Central Concepts of SR
Here are a few key points in Special Theory of Relativity that always cause confusion. Here I hope to shed some light on the solution to these “apparent” paradoxes.
1) “Observer A says that B’s clock goes slow, and observer B says that A’s clock goes slow. This is a logical contradiction. Thus, relativity should be abandoned”.
ØRelativity is designed, not only to describe the relation between the various parameters (t,x,y,z,etc.) in two reference frames, but mainly describe how events are perceived from different reference frames.
ØThus, without referencing an event, it is meaningless to talk about the relations between two frames. Based on these events, then it is possible to determine the frame with the ‘proper’ time. All other frames will measure a dilated time as compared to this ‘proper’ time, which will depend on the velocity of the respective frames with respect to the ‘proper’ time frame.
ØAll observers agree on the proper time.
ØThus, we notice that the wording of the problem is poor. Once the event under scrutiny is described, then all the apparent contradictions are resolved.
2)”Observer A says that B’s meter sticks are contracted along their direction of relative motion, and observer B says that A’s meter sticks are contracted. This is a logical contradiction. Thus, relativity should be abandoned.”
ØThis argument arises due to not specifying the metre stick which is being observed. When the metre stick in question is fixed (in the sense that its velocity is specified, thus allowing us to associate it with a frame in which the stick is at rest) the contradictions disappear and the two observers agree on the length of the stick.
ØThe length of the metre stick in the frame in which it is at rest (i.e one metre) will be the ‘proper’ length of the metre stick.
ØAll other observers in other reference frames will measure a contracted length, which happens due to lack of synchronization between the clocks in the moving frame with the clocks in the rest frame of the stick.
3)”Relativity does not even have a unique way to define space and time co-ordinates for the instantaneous position of an object. Lab and rocket observers (two frames used in the text) typically record different co-ordinates for this position and time. Thus, anything relativity says about the velocity of the object (and hence about its motion) is without meaning.”
ØRelativity is not a theory which “designs” the co-ordinates for the instantaneous space and time co-ordinates for one particular reference frame independent of other reference frames.
ØWhat the theory does is that once given the coordinates for two frames, it gives a way to plot the co-ordinates of one frame on the coordinates of the other frame, and vice versa.
ØThis is in tune with the purpose of the theory to explain how different observers moving with different relative velocities would observe an event.
4)”Relativity postulates that light travels with a standard speed regardless of the inertial frame from which its progress is measured. This postulate is certainly wrong. Anybody with common sense knows that travel at high speed in the direction of a receding light pulse will decrease the speed with which the pulse recedes. Hence, a flash of light cannot have the same speed for observers in relative motion. With this disproof of the basic postulate, all of relativity collapses.”
ØThis effect is very intuitive and something we experience in our daily life. We are so used to this fact that it is hard to accept that this is not the case when it comes to dealing with photons.
ØThe only way to substantiate the claim that the ‘law of addition of velocities’ (Galilean relativity) is false is to investigate the claims and validity of the new claim, that of the speed of light being a constant in all frames of reference.
ØOn doing that, it gives remarkably successful results and applies to all situations. In addition, it gives a law (‘law of combination of velocity’) which reduces down to the Galilean addition of velocities at small velocities compared to the speed of light (i.e. velocities we experience in our everyday lives).
ØHence, with the burden of truth, we have to accept the postulate of relativity that the speed of light is invariant.
5)”Relativity offers no way to describe an event without co-ordinates- and no way to speak about co-ordinates without referring to one or another particular reference frame. However, physical events have an existence independent of all choice of coordinates and all choice of reference frame. Hence relativity- with its coordinates and reference frames- cannot provide a valid description of these events.”
ØRelativity as a theory is not designed to give an explanation of the intrinsic workings of any event, i.e. it does not explain “why” an event occurs.
ØHowever, when recording where an event occurs, at what time it occurs, with that velocity it propagates in space, a coordinate system is mandatory, with which we can describe these values.
ØRelativity provides the connection between such coordinate systems moving at different relative speeds and how the parameters in the frames are inter-related with each other.
ØRelativity also never asserts that an event is dependent on which frame it is observed from. In fact, it considers events as independent from all reference frames and uses events to define coordinates.
6)”The mass of an object increases with increase in velocity “
ØThe important thing to understand first is that mass is an invariant quantity, just like the invariant interval (ds2=dt2-dx2-dy2-dz2). Likewise it is given by M2=E2-p2 (from this formula we get the famous E=mc2 for p=0).
ØThe fact that the mass of the particles does not increase is corroborated every time a particle accelerator is used to accelerate particles to near luminal velocities.
ØThis confusion arises from a lack of understanding the four-vector of momentum. The time component of this four vector, the energy is given by p0=E=γm . This expression gives the energy of the particle when viewed from a different reference frame. It is not an equation for the modified or “relativistic” mass.
ØAn object has a unique mass, which is its rest mass, which does not change with a change in velocity.