Other Functions

Other Functions

From the previous derivation we ended with the mass-time relation:

Using this relation, we can construct further relations--all of which are quite simple. First we start with the luminosity-time relation.

Luminosity-Time Relation

To begin with we state the definition of the luminosity of a black hole:

By simply plugging in the mass-time relation for the value of M we get a luminosity-time relation.

By simply plugging in all constants we then get the luminosity as a function of time

Energy-Time Relation

We can also derive an energy-time function where the E is the energy released by the blackhole.

To begin, we know the relation that

Where L is a function defined by

To simplify we can rewrite the constant as K

Thus providing us with

Pulling out the constant yields

And taking the derivative gives us

Reducing to

But we take away the negative sign because we know that this energy is the energy released by the blackhole. Plugging in K finally gives us

Temperature-Time Relation

We can state the definition of the temperature of the radiation of a blackhole

Plug in the function describing the mass of the blackhole

Then plug in constants and reduce to form

Radius-Time Relation

Stating the Schwarzchild Radius gives

Input the mass as a function of time

Then reduce constants and obtain the relation