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What do you mean by the term “invariant quantities”.Show that kinetic energy of a body is not variant.

I need the answer as a solution of a long question.

2 years ago

Anything that anybody measures or observes is invariant, independent of the selected frame of reference, assuming one is selected. The arbitrarily assigned co-ordinates (X, Y, Z, T) in a frame of reference are variants, and, in general, are different in each frame of reference. Consider two inertial observers with clocks in relative motion. Each one can measure the speed of the other one with some kind of instrument and they will both get the same answer, just in opposite directions. They also can observe the time dilation of the other one's clock and this also will be symmetrical. And finally, they can measure the length contraction of the other observer and his clock and this also will be symmetrical. But now let's use Special Relativity to assign co-ordinates to the two observers. We have total freedom to do this in any way we want. So let's suppose we assign co-ordinates such that the first observer is at rest. Now his clock is not experiencing time dilation or length contraction but the other one is. Now his speed is zero but the other observer has all the speed. But even in this rest frame for the first observer, the second observer will still measure the first one to be traveling at the relative speed and he will still observe the clock of the first observer to be running slow and to be compressed. Now let's pick a different frame of reference where the two observers are traveling at a constant speed in opposite directions, somewhat more than half the relative speed. With a little care, we can pick this frame in such a way that both observers always read the same time on their clocks but both would be time dilated by the same amount. In this frame, we say that each observer has a speed that is smaller than what they measure of the other one's speed. But at the same time this frame can show us how they each measure the other one's speed to be the same as it was before. And we can show how each observer sees his clock as running normal but the other one running slow and how his dimensions are normal but the other one's are contracted. Another way of explaining it is that we are superobservers and can see the whole big picture in relation to a frame of reference that assigns absolute values to dimensions, distances, times, velocities and lots of other things. In general, these values will depend on the particular frame of reference selected. But the observers in the scenario have no awareness of the big picture that we can "see". They can only see things through their own limited eyes, limited by the speed of light. And it's what they see that every arbitrary reference frame must agree on, these are the invariants.

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