Planck Era

Our understanding of the Planck era is poor because at this time a theory which encompasses both quantum mechanics and gravity might need to be developed before progress can be made. To see this, consider the following.

Compton Wavelength

Quantum mechanics is a strange and wonderful theory. It is a different way to view the world and makes many strange predictions and experiments have never been shown it to be incorrect. For now, let me concentrate on one aspect of the theory.

• Heisenberg Uncertainty Principle:

  dp x dx > h / (2 pi) .

  When you measure the position and momentum of a particle, you cannot measure the quantities with infinite precision. The product of the uncertainties must always be greater than some constant number (known as Planck's constant [denoted by h]) divided by 2 times pi. The uncomfortable notion is that the more precisely you measure the momentum, p, i.e., the smaller dp, in order to keep the product larger than some minimum value means that dx, the uncertainty in position, must be large (and vice versa). The upshot of this is that one cannot locate the position of a particle to infinite precision. One can only calculate the probability of where particles will be found.

  Roughly for a particle of mass m, let's find this uncertainty (spread) in the location of the particle. A nice number to use for dp is to let the uncertainty in the momentum be given by m x c, so that

  mc x dx greater than h ===> dx = Compton wavelength = W(Compton) ~ h / mc

  You cannot localize a particle any better than roughly its Compton wavelength.

Black Holes

Recall that a black hole has a singularity at its center which is cloaked by an event horizon. The radius of the event horizon is the

• Schwarschild radius ===> R(Sch) = 2 G M / c**2 .

Recall that it is good that the singularity is hidden from our Universe by the event horizon as interesting (and odd) things can happen near the singularity.

QM meets GR

So now ask the question of what happens if you merge the ideas of QM with the ideas of Einstein (GR)? Well, suppose that

• W(Compton) ~ R(Sch)

so that the uncertainty in the position of the singularity becomes comparable to the radius of the event horizon. This will uncloak the singularity!! On this scale, the structure of space-time becomes poorly defined and general relativity (as formulated) must break down.

• Planck Scales

  This breakdown occurs for masses known as the Planck mass which is defined as

  Planck mass = M(P) = sqrt[hc/(2G)] ~ 5.5 x 10**(-5) grams

  For reference purposes: m(proton) = 1.7 x 10**(-24) grams and m(electron) = 9.1 x 10**(-28) grams

  The W(Compton) and R(Sch) which correspond to the Planck mass is

  W(Compton) = h/mc = R(Sch) = 2GM/c**2 = 4 x 10**(-33) cm

  This distance is known as the Planck length and the light travel time across this distance is known as the Planck time and is given by

  Planck time = Planck length / c = 10**(-43) seconds !!!

What this means is that our current theory of gravity (GR) breaks down for times (and scales) less than 10**(-43) seconds, the Planck time. As a result, if we want to try and understand the earliest phases of the Universe we need to develop a theory which merges QM and GR, i.e., a theory of Quantum Gravity.