65 lines
1.2 KiB
Common Lisp
65 lines
1.2 KiB
Common Lisp
"Bondi metric..."
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; coordinate system
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(setq x0 u)
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(setq x1 r)
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(setq x2 theta)
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(setq x3 phi)
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; U, V, beta and gamma are functions of u, r and theta
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(setq g_uu (sum
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(product
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(V u r theta)
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(power r -1)
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(power e (product 2 (beta u r theta)))
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)
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(product
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-1
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(power (U u r theta) 2)
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(power r 2)
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(power e (product 2 (gamma u r theta)))
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)
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))
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(setq g_ur (product 2 (power e (product 2 (beta u r theta)))))
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(setq g_utheta (product
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2
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(U u r theta)
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(power r 2)
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(power e (product 2 (gamma u r theta)))
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))
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(setq g_thetatheta (product
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-1
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(power r 2)
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(power e (product 2 (gamma u r theta)))
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))
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(setq g_phiphi (product
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-1
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(power r 2)
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(power e (product -2 (gamma u r theta)))
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(power (sin theta) 2)
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))
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; metric tensor
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(setq gdd (sum
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(product g_uu (tensor u u))
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(product g_ur (tensor u r))
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(product g_ur (tensor r u))
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(product g_utheta (tensor u theta))
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(product g_utheta (tensor theta u))
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(product g_thetatheta (tensor theta theta))
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(product g_phiphi (tensor phi phi))
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))
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(gr) ; compute g, guu, GAMUDD, RUDDD, RDD, R, GDD, GUD and GUU
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"Is the Einstein tensor GUU divergence-free?"
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(zerop (contract23 (covariant-derivative-of-up-up GUU)))
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