) if at all, and give a strong upper limit to the maximum accretion of mass. We evaluate relativistic effects due to the peculiar motion relative to the CMBR and show that the existence of relativistic black holes with very high mass absorption is highly unlikely. Finally we demonstrate that thermodynamical equilibrium between black holes and the cosmic radiation can not exist for finite times, and therefore initially non-evaporating PBHs must jump to the evaporating regime. This analysis supports the several efforts performed to look for signatures of evaporating holes.

Evolution of primordial black holes in a radiation and phantom energy environment

General Relativity and Gravitation, 2007

In this work we extend previous work on the evolution of a Primordial Black Hole (PBH) to address... more In this work we extend previous work on the evolution of a Primordial Black Hole (PBH) to address the presence of a dark energy component with a super-negative equation of state as a background, investigating the competition between the radiation accretion, the Hawking evaporation and the phantom accretion, the latter two causing a decrease on black hole mass. It is found that there is an instant during the matter-dominated era after which the radiation accretion becomes negligible compared to the phantom accretion. The Hawking evaporation may become important again depending on a mass threshold. The evaporation of PBHs is quite modified at late times by these effects, but only if the Generalized Second Law of thermodynamics is violated.

Holographic principle bounds on primordial black-hole abundances

Classical and Quantum Gravity, 2003

The generalized Second Law of thermodynamics and the Holographic Principle are combined to obtain... more The generalized Second Law of thermodynamics and the Holographic Principle are combined to obtain the maximum mass of black holes formed inside a static spherical box of size R filled with radiation at initial temperature T i. The final temperature after the formation of black holes is evaluated, and we show that a critical threshold exists for the radiation to be fully consumed by the process. We next argue that if some form of Holographic Principle holds, upper bounds to the mass density of PBHs formed in the early universe may be obtained. The limits are worked out for inflationary and non-inflationary cosmological models. This method is independent of the known limits based on the background fluxes (from cosmic rays, radiation and other forms of energy) and applies to potentially important epochs of PBH formation, resulting in quite strong constraints to Ω pbh .

Some results on the evolution of primordial black holes

Brazilian Journal of Physics, 2005

We briefly review some recent results related to thermodynamics of semiclassical black holes appl... more We briefly review some recent results related to thermodynamics of semiclassical black holes applied to their primordial formation. Issues on the existence of an influx of ambient particles onto the PBHs, which may help grow them, are addressed. We revisit the integrated flux from evaporated PBHs contributing to the present backgrounds and show that there probe mass scales which are otherwise poorly constrained. Finally the Generalized Second Law of Thermodynamics and the Holographic Principle are combined to show that if some form of the latter holds, strong upper bounds to the mass density of PBHs formed in the early universe may be obtained, especially for inflationary cosmological models. This method is completely independent from those based on the background fluxes and applies to potentially important epochs of PBH formation, resulting in quite strong constraints to Ω pbh .

Fisica a escala de Planck usando o principio de incerteza generalizado: efeitos nas flutuações primordiais e buracos negros

Em escalas proximas à escala de Planck todas as teorias perturbativas de cordas produzem essencia... more Em escalas proximas à escala de Planck todas as teorias perturbativas de cordas produzem essenciamente a mesma relação de conmutação entre as coordenadas e impulsos (a chamada "álgebra deformada"), permitindo assim estudar a física resultante independentemente dos detalhes da teoria de cordas que seja considerada correta. Este resultado completamente geral, que inclui as interações gravitacionais junto com o resto dos campos pode ser considerada uma versão generalizada (GUP) do Princípio de Incerteza de Heisenberg. Aplicamos neste trabalho essas relações de conmutação para dois sistemas físicos bem definidos: buracos negros de massas próximas à massa de Planck, e flutuações quânticas em pequenas escalas antes do universo sofrer inflação. Obtemos dois resultados concretos dos efeitos do GUP : o primeiro é que o GUP impede a evaporação completa de buracos negros microscópicos na extensão do formalismo semiclássico, deixando assim remanescentes de pequena massa que já foram postulados como candidatos a matéria escura. O segundo resultado é o 'smoothing' das flutuações primordiais em pequenas escalas que levariam à produção de buracos negros primordiais após a inflação, impedindo assim a produção abundante destes últimos e predizendo abundancias atuais bem menores do que os limites disponíveis. Concluimos que, analogamente a utilização do Princípio de Incerteza de Heisenberg para estudar e determinar propriedades fundamentais das interações sem gravitação, o GUP e uma ferramenta poderosa para estudar uma ampla variedade de sistemas trans-Planckianos e predizer seu comportamento dispensando cálculos mais detalhados proprios da teoria quântica da gravitação.

