Transcript Planck Mass

Astronomy and Cosmology
week 5 – Tuesday 6 May 2003
LIGHT
• Star Date
• Light lecture
• Workshop: calculate Planck mass (Univ.5e Ch.28)
break
• Minilectures
• Thursday: Ch.6, Telescopes
• Friday: quiz on Ch.5 (not Ch.6)
• Next week: spectra and spectrometers
Calculating the Planck length and mass:
1.
You used energy conservation to find the GRAVITATIONAL
size of a black hole, the Schwartzschild radius R.
2. Next, use the energy of light to calculate the QUANTUM
MECH. size of a black hole, De Broglie wavelength l.
3. Then, equate the QM size with the Gravitational size to find
the PLANCK MASS Mp of the smallest sensible black hole.
4. Finally, substitute M into R to find PLANCK LENGTH Lp
5. and then calculate both Mp and Lp.
1. Gravitational size of black hole (BH):
R = event horizon
Gravitational energy  kinetic energy
GmM 1
2
 mv
r
2
2GM
You solved for r  2
v
The Schwarzschild radius, inside which not even light (v=c)
can escape, describes the GRAVITATIONAL SIZE of BH.
Rgrav
GM
 2
c
2. Quantum mechanical size of black hole
Energy of photon  wavelength of particle
E
hc
l
 pc

p  Mc 
h
l
Solve for wavelength l in terms of mass M :
l  ____________
The deBroglie wavelength, l, describes the smallest region of
space in which a particle (or a black hole) of mass m can be
localized, according to quantum mechanics.
3. Find the Planck mass, Mp
Schwartzschild radius  deBroglie wavelength
Rl
GM p
h

2
c
M pc
Solve for the Planck mass :
M p 2  ____________
If a black hole had a mass less than the Planck mass Mp,
its quantum-mechanical size could be outside its event horizon.
This wouldn’t make sense, so M is the smallest possible black hole.
4. Find the Planck length, Lp
hc
Substitute your Planck mass, M p 
, into either R or l :
G
GM p
R  2  ______________
c
h
l
 ______________
M pc
These both yield the Planck length, Lp. Any black hole smaller than
this could have its singularity outside its event horizon. That
wouldn’t make sense, so L is the smallest possible black hole we
can describe with both QM and GR, our current theory of gravity.
5. Calculate the Planck length and mass
Use these fundamental constants :
h  6 x 1034
3
kg m 2
m
m
, c  3 x 108  ms  , G  7 x 1011
s
s
kg s 2
hc
to evaluate the Planck mass, M p 
 _____________
G
and the Planck length L p 
GM p
c
2
 _________________
These are smallest scales we can describe with both QM and GR.