- 4025 - BIG BANG - how hot was it? The temperature of the Universe is inversely proportional to its size. As the Universe expands the temperature decrease. Today it has expanded and cooled to a temperature of 2.725 degrees Kelvin. That is 2.725 degrees above Absolute Zero.
------------------ 4025 - BIG BANG - how hot was it?
- What would happen if we
reversed the expansion and sent time collapsing back to the Big Bang. The temperature would get hotter and
hotter. The pressure would get greater
and greater.
-
- It turns out that the
temperature / pressure relationship is a linear relationship. Plotting
temperature versus pressure we get a straight line all the way back to the Big
Bang. Therefore, we should be able to
use the laws of physics and extrapolate back to the Big Bang to determine how
hot it was.
-
- The math used is called
“dimensional analysis.” It is a simple
concept. To learn it we will try it out
on a pendulum before we try it out on the Universe.
-
- Measurements in physics
and astronomy can involve 3 dimensions:
length (space), time, and mass (energy).
We refer to these as dimensions and they are measured in units: meters,
seconds, kilograms. Starting with the
pendulum let’s use these three dimensions to calculate the period of the
pendulum swing. The pendulum is a mass
hanging freely on a string and swinging back and forth. The period is the time for one cycle of one
swing.
-
- The period is “time“, it
is possibly a function of the three
dimensions ,length, time, and mass to some degree. The force on the mass is a function of
gravity. If we are on the surface of the
Earth the acceleration of gravity is 9.8 meters / second^2 ( 32 feet per second per second.). We might also guess that the period might be
a function of the length of the strength, and the weight of the mass. Guess:
-
------------------
Time = mass
* length *
acceleration of gravity
-
----------------
Acceleration of gravity = “g” = (length
/ time ^2) = (m /
sec^2)
-
--------------- Working
with the units of each dimension, try to derive the equation for the period of
a pendulum: To get to the equality
seconds = seconds.
-
---------------
second = kilograms
* meters *
meters / second^2
-
---------------- Working to
cancel the units on the right side of the equation to equal the left side of
the equation, which is seconds. Let’s
divide by the acceleration of gravity , (meters / seconds^2)^-1
-
------------- sec = kg * m / m / sec^2
-
-------------- Squaring the
left side to get sec^2. There are no
kilograms on the left side of the equation , so, mass must not be a
factor. The period of a pendulum is the
same regardless of the weight of the mass.
Therefore, we can guess the equation for the period of a pendulum to be:
-
-------------- ( period
)^2 =
length / acceleration of gravity
-
--------------- T^2 = L /
g
-
-------------- sec^2 = m^ /
m /sec^2
-
- And, that works. The period is directly proportional to the
length of the string. The longer the
string the longer the period. It is
inversely proportional to the acceleration of gravity. The pendulum would have a longer period
swinging on the Moon. The period would
be independent of the mass, as long as the string did not break.
-
- The pendulum illustrates
how you can solve a problem just by thinking through the dimensions involved
and the units of measurement. Now, you
are ready to derive the equation for the temperature at the instant of the Big
Bang:
-
- The three fundamental
dimensions are length, time and mass.
Our plan will be to use the 3 fundamental constants in Nature: “h“, “c“, and “G“. Use these to calculate the Energy of the
Big Bang from E= mc^2. Then use Boltzman’s
Constant to convert Energy to temperature.
First let’s define the 3 fundamental constants of Nature:
-
---------- h =
Planck’s Constant of Action.
Action is the product of Energy and Time.
-
-------------------- h =
6.6*10^-34 kilograms * meters^2 / second
-
-------------------- h
units =
kg * m^2 / sec
-
---------------------
c = (speed of light)
-
---------------------
c = 3*10^8 meters / second
-
-------------------- c
units =
(m / sec)
-
-------------------- G = Gravitational
Constant
-
---------------------
G = 6.7*10^-11 meters^3 / ( kilogram * seconds^2)
-
---------------------
G units = m^3 / ( kg*sec^2)
-
- Using E = mc^2 all we
need is Mass to calculate Energy. We
know the speed of light. Putting Mass in
terms of the units of the fundamental constants: To get the equation to balance we need
kilograms in the numerator. One way to
do that is to start with “ G” in the denominator:
-
---------------------
kilograms = 1/ G
= ( kg * sec^2) / m^3
-
- Now, to cancel out sec /
meter we multiply by “c” which is in m / sec:
-
------------------
kilograms = c / G
= ( kg * sec^2) / m^3 * m / sec
-
- Now, we still need to
cancel more sec / meter so multiply this by “h” which is kilograms * meters^2 /
sec.
-
------------------
kilograms = c * h / G
= ( kg * sec) / m^2 *
kg * m^2 / sec
-
---------------
kilograms = c * h / G
= ( kg ^2)
-
- Now, we need to square the left side of the
equation in order to get the equality and have all the units cancel out:
-
---------------
(kilograms)^2 = c * h / G
= ( kg ^2)
-
--------------- M^2 = c *
h / G
-
- Now, we have the Mass in
terms of the fundamental constants so we can substitute the values for the
constants and solve for the fundamental Mass:
-
------------ M^2 = (3*10^8 meters / second) *
(6.6*10^-34 kilograms * meters^2 / second) /
(6.7*10^-11 meters^3 / ( kilogram * seconds^2))
-
---------------- M^2 =
2.955*10^-15 kg
-
----------------- M = 5.4
*10^-8 kg
-
- Now, we can calculate E =
mc^2:
-
------------------ E = (5.4 *10^-8 kg) * ( 9 * 10^16 m^2/sec^2)
-
------------------ E = (50*10^8 kg * m^2/sec^2)
-
------------------ E = (5 *10^9 Joules.
-
- Now, using Boltzman’s
constant for converting Energy to Temperature:
( See Review # 1072 to learn about Boltzman’s Constant.
-
--------------- T =
(Kelvin / 1.381 *10^-23 Joules )
* Energy
-
-------------- T =
(Kelvin / 1.381 *10^-23 Joules )
* (5 *10^9 Joules)
-
----------------- T = 3.5 *
10^32 Kelvin
-
- Therefore the temperature
at the time of the Big Bang according to the fundamental constants was
350,000,000,000,000,000,000,000,000,000,000
degrees Kelvin. I think you
needed to know that.
-
------------------ Other Reviews that have background on this same
topic:
-
# 1345 - Absolute zero temperature strangeness.
-
# 1234 - What happens when you cool an atom
-
# 1072 - Boltzman’s Constant, Avogadro’s Number.
-
# 727 Absolute zero _
TEMPERATURE
-
# 406 Zero is nothing, but Absolute
Zero is something else.
-
-------------------------------------------------------------------------
- The same dimensional
analysis method can be used to calculate fundamental time:
-
------------------ T^2 = G
* h / c^5
-
------------------ T = 1.4
* 10^-43 seconds
-
--------------------------------------------------------------------------
- To calculate
fundamental distance:
-
----------------- L^2 = G *
h / c3
-
------------------- L =
1.62*10^-35 meters
-
May 23, 2023 BIG BANG
- how hot was it? 1346 4025
-------- Comments
appreciated and Pass it on to whomever is interested. ---
--- Some reviews are
at: -------------- http://jdetrick.blogspot.com -----
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corrections, request for copies or Index of all reviews
--- to: ------
jamesdetrick@comcast.net
------ “Jim Detrick” -----------
--------------------- ---
Saturday, May 27, 2023 ---------------------------------
-
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