Archive of Astronomy Questions and Answers

Why doesn't the electrostatic force lead to a black hole condition inside an electron?

The rest mass of an electron is 9.1 x 10^28 grams. But for a black hole radius of 10^-33 centimeters, you only need a mass of 10^-5 grams, so the rest mass of an electron would yield a black hole smaller than the Planck quantum limit. Now, because E = mc^2, we can have a look at the energy stored in the electric field of an electron and relate this to an effective mass, which according to general relativity should also contribute to its black hole size.

              
                        2
                      e
     Energy  =  --------------
                      
                     r

where  e = 4.8 x 10^-10  ESU.

Then, for example, we can derive the classical radius of the electron where its effective mass is equal to the local energy in its electric field:

             2       2
          m c    =  e  / radius

          

or                  2
                   e
      radius  =  ------
                      2
                  m c  

                  -13
 to get   2.8 x 10     centimeters..the so-called classical radius of the
electron. This also means that
                         -28 
    Mass(field)  = 9 x 10     grams/radius
                                          -13
where 'radius' is now in units of 2.8 x 10    centimeters. The  field strength
           -13
at 1.4 x 10    centimeters or 0.5 x classical radius, and so Mass(field) = 
            -28
2 x 9.1 x 10     grams.

Now, to get a black hole, the effective mass and radius must satisfy the relation:

                      2 G m
            R   =   ---------
                       2
                      c
               -33                       -5
which gives  10    centimeters for m = 10   grams.

Lets follow the effective mass of the electric field and its equivalent black hole radius for various values of the scale of the electron:


Physical Radius            m(field)            R(B. Hole)
..............................................................
      -13                         -28             -57
3 x 10     cm               9 x 10   gm         10    cm

      -20                         -21             -50
3 x 10     cm               9 x 10    gm        10    cm

      -30                         -11             -40
3 x 10     cm               9 x 10    gm        10    cm

      -33                         -8              -37
3 x 10     cm               9 x 10    gm        10    cm

...............................................................

So, the bottom line is that by the time you get to a scale equal to the supposed quantum limit to space-time itself, 3 x 10^-33 centimeters, the effective mass of the electron due to its field energy is still a factor of 10,000 too small for it to make a black hole with a horizon size larger than the Planck limit. In other words, the field energy does not have a large enough mass for the electron to collapse into its own black hole.