Calculation of the Energy
in the Universe.
Oops. Got
here by mistake, or making math calculations is not your forte? Then at
least scroll down to the bottom of the page to either (a) travel through the
Universe mathematically using powers of ten for practice or (b) travel through
the Universe visually to see how big it really is.
To verify, even with the approximate values
for the dimensions of the Universe we have, that the total energy of the
Universe is very close to zero, and probably will turn out to be exactly zero
when better numbers for the Universe are obtained, we need to compare the
positive and negative expressions
(a) E (positive) = mc ^{2}^{}
^{ }
and
(b) E (negative) =  m M _{u }G / R _{u}
_{ }
Insert or aside comment. Someone may question (b) above that the gravitational energy of the Universe is negative. Even when doing a high school physics problem, where you toss a ball of mass M straight up into the air with initial velocity V and you are asked to solve for the height H the ball will reach before it turns around and falls back to the ground, using energy equations rather than force equations, one typically writes the equation: ½ MV^{2
}= MgH That is equivalent to the equation: ½MV^{2
}– MgH = 0 where the two energy terms have opposite signs. If one considers the kinetic energy of ½ MV^{2 }as positive then the gravitational energy MgH is a negative energy contribution. The last equation is really a statement of Conservation of Energy and any system that starts with net energy ZERO must maintain that value, so if some energy goes positive then some other energy (gravitational) must go negative. 
We can eliminate m from both terms
(since it is a hypothetical particle anyway) and compare:
(a) the value of c ^{2}
^{ }
with
(b) the value of  M _{u }G / R _{u}
_{ }
We will use the MKS, or meters, kilograms,
seconds units of measure throughout, but in some instances the exact
dimensionality will not be stated as the expressions become clumsy, especially
when squared.
The first expression is the easiest. The velocity of light in this system of
measure is c = 3 x 10 ^{8
}meters/sec.
So
(a) c^{2 }= (3
x 10 ^{8 }meters/sec) ^{2
}= 9 x 10 ^{16 }meters^{2} per second^{2}.
^{ }
Questions arise to the size and mass of the
Universe as we have not yet seen the very ends of the Universe and so most
discussions simply refer to the Visible Universe, or the dimensions of all that
we can see, and therefore know exists.
The Hubble Telescope is now photographing
objects that are 12 billion light years away.
A light year is the distance light travels in one year. (So we also know that the Universe is at
least 12 billion years old as those objects were in existence 12 billion years
ago when that light first departed those stars on its way here.) We need to convert a light year into meters.
One light year in meters = (velocity of light
in meters/sec) times (number of seconds in a year).
Number of seconds in a year = (3600 sec/hour)
x (24 hours/day) x (365.25 days/year) =
3.156 x 10 ^{7 }seconds/year.
(Yes.
There are 365.25 days in a year  that's why we have leap years every
four years.) So
One light year = (3 x 10 ^{8 }meters/sec) x (3.156 x 10
^{7 }sec/year) = 9.47 x 10 ^{15
}meters.
12 billion years = 12 x 10 ^{9
}years so
12 billion light years = (12 x10 ^{9
}years) x (9.47 x 10 ^{15
}meters per year) = 1.14 x 10 ^{26
}meters so
R_{ u}_{ }= Radius of Visible Universe = 1.14 x
10 ^{26 }meters
As mentioned elsewhere, calculations have been
made on the number of protons (and hence atoms) that the Universe generated
during expansion as 10 ^{80
}protons (atoms.)
The mass of one proton = 1.67 x 10 ^{27
}kilograms.
So multiplying these two numbers together
gives us an estimate of the mass within the Visible Universe as:
M _{u}_{ }= (1.67 x 10 ^{27 }kilograms/proton) x (10 ^{80 }protons)^{ }= 1.67 x 10 ^{53 }kilograms.
It is interesting that calculations for the
mass of the Universe, using more sophisticated methods than we are using here,
including using the more complicated equations from Relativity, produce nearly the same answer.
That lends support to 10^{ 80 }atoms in the Universe being a very accurate number since that is
the "heart" of the calculation we just made.
Newton's Gravitational Constant in this system of units is G = 6.67 x 10 ^{ 11}
So we are now ready to insert (the blue)
values into the more complicated expression:
(b)  M _{u }G / R_{ u
}=  (1.67 x 10 ^{53}) x (6.67 x 10 ^{ 11}) / (1.14 x 10 ^{26}) =  9.77 x 10 ^{16 }
^{ }
If you will compare the two expressions in
red, the positive and negative energies of the Universe (after they are both
multiplied by the mass of our hypothetical particle sitting at the edge of the
Universe) , you will see that they are extremely close, considering what huge
numbers we were dealing with and our rounding off of numbers. Since the second one is considered as
negative energy, that means we have shown that the net energy in the
Universe is ZERO with a discrepancy of about 0.77 parts out of
9 or about 8%.
The most uncertain value used in our
calculations was in estimating the age of the Universe as 12 billion
years. That number was used; as that is
the farthest distance the Hubble Space Telescope (HST) has been able to see so
far. If the age of the Universe were
closer to 13 billion years, the two numbers we calculated would match
exactly. In fact, there is now enough
confidence in the concept itself, that the Universe was constructed from a net
energy of zero. that these calculations are often applied in reverse by saying
these calculations show that the Universe cannot be older than 13 billion
years. From a mathematical point of view
that is quite astounding, that such a simple requirement as "conservation
of energy" could be used to reveal the age of the Universe. If only all mathematical problems were that
easy.

Milky Way Galaxy. "Small stuff" compared to the Universe. The Milky Way is only about 100,000 light years across. Using the numbers above, can you calculate how many times larger the entire Universe is compared to our galaxy? 
Visual Note: It may strike you strange that you started out on the "edge" of a galaxy and ended up in the "center" of the Universe. Relativity indicates all position measurements are "relative," not "absolute" and so all observers in the Universe will measure the same "distance to the edge" and so will conclude they are in the center no matter what galaxy they reside in. This makes perfect sense when you realize that all points in the Universe started out as the same point in the embryonic Universe and so all final points came from the same original point and so behave as if they still are the original point at the center of the Universe. All of this, of course, makes no sense when you are trying to view the Universe from "outside," for as noted elsewhere, the outside of the Universe has no physical meaning. 
(a) If you need more practice using powers of 10 (exponents) then
travel through the size of the Universe while practicing with exponents by: clicking here> 

(b) Is Math not “your
thing?” Then click on either of the
two photos on the right to take a visual trip through the Universe,
starting from inside an oak leaf, where the tiniest known things – quarks –
reside, all the way out to the outer limits of the Universe, moving each time
via “powers of 10.” This computer
simulation represents a remake of the Powers of Ten movie shown in
science classes in the 1970s. 

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