Transformation of Coordinat in Special Relativity

yurik · sentient bipedal physicist Joined: 20 Feb 2007 Posts: 4

I'm talking about SR only (not GR!). Why Lorentz Transformation (LT) is so important in SR? Everybody who tries to deny or improve SR begins with LT. While it is true that we can describe physical model in arbitrary coordinates (we can use arbitrary transformation of coordinates) but the physical predictions of the model will be the same. Well, in arbitrary coordinates x=1 will be not 1 meter and t=1 will be not 1 sec. But we have to know the algebra and the reasoning because not only physical predictions in arbitrary coordinates will be the same but also all the results of SR can be obtained inside one coordinate system (without permanent going between "resting and moving observer"). Why nobody tries this dimension of SR? (Yes, Einstein, for some reason, avoided this approach).

Eugene Shubert

yurik · sentient bipedal physicist Joined: 20 Feb 2007 Posts: 4

We can have everything that is known about flat spacetime without LT. We can use metric tensor instead - use the methods developed in GR.

I meant that if two coordinate systems connected by LT then the resting physical meter will have the similar description in both. Not so if they connected by arbitrary transformation.

My question is: how anybody can judge SR without knowing (using) both alternative approaches in SR? SR should be started with metric tensor - not with "derivation of LT".

Eugene Shubert · Posted: Wed Feb 21, 2007 11:59 am Post subject:

yurik,

I have no problem with anyone teaching SR, starting with the Minkowski metric. Can you recommend a paper that does that with elegance and clarity? If someone is going to study GR, then there are distinct advantages to learning SR from the Minkowski metric. Teaching both views is a good idea. Ultimately, I think that everyone should decide for himself or herself what is the best approach to SR for a first course in this subject.

yurik · sentient bipedal physicist Joined: 20 Feb 2007 Posts: 4

As somebody has noticed, Minkovski is a scoundrel and books that present SR starting with Minkovski (actually Lorentz) metric are zensord out. There is no choice for the beginners but to do it by themselves. And it is difficult. Myself I was able to do it only after I new GR machinery in full detail.

Eugene Shubert · Posted: Fri Feb 23, 2007 10:56 am Post subject:

yurik,

Why don't you write a paper explaining SR beginning with the Minkowski metric? If I like it, I will put it on the internet for you.

yurik · sentient bipedal physicist Joined: 20 Feb 2007 Posts: 4

Special Relativity Starting with Lorentz/Minkowski Metrics

By Yuri N. Keilman

The very important principal of classical theoretical physics is the uniqueness of classical representation principal, which works in concordance with A. Einstein’s general covariance postulate. In theoretical physics the physical reality is represented uniquely by the mathematical identities—numbers (we relate scalars, vectors, tensors to the category of numbers or identities) which we call physical values. The Physical Laws are expressed by the mathematical relations between the physical values. The Einstein’s general covariance postulate applies to the physical values as well as to the physical laws. It reads:

“We shall be true to the principle of relativity in its broadest sense if we give such a form to the laws that they are valid in every such 4-dimensional system of coordinates, that is, if the equations expressing the laws are co-variant with respect to arbitrary transformations.”

The term “co-variant” has a number of different meanings. Here it means that the physical values have to be represented by scalars, vectors, or tensors which have definite transformation properties with respect to arbitrary transformations. This postulate Einstein formulated in a view of GR but it applies to SR as well. Only we should add that SR's metric tensor in cartesian coordinates is: gtt=1, gxx=-1, gyy=-1, gzz=-1 (Minkowski space, c=1). All the other components of the metric tensor are zero. From here we can obtain all the SR's consequences just inside one coordinate system – we do not need SR's postulates at all. We do not need LT.

Example: Tweens Paradox. Suppose the resting tween starts at the event A(0,0) and end up at the event C(t=T,x=0). Another tween starts at A, goes to the event B(t=T/2,x=VT/2) and then returns to the event C. The proper length of a straight line A-B is (T/2)sqrt(1-V^2) (you have to know how to use the metric tensor). The proper length of the back trip B-C is the same. The proper length of the straight line A-C (done by the resting tween) is T. The result is obvious.

Another example: Relativistic Mechanics. Newton's mass multiplied on acceleration has problem: mass can be considered as a 4-scalar but acceleration is 3-d vector. It is not acceptable (contradicts to the above postulate). Actually it is a mathematical mistake since in mathematics there is no vectors (the concept of a vector is a pure mathematical concept) that can depend on a parameter (time). Physicists, using time as a parameter, went into a math blunder. In theoretical physics it is always recommended to use mathematics that is independently developed by a good mathematician.