I confess that I've never heard of Paul Gilroy or the book Black Atlantic. To remedy this, I searched online and found this information which I'll share in case there are others here who also didn't recognize the man and the book or other books that he has written.

Here's some information from his Wikipedia page:

Paul Gilroy (born February 16, 1956) is a Professor at the London School of Economics.

Born in the East End of London to Guyanese and English parents (his mother was Beryl Gilroy). He was educated at University College School and obtained his bachelor's degree at Sussex University in 1978. He moved from there to Birmingham University where he completed his Ph.D. in 1986. Gilroy is a sociologically inclined scholar of Cultural Studies and Black Atlantic diasporic culture. He is the author of Ain't no Black in the Union Jack (1987), Small Acts (1993), The Black Atlantic (1993), Between Camps (2000) (also published as "Against Race" in the United States), and "After Empire" (2004) (published as Postcolonial Melancholia in the United States), among other works. Gilroy was also co-author of The Empire Strikes Back: race and racism in 1970s Britain (1982) a path-breaking, collectively-produced volume published under the imprint of the Centre for Contemporary Cultural Studies at Birmingham University where he was a doctoral student working with the Jamaican intellectual Stuart Hall. Other members of the group which produced that volume included Valerie Amos and Pratibha Parmar...

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Here's another excerpt from an article about the book Black Atlantic:

..."As a starting point, Paul Gilroy, a sociologist at Goldsmith's College in London, describes black identity in Europe and the New World as an ongoing process of travel and exchange across the Atlantic that tried to understand its position in relation to European modernity in his The Black Atlantic: Modernity and Double Consciousness (1992)...

More than simply understanding Black cultures from around the Atlantic basin as being marginal to or derived from dominant national cultures that result in specific subcultures like African-American or Anglo-African that have a closer relation to American or British culture-at-large than to each other, Gilroy argues that for a century and a half, black intellectuals have travelled and worked in a transnational frame that precludes anything but a superficial association with their country of origin. He argues, in other words, for a long, complex history of African-diasporic intellectual culture that is specifically transnational. Moreover, Gilroy shows how major figures from Frederick Douglass through W.E.B. DuBois to Richard Wright took up autonomous positions in relation to the great philosophers of modernity-- Hegel, Marx, & Nietzsche. Expanding on DuBois' crucial notion of "double consciousness," Gilroy argues for a modernity broad enough in scope not simply to include the marginal positions of slaves, but to put the "ungenteel" aspects of slavery and terror as crucial and systematic enough to understand them at the heart of modernity, itself: "A preoccupation with the striking doubleness that results from this unique position-- in an expanded West but not completely of it-- is a definitive characteristic of the intellectual history of the black atlantic" (58)."

http://www.english.emory.edu/Bahri/Gilroy.htm

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Well, it's clear that I have a lot more reading to do.