Mass as a Function

Mass as a Function

We can take the derivations earlier and take them much further to obtain a Mass-Time relationship for a blackhole. From there we can then establish Luminosity-Time, Temperature-Time and Schwarzchild Radius-Time relations.

First lets begin with the relationship between time, luminosity and energy.

Differentiating with respect to t

Yields

We can then expand the term L to

Dividing by c²

We can expand to get

Then move all of our M terms to one side of the equation

In order to simplify the rest of the derivation we assign the terms on the right side to K

To rid ourselves of the derivative we can integrate with respect to time from 0 to t

Yielding

From here we want to assign M(t) equal to the mass of a black hole today and M(0) equal to the mass of a black hole many years ago.

We can assign today's mass M(t) equal to zero hense the blackhole has just evaportated completely and then we can solve for M(t).

Thus

Solving gives

We can then substitute to obtain

Subtracting and then factoring gives

Since t₀ is essentially equal to zero at our respective point we can throw away the sign to

We can rewrite the subtraction as

And take the cube root

Plugging in the constants for K gives

If we then evaluate the left hand side since it is primarily constructed of constants we get a function for the Mass of the black hole with respect to time.

What does this mean?

It means that we assign time=0 as the time that a blackhole evaporates completely, we can find the mass of the blackhole by plugging in any value of t where t is the time before the blackhole evaporates in seconds.

For instance, if we wanted the mass of a blackhole 1 hour before it evaporates completely, we can plug in 3600 seconds (3600 seconds=1 hour) and obtain the mass.