Hawking radiation and the Stefan-Boltzmann law: The effective radius of the black-hole quantum atmosphere
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- 作者：Shahar Hoda ; b ; shaharhod@gmail.com" class="auth_mail" title="E-mail the corresponding author
- 刊名：Physics Letters B
- 出版年：2016
- 出版时间：10 June 2016
- 年：2016
- 卷：757
- 期：Complete
- 页码：121-124
- 全文大小：228 K
文摘
It has recently been suggested (S.B. Giddings (2016) [2]) that the Hawking black-hole radiation spectrum originates from an effective quantum “atmosphere' which extends well outside the black-hole horizon. In particular, comparing the Hawking radiation power of a mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316300429&_mathId=si1.gif&_user=111111111&_pii=S0370269316300429&_rdoc=1&_issn=03702693&md5=aa8b301899f7153a0ea2c9cfca0f0469" title="Click to view the MathML source">(3+1)mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><mo stretchy="false">(mo><mn>3mn><mo>+mo><mn>1mn><mo stretchy="false">)mo>math>-dimensional Schwarzschild black hole of horizon radius mmlsi2" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316300429&_mathId=si2.gif&_user=111111111&_pii=S0370269316300429&_rdoc=1&_issn=03702693&md5=6fa592709ae3fa8c24e2bd327617b54a" title="Click to view the MathML source">rHmathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><msub><mrow><mi>rmi>mrow><mrow><mtext>Hmtext>mrow>msub>math> with the familiar Stefan–Boltzmann radiation power of a mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316300429&_mathId=si1.gif&_user=111111111&_pii=S0370269316300429&_rdoc=1&_issn=03702693&md5=aa8b301899f7153a0ea2c9cfca0f0469" title="Click to view the MathML source">(3+1)mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><mo stretchy="false">(mo><mn>3mn><mo>+mo><mn>1mn><mo stretchy="false">)mo>math>-dimensional flat space perfect blackbody emitter, Giddings concluded that the source of the Hawking semi-classical black-hole radiation is a quantum region outside the Schwarzschild black-hole horizon whose effective radius mmlsi3" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316300429&_mathId=si3.gif&_user=111111111&_pii=S0370269316300429&_rdoc=1&_issn=03702693&md5=a254a42a13c7af7a055732f4398b835e" title="Click to view the MathML source">rAmathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll"><msub><mrow><mi>rmi>mrow><mrow><mtext>Amtext>mrow>msub>math> is characterized by the relation mmlsi4" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316300429&_mathId=si4.gif&_user=111111111&_pii=S0370269316300429&_rdoc=1&_issn=03702693&md5=8ab4ad8599401581ba83d435301d9bcd" title="Click to view the MathML source">Δr≡rA−rH∼rHmathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><mi mathvariant="normal">Δmi><mi>rmi><mo>≡mo><msub><mrow><mi>rmi>mrow><mrow><mtext>Amtext>mrow>msub><mo>−mo><msub><mrow><mi>rmi>mrow><mrow><mtext>Hmtext>mrow>msub><mo>∼mo><msub><mrow><mi>rmi>mrow><mrow><mtext>Hmtext>mrow>msub>math>. It is of considerable physical interest to test the general validity of Giddings's intriguing conclusion. To this end, we study the Hawking radiation of mmlsi41" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316300429&_mathId=si41.gif&_user=111111111&_pii=S0370269316300429&_rdoc=1&_issn=03702693&md5=f14a9a25ca65a5438b0c17b755e1990a" title="Click to view the MathML source">(D+1)mathContainer hidden">mathCode"><math altimg="si41.gif" overflow="scroll"><mo stretchy="false">(mo><mi>Dmi><mo>+mo><mn>1mn><mo stretchy="false">)mo>math>-dimensional Schwarzschild black holes. We find that the dimensionless radii mmlsi16" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316300429&_mathId=si16.gif&_user=111111111&_pii=S0370269316300429&_rdoc=1&_issn=03702693&md5=5ee7a6e343a26e7c3a4855a52076db1a" title="Click to view the MathML source">rA/rHmathContainer hidden">mathCode"><math altimg="si16.gif" overflow="scroll"><msub><mrow><mi>rmi>mrow><mrow><mtext>Amtext>mrow>msub><mo stretchy="false">/mo><msub><mrow><mi>rmi>mrow><mrow><mtext>Hmtext>mrow>msub>math> which characterize the black-hole quantum atmospheres, as determined from the Hawking black-hole radiation power and the mmlsi41" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316300429&_mathId=si41.gif&_user=111111111&_pii=S0370269316300429&_rdoc=1&_issn=03702693&md5=f14a9a25ca65a5438b0c17b755e1990a" title="Click to view the MathML source">(D+1)mathContainer hidden">mathCode"><math altimg="si41.gif" overflow="scroll"><mo stretchy="false">(mo><mi>Dmi><mo>+mo><mn>1mn><mo stretchy="false">)mo>math>-dimensional Stefan–Boltzmann radiation law, are a decreasing function of the number mmlsi7" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316300429&_mathId=si7.gif&_user=111111111&_pii=S0370269316300429&_rdoc=1&_issn=03702693&md5=44a3500973d9aee6a900da3900a30d7f" title="Click to view the MathML source">D+1mathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll"><mi>Dmi><mo>+mo><mn>1mn>math> of spacetime dimensions. In particular, it is shown that radiating mmlsi41" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316300429&_mathId=si41.gif&_user=111111111&_pii=S0370269316300429&_rdoc=1&_issn=03702693&md5=f14a9a25ca65a5438b0c17b755e1990a" title="Click to view the MathML source">(D+1)mathContainer hidden">mathCode"><math altimg="si41.gif" overflow="scroll"><mo stretchy="false">(mo><mi>Dmi><mo>+mo><mn>1mn><mo stretchy="false">)mo>math>-dimensional Schwarzschild black holes are characterized by the relation mmlsi8" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316300429&_mathId=si8.gif&_user=111111111&_pii=S0370269316300429&_rdoc=1&_issn=03702693&md5=1f34c083e551f9c712ec7f825e752037" title="Click to view the MathML source">(rA−rH)/rH≪1mathContainer hidden">mathCode"><math altimg="si8.gif" overflow="scroll"><mo stretchy="false">(mo><msub><mrow><mi>rmi>mrow><mrow><mtext>Amtext>mrow>msub><mo>−mo><msub><mrow><mi>rmi>mrow><mrow><mtext>Hmtext>mrow>msub><mo stretchy="false">)mo><mo stretchy="false">/mo><msub><mrow><mi>rmi>mrow><mrow><mtext>Hmtext>mrow>msub><mo>≪mo><mn>1mn>math> in the large mmlsi9" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316300429&_mathId=si9.gif&_user=111111111&_pii=S0370269316300429&_rdoc=1&_issn=03702693&md5=5583428eb866fdf031492a40050598a8" title="Click to view the MathML source">D≫1mathContainer hidden">mathCode"><math altimg="si9.gif" overflow="scroll"><mi>Dmi><mo>≫mo><mn>1mn>math> regime. Our results therefore suggest that, at least in some physical cases, the Hawking emission spectrum originates from quantum excitations very near the black-hole horizon.