Thermodynamics of black holes in finite boxes

We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black ho... more We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black hole mass evolution is analyzed in an initially empty box. Using the conservation of the energy and the Hawking evaporation flux, we deduce a minimal volume above which one black hole can loss all of its mass to the box, a result which agrees with the previous analysis made by Page. We then obtain analogous results using a box initially containing radiation, allowed to be absorbed by the black hole. The equilibrium times and masses are evaluated and their behavior discussed to highlight some interesting features arising. These results are generalized to N black holes + thermal radiation. Using physically simple arguments, we prove that these black holes achieve the same equilibrium masses (even that the initial masses were different). The entropy of the system is used to obtain the dependence of the equilibrium mass on the box volume, number of black holes and the initial radiation. The equilibrium mass is shown to be proportional to a positive power law of the effective volume (contrary to naive expectations), a result explained in terms of the detailed features of the system. The effect of the reflection of the radiation on the box walls which comes back into the black hole is explicitly considered. All these results (some of them counter-intuitive) may be useful to formulate alternative problems in thermodynamic courses for graduate and advanced undergraduate students. A handful of them are suggested in the Appendix.

Thermodynamics of black holes in a finite box

American Journal of Physics, 2003

We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black ho... more We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black hole mass evolution is analyzed in an initially empty box. Using the conservation of the energy and the Hawking evaporation flux, we deduce a minimal volume above which one black hole can loss all of its mass to the box, a result which agrees with the previous analysis made by Page. We then obtain analogous results using a box initially containing radiation, allowed to be absorbed by the black hole. The equilibrium times and masses are evaluated and their behavior discussed to highlight some interesting features arising. These results are generalized to N black holes + thermal radiation. Using physically simple arguments, we prove that these black holes achieve the same equilibrium masses (even that the initial masses were different). The entropy of the system is used to obtain the dependence of the equilibrium mass on the box volume, number of black holes and the initial radiation. The equilibrium mass is shown to be proportional to a positive power law of the effective volume (contrary to naive expectations), a result explained in terms of the detailed features of the system. The effect of the reflection of the radiation on the box walls which comes back into the black hole is explicitly considered. All these results (some of them counter-intuitive) may be useful to formulate alternative problems in thermodynamic courses for graduate and advanced undergraduate students. A handful of them are suggested in the Appendix.

Dynamics of black hole motion

Physical Review D, 1999

We evaluate in the proper and cosmological frames the effects of nonzero velocities on the mass g... more We evaluate in the proper and cosmological frames the effects of nonzero velocities on the mass gain or loss of primordial black holes. An upper limit of twice the initial Lorentz factor is derived from the mass gain of the black holes, a value that may be enough to preclude their evaporation. Next, we analyze accelerated black holes and find that the Unruh effect can delay the onset of the evaporation regime. Finally, we reassess the equilibrium between black holes and the relic thermal radiation. ͓S0556-2821͑99͒00718-3͔

Evolution of primordial black holes in a radiation and phantom energy environment

by Daniel Guariento and Paulo Custodio

General Relativity and Gravitation 40(8), 1596-1602, Aug 2008

In this work we extend previous work on the evolution of a primordial black hole (PBH) to address... more In this work we extend previous work on the evolution of a primordial black hole (PBH) to address the presence of a dark energy component with a super-negative equation of state as a background, investigating the competition between the radiation accretion, the Hawking evaporation and the phantom accretion, the latter two causing a decrease on black hole mass. It is found that there is an instant during the matter-dominated era after which the radiation accretion becomes negligible compared to the phantom accretion. The Hawking evaporation may become important again depending on a mass threshold. The evaporation of PBHs is quite modified at late times by these effects, but only if the generalized second law of thermodynamics is violated.

