When we looked at Quantum Mechanics, we talked about its various interpretations. The reason we have such interpretations, I said, é que o QM lida com uma realidade à qual não temos acesso,,en,através de nossos aparatos sensoriais e perceptivos,,en,A Relatividade Especial trata de macro objetos em movimento,,en,e não temos nenhum problema em imaginar essas coisas,,en,Então, por que precisaríamos ter uma interpretação,,en,A resposta é sutil,,en,Albert Einstein,,de,relatividade especial,,en,Velocidade da luz,,en,A velocidade da luz sendo uma constante soa como uma declaração simples,,en,Mas há mais do que isso,,en,um pouco mais,,en,Vejamos o que essa constância realmente significa,,en,diz que se você está em algum lugar,,en,e há um raio de luz indo da sua direita para a esquerda,,en,tem uma velocidade,,en,c,,en,E outro raio de luz indo da esquerda para a direita também tem uma velocidade,,en,Tão longe,,en,tão bom,,en,Agora, digamos que você esteja em um foguete,,en,movendo da direita para a esquerda,,en,Teoria da Relatividade Especial,,en, through our sensory and perceptual apparatuses. On the other hand, Special Relativity is about macro objects in motion, and we have no problem imagining such things. So why would we need to have an interpretation? The answer is a subtle one.
What do we mean when we say we observe an object in motion? Let’s say we are at a position, say, . The object is at position at time , and at at . Although most people may not notice it, in the last sentence, I already embedded an unreasonable number of assumptions about space and time, which will become clear very soon.
Here are three possible views we could have when we make a statement about its speed.
- We could say that when the object is at , it emits a ray of light, which reaches us at at time . Similarly for and . Note that we are defining and (and assuming what and mean) with this statement. Then the velocity of the object becomes . We will call it the perceived velocity. It is how the object appears to move, from our perspective.
- Another thing we could say is this: Since we know that light takes a certain amount of time to reach us from and , the real time for the object is and . And then the real velocity would be . Let’s call this quantity its absolute velocity, or its velocity in the absolute reality. Note that we are again assuming what and mean, but handling the time at the object a bit better.
- A third way of looking at it is this: Let’s say we send out a pulse of light to the object at time . It gets reflected back to us at our time . So, at time , the object is at a distance . If the object is going directly away from us (this is important), we can define another distance at in exactly the same way. Then its speed would be .
Special Relativity takes the third view (or something very close to it). Based on this view, SR defines a coordinate transformation between two frames of reference. But having done that, it insists that the transformation is real, and not a perceptual effect. And, this is the subtle point, we still have to do computations of the kind described in the view number 2 above in order to understand motion. Since this post is meant for a non-technical crowd, I won’t belabor the point, but how it pans out mathematically is quite interesting.
Here is a real lecture on the co-ordinate transformation part of SR by Prof Susskind of Stanford. Although it looks formal and rigorous, it does have the same subtle issues. If you watch the derivation of the transformation from to carefully, from around the timestamp 47:05 in the video, you will see that he makes the statement and . This statement is true only on the axis and axis (or when and ) respectively, which he explicitly states, but then he silently goes on to assume it is true for all and . In other words, what is written is the same as and . It does not follow that . So the coordinate transformation is derived with insufficient mathematical or physical reasoning.
In the original paper, Einstein sweeps this difficulty under the rug by saying that it follows from the obvious isotropy we ascribe to space, despite the fact that space is not isotropic in SR coordinate transformation; it contracts only in the direction of motion. This is why I added the caveat that the third view of speed and motion applies only when the object is flying away from the observer — it is speed, a scalar, not velocity, the vector. I believe there is another way of deriving a different set of transformation rules without using a non-existent isotropy argument.
My belief is that one has to decouple the constancy of the speed of light (and the resultant covariant transformation of space-time as required by Maxwell’s equations) from the perceptual effects of light travel time. Such an approach would be equivalent to developing a framework based on second one of the views of speed and motion listed above. I did this in my book and my published papers. In fact, I went one step further, and investigated the nature of space, and what we mean by it — space is a cognitive model created in our brain and mind to represent the electrical signals generated by light falling on our retinas. Let me quote Einstein again.
Developing a new framework, however, is tantamount to redefining the whole of modern physics based on a different theory of relativity. It is a tall order.