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Revisiting Relativity

Writer's picture: Mishkat BhattacharyaMishkat Bhattacharya

I am teaching Modern Physics this semester, a course I have taught three times before. The fun of re-teaching is many-sided: I get to have fresh insights into the material, the students push me with their questions [I always say students come to university to educate the professors; just don't ask me why they have to pay tuition to do that-:)], and I get to appreciate just how strange relativity and quantum mechanics are.


Below I present an example from each of these categories:


i) (Re-) Fresh (-ing) insight


The first thing we learn in special relativity are the Lorentz transforms, which famously link position with time through the speed of light. The transforms for time (dilation) or space (contraction) intervals can be derived by considering two frames moving with uniform velocities with respect to each other and by assuming the velocity of light is the same in both frames.


This is an exercise in geometry and algebra and takes several steps and some focus to execute. I wanted a more intuitive explanation of how the velocity of light joins space to time, an argument compact enough that I could carry it around 'in my pocket'.


Here's one that I thought of, and maybe it is well known to some people, but I will share it nonetheless. Both in Newtonian and Einsteinian mechanics, velocity is given by the time derivative of position:


v = dx/dt


Now if space and time are allowed to vary independently, then dx and dt can take any values. Sure, dx and dt are small quantities from the point of view of calculus. But we can choose dt so much smaller than dx that v turns out to be even greater than the speed of light.


But if we enforce the requirement that v cannot exceed the speed of light (c), then it follows immediately that dx cannot vary independently of dt. In other words, space and time are linked to each other through the constancy of the speed of light.


ii) One-way speed of light


I was not even aware that this was an issue until an undergraduate in my class pointed it out to me a week ago. There are contentions in the literature saying that measuring the one-way speed of light is dependent on the synchronization convention between the clocks at the start and finish points.


The way to get around the synchronization problem is to use a mirror and reflect the light back to the clock at the starting point. This clock is obviously synchronized with itself. However, this configuration opens up the possibility of different velocities for light in different directions: what if the velocity was c/2 to the mirror and instantaneous back to the clock after reflection? What we would be measuring in that case was the average velocity of light.


There is a YouTube video saying the one-way measurement cannot be done, as well as a Wikipedia article. However, a little digging revealed a cool classroom experiment that measured the one-way speed of light to be within 0.4% accuracy of the accepted value of c.


The trick is to modulate the light intensity and to keep track of the relative phase between the modulator at the starting point and the detector at the end point. This relative phase changes as the path length for the light travel is varied. The time can be calculated from the relation


phase = modulation frequency x time


It is an elegant experiment which does not require absolute path lengths (x) or times (t) to be measured since only the slope of the graph is relevant: c = dx/dt!


iii) How strange: mass is a form of energy


In Newtonian mechanics, mass is conserved. In class I remind the students about this by considering two apples, then transforming them (on the white board) into apple pie, then into applesauce, and finally into apple juice. In all cases the mass is 2m_a where m_a is the mass of one apple. (Of course, we are working in the approximation where ingredients put in and fluids drained out are being ignored, for the picky people, this is just another 'I can carry it in my pocket' argument).


In relativity mass is a form of energy that can be converted into other forms of energy (just as kinetic energy can be converted to potential energy in classical mechanics). For example a neutron and a proton have some of their rest mass converted into the potential energy of binding when they combine into a deuteron.


So in relativity mass is not conserved, but total relativistic energy is. In case some of you are looking for a 'fruit'ful analogy for relativistic fundamental particles, you can try the fruit quark here -:).










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