Dynamics of black hole motion

We evaluate in the proper and cosmological frames the effects of nonzero velocities on the mass g... more We evaluate in the proper and cosmological frames the effects of nonzero velocities on the mass gain or loss of primordial black holes. An upper limit of twice the initial Lorentz factor is derived from the mass gain of the black holes, a value that may be enough to preclude their evaporation. Next, we analyze accelerated black holes and find that the Unruh effect can delay the onset of the evaporation regime. Finally, we reassess the equilibrium between black holes and the relic thermal radiation. S0556-28219900718-3

Cosmological quintessence accretion onto primordial black holes : conditions for their growth to the supermassive scale

In this work we revisit the growth of small primordial black holes (PBHs) immersed in a quintesse... more In this work we revisit the growth of small primordial black holes (PBHs) immersed in a quintessential field and/or radiation to the su-permassive black hole (SMBHs) scale. We show the difficulties of scenarios in which such huge growth is possible. For that purpose we evaluated analytical solutions of the differential equations (describing mass evolution) and point out the strong fine tuning for that conclusions. The timescale for growth in a model with a constant quintessence flux is calculated and we show that it is much bigger than the Hubble time.The fractional gain of the mass is further evaluated in other forms, including quintessence and/or radiation. We calculate the cosmological density Ω due to quintessence necessary to grow BHs to the supermassive range and show it to be much bigger than one. We also describe the set of complete equations analyzing the evolution of the BH+quintessence universe, showing some interesting effects such the quenching of the BH mass growth due to the evolution of the background energy. Additional constraints obtained by using the Holographic Bound are also described. The general equilibrium conditions for evaporating/accreting black holes evolving in a quintessence/radiation universe are discussed in the Appendix.

Thermodynamics of black holes in finite boxes

We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black ho... more We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black hole mass evolution is analyzed in an initially empty box. Using the conservation of the energy and the Hawking evaporation flux, we deduce a minimal volume above which one black hole can loss all of its mass to the box, a result which agrees with the previous analysis made by Page. We then obtain analogous results using a box initially containing radiation, allowed to be absorbed by the black hole. The equilibrium times and masses are evaluated and their behavior discussed to highlight some interesting features arising. These results are generalized to N black holes + thermal radiation. Using physically simple arguments, we prove that these black holes achieve the same equilibrium masses (even that the initial masses were different). The entropy of the system is used to obtain the dependence of the equilibrium mass on the box volume, number of black holes and the initial radiation. The equilibrium mass is shown to be proportional to a positive power law of the effective volume (contrary to naive expectations), a result explained in terms of the detailed features of the system. The effect of the reflection of the radiation on the box walls which comes back into the black hole is explicitly considered. All these results (some of them counter-intuitive) may be useful to formulate alternative problems in thermodynamic courses for graduate and advanced undergraduate students. A handful of them are suggested in the Appendix.

Cosmological Quintessence Accretion Onto Primordial Black Holes: Conditions for Their Growth to the Supermassive Scale

International Journal of Modern Physics D, 2005

In this work we revisit the growth of small primordial black holes (PBHs) immersed in a quintesse... more In this work we revisit the growth of small primordial black holes (PBHs) immersed in a quintessential field and/or radiation to the supermassive black hole (SMBHs) scale. We show the difficulties of scenarios in which such huge growth is possible. For that purpose we evaluated analytical solutions of the differential equations (describing mass evolution) and point out the strong fine tuning for the conclusions. The timescale for growth in a model with a constant quintessence flux is calculated and we show that it is much bigger than the Hubble time. The fractional gain of the mass is further evaluated in other forms, including quintessence and/or radiation. We calculate the cosmological density Ω due to quintessence necessary to grow BHs to the supermassive range and show it to be much bigger than one. We also describe the set of complete equations analyzing the evolution of the BH + quintessence universe, showing some interesting effects such the quenching of the BH mass growth du...

Thermodynamics of Black Holes in Finite Boxes, 2010

We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black ho... more We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black hole mass evolution is analyzed in an initially empty box. Using the conservation of the energy and the Hawking evaporation flux, we deduce a minimal volume above which one black hole can loss all of its mass to the box, a result which agrees with the previous analysis made by Page. We then obtain analogous results using a box initially containing radiation, allowed to be absorbed by the black hole. The equilibrium times and masses are evaluated and their behavior discussed to highlight some interesting features arising. These results are generalized to N black holes + thermal radiation. Using physically simple arguments, we prove that these black holes achieve the same equilibrium masses (even that the initial masses were different). The entropy of the system is used to obtain the dependence of the equilibrium mass on the box volume, number of black holes and the initial radiation. The equilibrium mass is shown to be proportional to a positive power law of the effective volume (contrary to naive expectations), a result explained in terms of the detailed features of the system. The effect of the reflection of the radiation on the box walls which comes back into the black hole is explicitly considered. All these results (some of them counter-intuitive) may be useful to formulate alternative problems in thermodynamic courses for graduate and advanced undergraduate students. A handful of them are suggested in the Appendix.

