Thursday, November 18, 2004

4-Scalar Equations

As with the equation dt/k*(x)=dt'/k*(x)' (or a 4-scalar field law), this will have to sustain a certain equation. Much like when working with light cones, we have to follow the general guid lines that the Light Cone dynamical laws bring. For example, the scalar law will only sustain the general coordinate transformation of x. But, much like to describe all coordinates, a tensor is required.
But, first let's describe all of the scalars and 4-Scalars locally. If we had an event E at a certain time t, then the scalar would be:


In this case, the only local coordinate would be (x). But, notice that the 4-scalar position is the only thing that is keeping it from just saying that E=t. This is logically consistant with the scalar law, for what would the transformation look like? Locally, it would look something like this:


Notice, that we revearsed the law by saying (k*(x))/dt other than dt/(k*(x)). Why is that? Well, first notice that t=t'-dt, and that dt=dt'-t. This provides symmetry. But, why we switched the law is because of a simple mathematical rule. Inthis case, if we used the format t(dt)/k*(x) would only provide that t is the dependant variable, while it should be (in this case) the independant. But, it is not what it seems. The 4-Scalar position is the key to this calculation. Since the 4-Scalar is dependent off the dt, that makes the scalar the dependant variable of this equation! So, now, we find that the argument presented above isn't valid in this case! So, t and the dt are independant of each other.

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