Simple Derivation for Relativistic Mass
Short Description
A more straight way to understand relativistic mass...
Description
A Simple Derivation for Relativistic Mass Whenever you look for the reason why relativistic mass has to be the way it is (rest mass multiplied by the factor ), you will find just some direct √
affirmations, saying that this is due to momentum conservation or, sometimes, also a basic experiment about the perpendicular components of momentum during a simple collision between two particles. That is a very superficial explanation and not a demonstration, because the main important influences of relativistic effects are exactly in the parallel direction (direction of the relative movement). So, this way, it is like to consider a particular case and just from this, conclude the general case! If you try to go further, you may find equations involving energy and momentum transformations, and get expressions with radices of radices like this: ( )
√{
[(
)
(
( )(
)
)
(
( )(
)
) ]}
and so on… The problem has to be made simpler! The conservation of momentum is an inevitable Physical principle and must be sustained by relativity. The reason is that the momentum conservation (that is, momentum conservation in one system must imply momentum conservation in all systems in relative movement) keeps physical coherence among all the systems in relative movement. This means that, if a ball breaks a window in a system, it will also break the window in another system moving relativity to this one. It may break it in a slow motion way, but it will break it!
A Simple Derivation for Relativistic Mass This is the reason why relativity must preserve momentum conservation! The problem arises with the Lorentz Transformations of coordinates. As we have seen, the main equation, that synthetizes all this relativistic transformation, is given by the expression:
that is obtained by the thought experiment of the light ray reflected in a mirror located perpendicular to the moving observer. From this expression we could derive physically the Lorentz transformations for all coordinates ( where is the relative velocity between systems) given by: ( )( ( )(
) )
As always in relativity, the most important thing is “time”, the relation that connects the time from a system to the time of another system in movement. It’s important to bear in mind that these transformations also give the relation between coordinate’s differentials. Now let’s suppose a general collision of two particles in one system moving with speed with respect to “ ” (the conservation for more complex collisions and systems are just a compound of many two particle systems):
A Simple Derivation for Relativistic Mass ⃗⃗⃗
⃗⃗⃗
⃗⃗⃗
⃗⃗⃗
Using the differential expression for the speeds, we get: ⃗⃗
⃗⃗
If we applied now, as usual, the coordinate transformation for speeds, in order to get what system , moving with speed relatively to , measures for momentum, we would get expressions like those complicated equations with radices of radices…. Let’s instead consider just one of the perpendicular components of momentum ( ):
(2)
Having in mind the fundamental equation of relativity, that is:
√ It is easy to see that, for
: ( √
)
A Simple Derivation for Relativistic Mass We can see immediately that, if we multiply the mass of each particle by its respective factor, all the terms in the equation will result with the proper time of each particle, in each part of the collision, at the respective denominator. So, multiplying by the respective factor, we obtain: (
)
( (
) )
(
Now, applying the coordinate transformation
) , yields: (3)
But the “proper time” of each particle in each part of the collision is a common quantity (“invariant”) for all the systems in relative movement. This means two main things: First – The expression given in equation number (3) is valid in all the systems in relative movement. Second – The expression given in equation number (2) is not valid in all the systems in relative movement. From those conclusions, it is possible to conclude that: First – The classical definition of momentum (2) does not satisfy the necessity of invariance of the momentum conservation. Second – Once the coordinate has the same rule of transformation (perpendicular coordinate), the expression given by equation (3) is also valid in coordinate for “ ”, implying that the conservation in perpendicular momentum components are invariant under the condition of multiplying the masses by the respective factors.
A Simple Derivation for Relativistic Mass It´s just necessary now to verify that this condition (multiplying the masses by the respective factors) guarantees invariance of momentum conservation along it´s component (paralel direction) for “ ”: The component
of the momentum is given by:
Multiplying the masses by the respective factors and applying the Lorentz transformation for “ ” coordinate, we obtain: ( )(
)
( )(
( )(
)
)
( )(
)
Canceling the common term ( ) and bear in mind that: ( )
( ) We get: (
) (
(
( )
) (
( (
)
) )
) (
(
) (
( )
( )
) (
( (
)
)
) (
)
)
A Simple Derivation for Relativistic Mass In order to get for the conservation of momentum (in the same way we got it for ), that is, in order to have: (
)
(
)
(
)
(
)
the expression: (
)
(
)
(
)
(
)
(4)
must hold. And, indeed, this will be the same condition that the inverse transformation.
would get in the case of
Observation: This is the expression for the relativistic mass conservation, which is the condition for the invariance of the conservation of relativistic momentum (this condition is automatically satisfied by the classical momentum, that does not take into account the facto ( ) It is clear along those equations the main role of the proper time as common key for all the system regarding the measure of momentum conservation. Also is important to note the condition for the conservation of momentum, equation (4), where we have the relativistic conservation of mass, to guarantee the conservation of relativistic momentum. So relativistic momentum logically become
√ Where
and
A Simple Derivation for Relativistic Mass Therefore, in order to obtain the relativistic energy, we should consider in the respective equation ( ) ∫ ∫ ( ) in place of
(rest mass).
From this point, considering, as we have showed, the relativistic mass deduce the Einstein’s famous equation for relativistic energy: (
)
(
) (
)
√
(
(
) (
)
(
)
)
, we
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