The present research paper derives a formula for gravitational force acting between the black hole and light particle passing near the radius of event horizon of black holes and calculates also their values of different test black holes existing in only X-ray binaries (XRBs).

Dipo Mahto

Md Shams Nadeem

Mahendra Ram

Kumari Vineeta

Gravitational Force between the Black Hole and Light Particle in XRBs

We use the theory of teleparallelism equivalent to general relativity based on noncommutative spacetime coordinates. In this context, we write the corrections of the Schwarzschild solution. We propose the existence of a Weitzenb&#xf6;ck spacetime that matches the corrected metric tensor. As an important result, we find the corrections of the gravitational energy in the realm of teleparallel gravity due to the noncommutativity of spacetime. Then we interpret such corrections as a manifestation of quantum theory in gravitational field.

S. C. Ulhoa

R. G. G. Amorim

On Noncommutative Corrections of Gravitational Energy in Teleparallel Gravity

We study how may vary the gravitational and the cosmological &#x201c;constants,&#x201d; (<svg     style="vertical-align:-0.23206pt;width:12.4375px;" id="M1" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242&#xA;q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z" /></g>

		
	
</svg> and <svg     style="vertical-align:-0.0pt;width:11.7875px;" id="M2" height="11.425" version="1.1" viewBox="0 0 11.7875 11.425" width="11.7875"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.363)"><path id="x39B" d="M671 0h-247v28q49 7 60 18.5t0 45.5l-155 438h-2q-21 -61 -74 -204.5t-75 -210.5q-15 -51 -4.5 -65t67.5 -22v-28h-221v28q50 9 69 26.5t45 84.5l202 517l32 9l194 -535q23 -61 42 -78.5t67 -23.5v-28z" /></g>

		
	
</svg>) in several scalar-tensor theories with Bianchi III, <svg     style="vertical-align:-3.3907pt;width:24.200001px;" id="M3" height="15.4" version="1.1" viewBox="0 0 24.200001 15.4" width="24.200001"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path id="x56" d="M687 650v-28q-52 -6 -71.5 -23t-47.5 -83q-142 -346 -208 -527h-31q-119 325 -207 545q-21 54 -39.5 68.5t-65.5 19.5v28h245v-28q-52 -5 -61.5 -15.5t3.5 -43.5q55 -157 166 -437h2q120 303 158 421q14 43 3.5 55.5t-71.5 19.5v28h225z" /></g><g transform="matrix(.017,-0,0,-.017,12.013,11.112)"><path id="x49" d="M303 0h-265v28q62 5 76 19.5t14 77.5v400q0 63 -14 77.5t-76 19.5v28h265v-28q-62 -5 -76 -19.5t-14 -77.5v-400q0 -64 14 -78t76 -19v-28z" /></g>

			<g transform="matrix(.012,-0,0,-.012,17.812,15.187)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g>

		
	
</svg>, and <svg     style="vertical-align:-3.3907pt;width:24.637501px;" id="M4" height="15.4" version="1.1" viewBox="0 0 24.637501 15.4" width="24.637501"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x56"/></g><g transform="matrix(.017,-0,0,-.017,12.013,11.112)"><use xlink:href="#x49"/></g>

			<g transform="matrix(.012,-0,0,-.012,17.812,15.187)"><path id="x210E" d="M499 88q-41 -46 -83.5 -73t-68.5 -27q-41 0 -19 98l52 225q10 43 8 59.5t-16 16.5q-37 0 -112 -65t-117 -135q-14 -42 -38 -179q-36 -7 -75 -20l-7 8l127 594q8 40 2.5 49t-35.5 9h-35l5 25q36 3 71.5 13t56.5 18t25 8q11 0 3 -38l-88 -417h2q77 102 195 167q44 24 79 24&#xA;q61 0 30 -130l-52 -220q-8 -33 4 -33q8 0 31 14.5t43 33.5z" /></g>

		
	
