Invariant quantities

  1. I'm really confused about invariant quantities.Could someone explain which quantities are invariant in special relativity and how are they recognised?
  2. jcsd
  3. ghwellsjr

    ghwellsjr 5,122
    Science Advisor
    Gold Member

    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.
  4. thanks.good explanations.I got it.
    But it seems velocity is different.
    imagine In my frame of reference,a ball is moving with the velocity v.But at yours its u.what should we agree on?the one which is measured in the ball's frame and thats what?
    I'm wondering is there anything else?
    Well looks like if we just stick to a proper frame we won't have to deal with such problems.
    yeah I understood.
    thanks a lot
  5. ghwellsjr

    ghwellsjr 5,122
    Science Advisor
    Gold Member

    I'm not sure what you mean by "proper frame". Measurements made by observers are called "proper", like the proper time on an observer's clock. Times that are defined or analyzed by a frame of reference are called "coordinate", such as coordinate time.

    If you measure a ball to be moving at velocity v then it is a proper invariant measurement and every frame of reference can show that you will get the value v when you make that measurement. This, of course, includes a frame in which you are at rest. Your clock may be dilated and your rulers contracted in these other frames, but those two effects will result in you getting the same value for your measurement of the ball's velocity. In those other frames, the velocity of the ball will generally have a different value but it is called a coordinate velocity.

    Now when you look at the ball and ask what velocity it measures of itself, you will conclude from your frame of reference that it thinks it is 0, even though you assign it a velocity of v in your stationary frame of reference.

    By the way, when you are considering frames of reference, you should do your analysis of the entire scenario using just one frame of reference, it doesn't matter which one. Then if you want to use another one, you can transform all the events (space and time coordinates) from the first frame to a second frame, but you should never be concerned about the fact that the coordinate values that you get for the same parameter in two different frames have different values. These values have no meaning except in reference to the frame of reference. That's why it is called a frame of reference.
  6. jtbell

    Staff: Mentor

    I've added boldface to a word that I consider to be key in the following statement.

    To elaborate on this, suppose that you measure the velocity of the ball (in your reference frame) using a device that has a digital display of the measured velocity. If it reads "5.00 m/s," then everybody who looks at your device, regardless of whether they are moving relative to you or not, will agree that the device reads "5.00 m/s".
  7. We agree that you will measure a velocity v and that I will measure a velocity u and we agree that the ball measures its own velocity to be 0. So we each agree on the result of any given measurement. What we do not agree on is whether or not the thing that the other people measured represents the ball's velocity. Similarly with any frame variant quantity like position, length, duration, energy, momentum, etc.

    Frame invariant quantities mean that not only does everyone agree what the measurement is, but they also agree that it represents the same thing.
  8. ghwellsjr

    ghwellsjr 5,122
    Science Advisor
    Gold Member

    When you measure the velocity of a ball, you do not have to establish or reference any reference frame to make your measurements. You just measure it by any means at your disposal. If you or anybody else observes you making that measurement, they will agree that you have made the correct measurements using your instruments and made the correct calculation to arrive at your correct assessment of the velocity, even if they cannot see the values on your measuring devices. They can see what you will measure because they will see that your rulers are length contracted and they will see that your clocks are time dilated and they can independently verify that your measurements are correct. This, of course, assumes that the ball is not accelerating in which case the problem is in a different ballpark.
  9. This might have already been stated above, so if it has I apologize, however in special relativity the speed of light is invariant. It is invariant given that it does not change regardless of the observer or his or her motion.
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