This article describes the Robertson-Walker equations from the Cartan's structure equations.

Primordial black holes in the very early Universe

Physical Review D, 1989

ABSTRACT

Efeitos astrofísicos da matéria escura: evolução e sobrevivência de buracos negros

A inflação e fósseis primordiais: pertubações de densidade e buracos negros primordialmente formados pelo inflaton

Survival of small PBHs in the Very Early Universe

viXra, 2017

The formation of Primordial Black Holes is a robust prediction of several gravitational theories.... more The formation of Primordial Black Holes is a robust prediction of several gravitational theories. Whereas the creation of PBHs was very active in the remote past, such process seem to be very negligible at the present epoch. In this work, we estimate the effects from the radiation surrounding PBHs due to the absorption term in the equations that describe how their masses depend on time. The Hawking radiation contributes with mass loss and the absorption term contributes with gain, but a interesting competition between these terms is analysed. These effects are included in the equations describing PBHs and its mass density as the universe evolves in time and the model is able to describes the evolution of the numerical density of PBHs and the mass evolution and comparisons with cosmological constraints set upper limits in their abundances. We evaluate the effect of this accretion onto PBHs and we get some corrections for the initial masses that indicates some deviations from de- faul...

Supermassive Black Holes May Be Limited by the Holographic Bound

General Relativity and Gravitation, 2003

The Evolution of Primordial Black Hole Masses in the Radiation-Dominated Era

General Relativity and Gravitation, 2002

We discuss in this work the behaviour of primordial black holes (PBHs) in the radiative era. Taki... more We discuss in this work the behaviour of primordial black holes (PBHs) in the radiative era. Taking into account the Hawking evaporation and the absorption of energy we revisit the complete differential equation for the evolution of the mass of a PBH. We show that the mass can grow in this cosmological phase in a very slow fashion (even when considering the very high temperature of the radiation) if at all, and give a strong upper limit to the maximum accretion of mass. We evaluate relativistic effects due to the peculiar motion relative to the CMBR and show that the existence of relativistic black holes with very high mass absorption is highly unlikely. Finally we demonstrate that thermodynamical equilibrium between black holes and the cosmic radiation can not exist for finite times, and therefore initially non-evaporating PBHs must jump to the evaporating regime. This analysis supports the several efforts performed to look for signatures of evaporating holes.

Evolution of primordial black holes in a radiation and phantom energy environment

General Relativity and Gravitation, 2007

In this work we extend previous work on the evolution of a Primordial Black Hole (PBH) to address... more In this work we extend previous work on the evolution of a Primordial Black Hole (PBH) to address the presence of a dark energy component with a super-negative equation of state as a background, investigating the competition between the radiation accretion, the Hawking evaporation and the phantom accretion, the latter two causing a decrease on black hole mass. It is found that there is an instant during the matter-dominated era after which the radiation accretion becomes negligible compared to the phantom accretion. The Hawking evaporation may become important again depending on a mass threshold. The evaporation of PBHs is quite modified at late times by these effects, but only if the Generalized Second Law of thermodynamics is violated.

Holographic principle bounds on primordial black-hole abundances

Classical and Quantum Gravity, 2003

The generalized Second Law of thermodynamics and the Holographic Principle are combined to obtain... more The generalized Second Law of thermodynamics and the Holographic Principle are combined to obtain the maximum mass of black holes formed inside a static spherical box of size R filled with radiation at initial temperature T i. The final temperature after the formation of black holes is evaluated, and we show that a critical threshold exists for the radiation to be fully consumed by the process. We next argue that if some form of Holographic Principle holds, upper bounds to the mass density of PBHs formed in the early universe may be obtained. The limits are worked out for inflationary and non-inflationary cosmological models. This method is independent of the known limits based on the background fluxes (from cosmic rays, radiation and other forms of energy) and applies to potentially important epochs of PBH formation, resulting in quite strong constraints to Ω pbh .

