1.5 Energy and Momentum Considerations (Special Relativity)
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This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
1.5 Energy and Momentum Considerations (Special Relativity)
Another consequence of relativity is a relationship between mass,
energy, and momentum. Note that velocity involves the question of how far
you go and how long it takes. Obviously, if relativity affects the way
observers view lengths and times relative to one another, one could expect
that any Newtonian concepts involving velocity might need to be re-thought.
For example, because of relativity we can no longer simply add velocities to
transform from one frame to another as we did with the ball and the train
earlier. (However, for small velocities like we see every day, the
differences which comes in because of relativity are much to small for us to
notice).
Further, consider momentum (which in Newtonian mechanics is defined as
mass times velocity). With relativity, this value is no longer conserved in
different reference frames when an interaction takes place. The quantity
that is conserved is relativistic momentum which is defined as
(Eq 1:6)
p = gamma * m * v
where gamma is defined in the previous section.
By further considering conservation of momentum and energy as viewed
from two frames of reference, one can find that the following equation must
be true for the total energy of an unbound particle:
(Eq 1:7)
E^2 = p^2 * c^2 + m^2 * c^4
Where E is energy, m is mass, and p is the relativistic momentum as defined
above.
Now, by manipulating the above equations, one can find another way to
express the total energy as
(Eq 1:8)
E = gamma * m * c^2
Notice that even when an object is at rest (gamma = 1) it still has an
energy of
(Eq 1:9)
E = m * c^2
Many of you have seen something like this stated in context with the theory
of relativity ("E equals m c squared"). It says that because of the
relationship between space and time for different observers as discovered by
special relativity, we must conclude that an object possesses an internal
energy contained in its mass--mass itself contains energy, or, to put it
more eloquently, mass is simply a convenient form of energy.
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