(a) Derive the following useful conversion factors from the SI values of G and c: G/c2 = 7.425 Ã 1028mkg1 = 1, c5/G = 3.629

(a) Derive the following useful conversion factors from the SI values of G and c:
G/c2 = 7.425 × 10ˆ’28mkgˆ’1 = 1,
c5/G = 3.629 × 1052J sˆ’1 = 1.
(b) Derive the values in geometrized units of the constants in Table 8.1 from their given values in SI units.
(c) Express the following quantities in geometrized units:
(i) a density (typical of neutron stars) ρ = 1017 kgmˆ’3;
(ii) a pressure (also typical of neutron stars) p = 1033 kg sˆ’2 mˆ’1;
(iii) the acceleration of gravity on Earth's surface g = 9.80msˆ’2;
(iv) the luminosity of a supernova L = 1041 J sˆ’1.
(d) Three dimensioned constants in nature are regarded as fundamental: c, G, and h, with c = G = 1, h has units m2, so h1/2 defines a fundamental unit of length, called the Planck length. From Table 8.1, we calculate h1/2 = 1.616 × 10ˆ’35m. Since this number involves the fundamental constants of relativity, gravitation, and quantum theory, many physicists feel that this length will play an important role in quantum gravity. Express this length in terms of the SI values of c, G, and h. Similarly, use the conversion factors to calculate the Planck mass and Planck time, fundamental numbers formed from c, G, and h that have the units of mass and time respectively. Compare these fundamental numbers with characteristic masses, lengths, and timescales that are known from elementary particle theory.

Table 8.1 Comparison of SI and geometrized values of fundamental constants Constant SI value Geometrized value 2.998 x 108 ms-I 6.674 x 10-1Im3 kgs2 1.055 x 10-34 kgm2s-1 9.109 x 10-31 kg 1.673 10-27 kg 1.988 x 1030 kg 5.972 x 1024kg 3.84 x 1026 kg ms-3 2.612 x 10-70 m2 6.764 x 10-58 m 11.242 x 10-54 m 1.476 x 103 m 4.434 x 10-3 m 1.06 x 10-26 lne Lo