Some results on the evolution of primordial black holes

Brazilian Journal of Physics, 2005

We briefly review some recent results related to thermodynamics of semiclassical black holes appl... more We briefly review some recent results related to thermodynamics of semiclassical black holes applied to their primordial formation. Issues on the existence of an influx of ambient particles onto the PBHs, which may help grow them, are addressed. We revisit the integrated flux from evaporated PBHs contributing to the present backgrounds and show that there probe mass scales which are otherwise poorly constrained. Finally the Generalized Second Law of Thermodynamics and the Holographic Principle are combined to show that if some form of the latter holds, strong upper bounds to the mass density of PBHs formed in the early universe may be obtained, especially for inflationary cosmological models. This method is completely independent from those based on the background fluxes and applies to potentially important epochs of PBH formation, resulting in quite strong constraints to Ω pbh .

Fisica a escala de Planck usando o principio de incerteza generalizado: efeitos nas flutuações primordiais e buracos negros

Em escalas proximas à escala de Planck todas as teorias perturbativas de cordas produzem essencia... more Em escalas proximas à escala de Planck todas as teorias perturbativas de cordas produzem essenciamente a mesma relação de conmutação entre as coordenadas e impulsos (a chamada "álgebra deformada"), permitindo assim estudar a física resultante independentemente dos detalhes da teoria de cordas que seja considerada correta. Este resultado completamente geral, que inclui as interações gravitacionais junto com o resto dos campos pode ser considerada uma versão generalizada (GUP) do Princípio de Incerteza de Heisenberg. Aplicamos neste trabalho essas relações de conmutação para dois sistemas físicos bem definidos: buracos negros de massas próximas à massa de Planck, e flutuações quânticas em pequenas escalas antes do universo sofrer inflação. Obtemos dois resultados concretos dos efeitos do GUP : o primeiro é que o GUP impede a evaporação completa de buracos negros microscópicos na extensão do formalismo semiclássico, deixando assim remanescentes de pequena massa que já foram postulados como candidatos a matéria escura. O segundo resultado é o 'smoothing' das flutuações primordiais em pequenas escalas que levariam à produção de buracos negros primordiais após a inflação, impedindo assim a produção abundante destes últimos e predizendo abundancias atuais bem menores do que os limites disponíveis. Concluimos que, analogamente a utilização do Princípio de Incerteza de Heisenberg para estudar e determinar propriedades fundamentais das interações sem gravitação, o GUP e uma ferramenta poderosa para estudar uma ampla variedade de sistemas trans-Planckianos e predizer seu comportamento dispensando cálculos mais detalhados proprios da teoria quântica da gravitação.

Thermodynamics of black holes in finite boxes

We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black ho... more We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black hole mass evolution is analyzed in an initially empty box. Using the conservation of the energy and the Hawking evaporation flux, we deduce a minimal volume above which one black hole can loss all of its mass to the box, a result which agrees with the previous analysis made by Page. We then obtain analogous results using a box initially containing radiation, allowed to be absorbed by the black hole. The equilibrium times and masses are evaluated and their behavior discussed to highlight some interesting features arising. These results are generalized to N black holes + thermal radiation. Using physically simple arguments, we prove that these black holes achieve the same equilibrium masses (even that the initial masses were different). The entropy of the system is used to obtain the dependence of the equilibrium mass on the box volume, number of black holes and the initial radiation. The equilibrium mass is shown to be proportional to a positive power law of the effective volume (contrary to naive expectations), a result explained in terms of the detailed features of the system. The effect of the reflection of the radiation on the box walls which comes back into the black hole is explicitly considered. All these results (some of them counter-intuitive) may be useful to formulate alternative problems in thermodynamic courses for graduate and advanced undergraduate students. A handful of them are suggested in the Appendix.