</svg> symmetries. By working under the hypothesis of self-similarity we find exact solutions for two different theoretical models, which are the Jordan-Brans-Dicke (JBD) with <svg     style="vertical-align:-3.56265pt;width:37.025002px;" id="M5" height="16.625" version="1.1" viewBox="0 0 37.025002 16.625" width="37.025002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x39B"/></g><g transform="matrix(.017,-0,0,-.017,11.724,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,17.605,12.138)"><path id="x1D719" d="M542 242q0 -106 -80 -176.5t-192 -72.5l-55 -251l-6 -3q-18 7 -28 38t1 91l23 124q-85 16 -133.5 72.5t-48.5 137.5q0 95 62.5 159.5t166.5 97.5l14 -31q-77 -32 -115 -82.5t-38 -128.5q0 -63 31.5 -107t74.5 -59l125 597l56 43l16 -10l-40 -168q-9 -40 3 -45&#xA;q77 -33 120 -85.5t43 -140.5zM469 232q0 67 -37 115t-77 62l-76 -370q89 -4 139.5 52.5t50.5 140.5z" /></g><g transform="matrix(.017,-0,0,-.017,27.21,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g><g transform="matrix(.017,-0,0,-.017,33.091,12.138)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g>

		
	
</svg> and the usual JBD model with potential <svg     style="vertical-align:-3.56265pt;width:34.462502px;" id="M6" height="16.625" version="1.1" viewBox="0 0 34.462502 16.625" width="34.462502"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D448" d="M751 650l-6 -28q-53 -1 -69 -12.5t-24 -38.5q-23 -85 -43 -192l-13 -69q-34 -181 -130 -262q-79 -65 -185 -65t-171 57q-87 77 -54 258l42 232q12 58 0.5 73t-75.5 19l8 28h263l-6 -28q-63 -5 -78 -18t-26 -74l-41 -226q-25 -135 22 -201t139 -66q99 0 161.5 68t91.5 217&#xA;l11 57q28 149 28 193q0 28 -19 38t-79 12l6 28h247z" /></g><g transform="matrix(.017,-0,0,-.017,13.033,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,18.914,12.138)"><use xlink:href="#x1D719"/></g><g transform="matrix(.017,-0,0,-.017,28.518,12.138)"><use xlink:href="#x29"/></g>

		
	
</svg> (that mimics the behaviour of <svg     style="vertical-align:-3.56265pt;width:39.037498px;" id="M7" height="16.625" version="1.1" viewBox="0 0 39.037498 16.625" width="39.037498"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x39B"/></g><g transform="matrix(.017,-0,0,-.017,11.724,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,17.605,12.138)"><use xlink:href="#x1D719"/></g><g transform="matrix(.017,-0,0,-.017,27.21,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,33.091,12.138)"><use xlink:href="#x29"/></g>

		
	
</svg>. We compare both theoretical models, and some physical and geometrical properties of the solutions are also discussed putting special emphasis on the study of the isotropization of the solutions.

J. A. Belinch&#xf3;n

Scalar-Tensor Bianchi VI Models

We develop a simple model for a self-gravitating spherically symmetric relativistic star which begins to collapse from an initially static configuration by dissipating energy in the form of radial heat flow. We utilize the model to show how local anisotropy affects the collapse rate and thermal behavior of gravitationally evolving systems.

Ranjan Sharma

Shyam Das

Collapse of a Relativistic Self-Gravitating Star with Radial Heat Flux: Impact of Anisotropic Stresses

We have developed the recent investigations on the second-order phase transition in the holographic superconductor using the probe limit for a nonlinear Maxwell field strength coupled to a massless scalar field. By analytical methods, based on the variational Sturm-Liouville minimization technique, we study the effects of the spacetime dimension and the nonlinearity parameter on the critical temperature and the scalar condensation of the dual operators on the boundary. Further, as a motivated result, we analytically deduce the DC conductivity in the low and zero temperatures regime. Especially in the zero temperature limit and in two dimensional toy model, we thoroughly compute the conductivity analytically. Our work clarifies more features of the holographic superconductors both in different space dimensions and on the effect of the nonlinearity in Maxwell's strength field.

Davood Momeni

Muhammad Raza

Ratbay Myrzakulov

More on Superconductors via Gauge/Gravity Duality with Nonlinear Maxwell Field

The gravitational field of conical mass distributions is formulated using the general theory of relativity. The gravitational metric tensor is constructed and applied to the motion of test particles and photons in this gravitational field. The expression for gravitational time dilation is found to have the same form as that in spherical, oblate spheroidal, and prolate spheroidal gravitational fields and hence confirms an earlier assertion that this gravitational phenomena is invariant in form with various mass distributions. It is shown using the pure radial equation of motion that as a test particle moves closer to the conical mass distribution along the radial direction, its radial speed decreases.

