SPECIAL THEORY OF RELATIVITY-CONSEQUENCES OF SPECIAL THEORY OF RELATIVITY-RESULTS OF THEORY OF RELATIVITY-TIME DILATION-MASS VARIATION-LENGTH CONTRATION-MASS ENERGY EQUATION

CONSEQUENCES OF SPECIAL THEORY OF RELATIVITY

We observe that in the development of special theory of relativity,frames of reference in relative motion with a constant speed V have been used. If the speed V becomes large enough to approach the velocity of light C, then the Galilean's transformations are found to be noticeably wrong. To correct the state of affairs it will be necessary to introduce a factor called 'Lorentz Factor' or 'Relativistic factor'.

Lorentz Factor is equal to:

This factor is in fact a measure of departure of Galilean's transformation. If

is much smaller than as it is in our common situations,then

is so small that the relativistic factor is essentially equal to unity. Under these conditions the classical and the relativistic physics predict nearly identical results. However when V approaches c (e.g.: V = C/5), Then the Galilean transformation will be incorrect.

Based on these considerations, if we interpret the result of special theory of relativity we end up in some very interesting consequences. Without going to make actual mathematical calculation, We may summarize the important consequences of the theory of special relativity which are as under:

MASS VARIATION

According to the special theory of relativity, the mass of an object in a frame of reference at rest is called its rest mass m_o. if this mass is measured by an observation moving with a constant speed V relative to the object, then it will not remain constant if the speed V is comparable to C. The mass m in the moving frame will very according to the mass variation given by:

This mass variation formula shows that mass changes with the velocity and not in general a constant nor the same for all observes but it is quantity that:
(a) depend upon the reference frame from which the body is being observed.
(b) is greater then or equal to the rest mass m_o when the body is at rest in the frame of reference from which the body is being observed.

LENGTH CONTRACTION

In the theory of special relativity it has been found that the measurement of length of a rod in a stationary frame of reference is not the same when the rod is measured by the observer in the moving frame of reference with the velocity relative to the rod, provided the measurement is made along the direction of motion.
Hence, if L_o is the length of rod in the frame at rest, and L is the length of same rod in the moving frame, then:

Since v/c is less then unity, the length L is less then L_o i.e. there is a contraction in length along the direction of motion. This is called the Lorentz-Fitzgerald contraction.
above equation tells us that an observer past whom a system is moving with a speed v measures object in the moving system to be shortened in length along the direction of motion by a factor:

It is important to note that only the dimension along the line of motion is changed and there is no change in the other two perpendicular directions.
With the development of special theory of relativity it became apparent that there is no physical contraction of the moving objects. There is, however, an apparent contraction of body for an observer where there is a relative motion of the object and the observer. In the natural sense the observer in moving frame can not detect the contraction because in this frame it does not exist; where is in the rest frame, it does exist, but the measuring rod in the moving system has shrunk too further we must note that for moderate velocities (v/c<<1)of the objects the contraction in length is negligible as observed in our every day observation.

TIME DILATION

Time is regarded as an absolute quantity in classical mechanics whereas in the special theory of relativity it is considered to be a relative entity based on the measurement of time in frame of references in relative motion.
The time interval between two events taking place at the same point in space as timed with a clock at rest with respect to that point is called the proper time interval and is denoted Dt_o=T_o.Time measured with a clock in motion with respect to the events is known as relativistic time it is represented by Dt=T. Both of the time intervals T_o & T refer to the time elapsed between the same pair of events occurring in the two frames moving with a relative speed v. then, according to special relativity the two times are related by the formula:

Above equation represents , what we call as the time dilation phenomena. According to the time dilation formula we mean that from the point of view of an observer at rest, the time of the observer in motion is dilated i.e. the clocks in moving frame run slowly and the Lorentz factor

Gives us the ratio of the rates of clocks for normal speeds, this factor is so close to unity (1.00) that we are quite unable to detect time dilation effect, but for speed comparable to the speed of light c the time dilation effect is quite significant.
We can now conclude that for every observer his own clock in his frame of reference run faster than do any other clocks which are moving relative to him. We may also note that every observer may consider himself to be at rest and consider all that moves as moving relative to him. This is actually an outcome of the principle of special relativity stated earlier: Every observer is equivalent to every other observer.

MASS ENERGY RELATION

In the begining of this section we have stated the postulates of relativity that the speed of light is a universal constant. We can not reach speeds greater than the speed of light by the relativistic addition of velocities. The equation is how to reconcile with this result of special relativity with Newton's second law, F=ma? It would be seen that any constant force, no matter how small, applied for a considerably very long time, should continuously accelerate any mass 'm' at a rate a=f/m until the speed was arbitrarily very large. Einstein, concluded that energy has inertia i.e. the more energy a body possess, the more inertia that body will display. Since, inertia is a property of matter, which is associated with mass. Thus from Einstein's argument mass is simply a property attributed to the total energy of the body and only the total energy is required, to know the total mass of the body.Thus, in special theory of relativity total energy and mass are related by the famous Einstein's equation.

E=mc²

From this relation between mass and energy it has been predicted that any process that changed the mass by a detectable amount would involve huge amounts of energy. For example, a mass change of 1.00 gram is equal to an energy change of 9 x 10¹³ joules.

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