Thermodynamics of black holes in a finite box

American Journal of Physics, 2003

We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black ho... more We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black hole mass evolution is analyzed in an initially empty box. Using the conservation of the energy and the Hawking evaporation flux, we deduce a minimal volume above which one black hole can loss all of its mass to the box, a result which agrees with the previous analysis made by Page. We then obtain analogous results using a box initially containing radiation, allowed to be absorbed by the black hole. The equilibrium times and masses are evaluated and their behavior discussed to highlight some interesting features arising. These results are generalized to N black holes + thermal radiation. Using physically simple arguments, we prove that these black holes achieve the same equilibrium masses (even that the initial masses were different). The entropy of the system is used to obtain the dependence of the equilibrium mass on the box volume, number of black holes and the initial radiation. The equilibrium mass is shown to be proportional to a positive power law of the effective volume (contrary to naive expectations), a result explained in terms of the detailed features of the system. The effect of the reflection of the radiation on the box walls which comes back into the black hole is explicitly considered. All these results (some of them counter-intuitive) may be useful to formulate alternative problems in thermodynamic courses for graduate and advanced undergraduate students. A handful of them are suggested in the Appendix.

Dynamics of black hole motion

Physical Review D, 1999

We evaluate in the proper and cosmological frames the effects of nonzero velocities on the mass g... more We evaluate in the proper and cosmological frames the effects of nonzero velocities on the mass gain or loss of primordial black holes. An upper limit of twice the initial Lorentz factor is derived from the mass gain of the black holes, a value that may be enough to preclude their evaporation. Next, we analyze accelerated black holes and find that the Unruh effect can delay the onset of the evaporation regime. Finally, we reassess the equilibrium between black holes and the relic thermal radiation. ͓S0556-2821͑99͒00718-3͔

Evolution of primordial black holes in a radiation and phantom energy environment

by Daniel Guariento and Paulo Custodio

General Relativity and Gravitation 40(8), 1596-1602, Aug 2008

In this work we extend previous work on the evolution of a primordial black hole (PBH) to address... more In this work we extend previous work on the evolution of a primordial black hole (PBH) to address the presence of a dark energy component with a super-negative equation of state as a background, investigating the competition between the radiation accretion, the Hawking evaporation and the phantom accretion, the latter two causing a decrease on black hole mass. It is found that there is an instant during the matter-dominated era after which the radiation accretion becomes negligible compared to the phantom accretion. The Hawking evaporation may become important again depending on a mass threshold. The evaporation of PBHs is quite modified at late times by these effects, but only if the generalized second law of thermodynamics is violated.

Dynamics of black hole motion

We evaluate in the proper and cosmological frames the effects of nonzero velocities on the mass g... more We evaluate in the proper and cosmological frames the effects of nonzero velocities on the mass gain or loss of primordial black holes. An upper limit of twice the initial Lorentz factor is derived from the mass gain of the black holes, a value that may be enough to preclude their evaporation. Next, we analyze accelerated black holes and find that the Unruh effect can delay the onset of the evaporation regime. Finally, we reassess the equilibrium between black holes and the relic thermal radiation. S0556-28219900718-3

Cosmological quintessence accretion onto primordial black holes : conditions for their growth to the supermassive scale

In this work we revisit the growth of small primordial black holes (PBHs) immersed in a quintesse... more In this work we revisit the growth of small primordial black holes (PBHs) immersed in a quintessential field and/or radiation to the su-permassive black hole (SMBHs) scale. We show the difficulties of scenarios in which such huge growth is possible. For that purpose we evaluated analytical solutions of the differential equations (describing mass evolution) and point out the strong fine tuning for that conclusions. The timescale for growth in a model with a constant quintessence flux is calculated and we show that it is much bigger than the Hubble time.The fractional gain of the mass is further evaluated in other forms, including quintessence and/or radiation. We calculate the cosmological density Ω due to quintessence necessary to grow BHs to the supermassive range and show it to be much bigger than one. We also describe the set of complete equations analyzing the evolution of the BH+quintessence universe, showing some interesting effects such the quenching of the BH mass growth due to the evolution of the background energy. Additional constraints obtained by using the Holographic Bound are also described. The general equilibrium conditions for evaporating/accreting black holes evolving in a quintessence/radiation universe are discussed in the Appendix.

