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I think:

The book is set aboard a spheric spacecraft which has very odd physical rules. It travels at the speed of light but the speed of light is the speed of sound aboard this ship. When people walk around its like walking through mud since the head which is more outside moves faster than the feet. One scene was a girl who watches her back in a mirror as the light is not fast enough to update the mirror image in time.

Does anyone know of this book?

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    Do you remember anything else, such as when you read the book? Was it a short story or a full novel? – tobiasvl Mar 10 '17 at 11:01
  • It was a novel and it was some years ago, I don't recall how old the book was :-( – Asharon Mar 10 '17 at 11:13
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    Slide rules where often called "slip sticks" as well. – SQB Mar 10 '17 at 11:44
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    One possibility that comes to mind is Redshift Rendezvous by John Stith. Do you remember any other details of the story, plot, et cetera? – Jeff Zeitlin Mar 10 '17 at 13:09
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    Never having read it, I think @JeffZeitlin is correct because I vividly remember hearing a description of a book with a scene where a girl looks at her back in a mirror because of the speed-of-light issue and although I couldn't 100% remember the title my mind was groping around the word "Redshift." – starpilotsix Mar 10 '17 at 14:04
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The only story I've ever read that matches this description is Redshift Rendezvous by John E. Stith. The ship achieves 'faster than light' travel in our universe by shifting into a 'hyperspace' where the speed of light is only 10MPH, and things are much closer together. There are several subplots intertwined in the story, but in all cases, the odd physics play a role in resolving them.

Some of the odd effects include being able to look at your back, as described in the question; being able to create a sonic boom by running in the corridors - this, by the way, is explicitly against ship rules; having to wear a protective field in which c is normal (if your field fails or is shut off, you die - chemical processes are sensitive to the value of c); seeing things outside your field in reds and greys; and so on. While none of these effects are completely nonsensical, a bit of thought will suggest that Stith's science may have been a bit weak, and the effects would not actually occur as described. However, remember that in fiction, plot generally trumps realism and accuracy.

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