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Unit Error in Oppenheimer-Snyder 1939 paper?

  1. May 22, 2015 #1
    Is there an error, perhaps typographical, in the famous 1939 Oppenheimer-Snyder paper "On Continued Gravitational Contraction" (Phys Rev v56 Sept 1, 1956 pp 455-459)?

    Do the units balance in equation 37? The last term inside the last parantheses

    ##\frac{3r_0^{1/2} \tau}{2R_b^2}##

    where ##r_0##, ##\tau## and ##R_b## all have units of length (or time), isn't unitless, as it should be to be subtracted from 1.
     
    Last edited: May 22, 2015
  2. jcsd
  3. May 22, 2015 #2

    fzero

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    First of all, the curly brackets means that the 2nd line of (37) is still under the logarithm. The factor

    $$\frac{3r_0^{1/2} \tau}{2R_b^2}$$

    should be

    $$\frac{3r_0^{1/2} \tau}{2R_b^{3/2}}.$$

    This term comes from the ##R_b r/(r_0 R)## term in the expression for ##M(y)## in (32). We are to set ##R=R_b## and use (27) for ##r##, which gives

    $$ \frac{ R_b r}{r_0 R_b} = \frac{ 1}{r_0 } \left( R_b^{3/2} - \frac{3}{2} r_0^{1/2}\tau \right)^{2/3} = \frac{ R_b}{r_0 } \left( 1- \frac{3r_0^{1/2}\tau}{2 R_b^{3/2}} \right)^{2/3}.$$
     
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