Thermodynamics of black holes in finite boxes

We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black ho... more We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black hole mass evolution is analyzed in an initially empty box. Using the conservation of the energy and the Hawking evaporation flux, we deduce a minimal volume above which one black hole can loss all of its mass to the box, a result which agrees with the previous analysis made by Page. We then obtain analogous results using a box initially containing radiation, allowed to be absorbed by the black hole. The equilibrium times and masses are evaluated and their behavior discussed to highlight some interesting features arising. These results are generalized to N black holes + thermal radiation. Using physically simple arguments, we prove that these black holes achieve the same equilibrium masses (even that the initial masses were different). The entropy of the system is used to obtain the dependence of the equilibrium mass on the box volume, number of black holes and the initial radiation. The equilibrium mass is shown to be proportional to a positive power law of the effective volume (contrary to naive expectations), a result explained in terms of the detailed features of the system. The effect of the reflection of the radiation on the box walls which comes back into the black hole is explicitly considered. All these results (some of them counter-intuitive) may be useful to formulate alternative problems in thermodynamic courses for graduate and advanced undergraduate students. A handful of them are suggested in the Appendix.

Relatividade Geral

Introdução à Relatividade Geral - Maple, 2020

Aqui está um resumo sobre o conteúdo do meu curso na Udemy: Introdução à Física dos Buracos Negro... more

Factoring Integers_in_Primes_usingCSahrp.pdf

It is a draft of a no-recursive algorithm to factorize integers in primes. It is necessary to mod... more

Introdução aos Buracos Negros e Relatividade Geral - seção 2

Este capítulo é uma breve introdução ao cálculo de tensores e equações de Einstein, uma preparaçã... more

Introdução aos Buracos Negros e Relatividade Geral - seção 1

Nesta primeira seção apresento um resumo sobre Mecânica Relativística e uma rápida revisão sobre ... more

High-Energy Production of Black Holes in the Very Early Universe

The formation of Primordial Black Holes is a robust prediction of several gravitational theories.... more The formation of Primordial Black Holes is a robust prediction of several gravitational theories. Whereas the creation of PBHs was very active in the remote past, such process seem to be very negligible at the present epoch. In this work, we estimate the fraction of PBHs (in terms of the critical density) formed from the high-energy collision of particles before Inflation, when the temperatures were close to Planck values. We consider that the Universe has dimension D, and we evaluate the e-folds number in order get a universe free from such Primordial Black Holes. Finally, the Holographic constraint is used to estimate upper bounds to temperature and mass of PBHs with monochromatic spectrum.

Plataforma_NET.pdf

introdução básica à programação em C#

Survival of small PBHs in the Very Early Universe

The formation of Primordial Black Holes is a robust prediction of several gravitational theories.... more The formation of Primordial Black Holes is a robust prediction of several gravitational theories. Whereas the creation of PBHs was very active in the remote past, such process seem to be very negligible at the present epoch. In this work, we estimate the effects from the radiation surrounding PBHs due to the absorption term in the equations that describe how their masses depend on time. The Hawking radiation contributes with mass loss and the absorption term contributes with gain, but a interesting competition between these terms is analysed. These effects are included in the equations describing PBHs and its mass density as the universe evolves in time and the model is able to describes the evolution of the numerical density of PBHs and the mass evolution and comparisons with cosmological constraints set upper limits in their abundances. We evaluate the effect of this accretion onto PBHs and we get some corrections for the initial masses that indicates some deviations from default values for the time scale for evaporation. The scale time of the PBHs in the early universe is modified due to the energy accretion and we can estimate how these contributions may alter the standard model of PBHs.

Transforming metric s in new coordinates

In this small lecture we deduce the transformation between Scharschild into Kruskal-Szekeres metr... more

Relatividade Geral Conceitos básicos - atualizado - Maio 2023

Relatividade Geral, 2023

Survival of small PBHs in the Very Early Universe

The formation of Primordial Black Holes is a robust prediction of several gravitational theories.... more The formation of Primordial Black Holes is a robust prediction of several gravitational theories. Whereas the creation of PBHs was very active in the remote past, such process seem to be very negligible at the present epoch. In this work, we estimate the effects from the radiation surrounding PBHs due to the absorption term in the equations that describe how their masses depend on time. The Hawking radiation contributes with mass loss and the absorption term contributes with gain, but a interesting competition between these terms is analysed. These effects are included in the equations describing PBHs and its mass density as the universe evolves in time and the model is able to describes the evolution of the numerical density of PBHs and the mass evolution and comparisons with cosmological constraints set upper limits in their abundances. We evaluate the effect of this accretion onto PBHs and we get some corrections for the initial masses that indicates some deviations from default values for the time scale for evaporation. The scale time of the PBHs in the early universe is modified due to the energy accretion and we can estimate how these contributions may alter the standard model of PBHs.