Chifu Ebenezer Ndikilar

Gravitational Fields of Conical Mass Distributions

The flat Friedmann universes filled by stiff fluid and a nonminimally
coupled material scalar field with polynomial potentials of the
fourth degree are considered in the framework of the
Einstein-Cartan theory. Exact general solution is obtained for arbitrary positive values of the coupling constant <svg     style="vertical-align:-2.57985pt;width:7.7375002px;" id="M1" height="15.9125" version="1.1" viewBox="0 0 7.7375002 15.9125" width="7.7375002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.638)"><path id="x1D709" d="M425 682q0 -40 -49 -59q-36 -13 -110 -13q-31 -18 -52.5 -49.5t-21.5 -67.5q0 -34 17.5 -57.5t45.5 -28.5q63 15 97 15q18 0 18 -16q0 -13 -22 -36t-34 -23q-50 0 -93 8q-47 -20 -80.5 -66.5t-33.5 -96.5q0 -76 84 -103q33 -11 75 -21q101 -26 101 -93q0 -51 -39 -95.5&#xA;t-76 -68.5l-22 20q60 69 60 111q0 21 -14 34t-55 22q-21 4 -63 16q-68 19 -101.5 52.5t-33.5 96.5q0 125 157 206q-27 10 -46 36t-19 58q0 84 103 152q-43 3 -71.5 21t-28.5 45q0 37 65 59l17 -6q-9 -14 -9 -33q0 -21 17.5 -36t51.5 -19q82 53 147 53q18 0 18 -17z" /></g>

		
	
</svg>. A comparative analysis of the cosmological models with and without stiff fluid is carried out. Some effects of stiff fluid are
elucidated. It is shown that singular models with a de Sitter asymptotic and with the power-law <svg     style="vertical-align:-2.3205pt;width:34.637501px;" id="M2" height="19.5375" version="1.1" viewBox="0 0 34.637501 19.5375" width="34.637501"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,16.588)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,5.944,16.588)"><path id="x1D461" d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" /></g>

			<g transform="matrix(.012,-0,0,-.012,11.738,8.425)"><path id="x34" d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" /></g><g transform="matrix(.012,-0,0,-.012,17.449,8.425)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.012,-0,0,-.012,22.351,8.425)"><path id="x33" d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5&#xA;q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" /></g>

			<g transform="matrix(.017,-0,0,-.017,28.688,16.588)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g>

		
	
</svg> asymptotic at late times are possible. It is found that <svg     style="vertical-align:-2.57985pt;width:50.487499px;" id="M3" height="15.9125" version="1.1" viewBox="0 0 50.487499 15.9125" width="50.487499"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.638)"><use xlink:href="#x1D709"/></g><g transform="matrix(.017,-0,0,-.017,12.387,12.638)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,27.091,12.638)"><use xlink:href="#x33"/></g><g transform="matrix(.017,-0,0,-.017,35.25,12.638)"><use xlink:href="#x2F"/></g><g transform="matrix(.017,-0,0,-.017,42.253,12.638)"><path id="x38" d="M249 635q70 0 116 -43t46 -105q0 -46 -28 -80q-22 -25 -80 -64q62 -35 97 -75t35 -99q0 -81 -63 -131t-135 -50q-83 0 -137.5 45.5t-54.5 123.5q0 52 45 95q29 28 89 64q-109 62 -109 155q0 66 50.5 115t128.5 49zM238 603q-42 0 -67.5 -31t-25.5 -72q0 -50 32.5 -79.5&#xA;t98.5 -62.5q61 48 61 124q0 59 -29.5 90t-69.5 31zM248 20q46 0 76.5 33.5t30.5 89.5q0 50 -39 85.5t-110 71.5q-81 -54 -81 -137q0 -67 35.5 -105t87.5 -38z" /></g>

		
	
</svg> is a specific value of the coupling constant. It is demonstrated that the bouncing models without the particle horizon and with an accelerated expansion by a de Sitter law of an evolution at late times are admissible.

Almaz Galiakhmetov

Stiff Fluid in Accelerated Universes with Torsion

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Journal of Gravity