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## Does SR = invariance of dτ

 Quote by strangerep You've proved nothing that is not already contained in the definition of the Dirac delta itself. I can just as well write down: $$\int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} ~=~ 1 ~=~ \int_{ - \infty }^{ + \infty } {\delta (x - y_0)\,dx} ~=~ \int_{ - \infty }^{ + \infty } {\delta (y - y_0)\,dy}$$ where, in the last step, I simply renamed the dummy integration variable.
Of course you can always change the dummy variable in any equation and get the same answer. But then you're stuck having to interpret the new dummy variable as if it were the exact same thing as the old dummy variable.

I don't think you can change x0 to y0 without specifying a function y(x). So I don't think you can go from the integral on the left to the middle integral. But if you do, then it is no longer a dummy variable change; you're actually specifying a function from x to y.

What I think I've shown is when there is a relationship between x and y, namely, y=y(x), you get the same equation back again. Then we can interpret x and y as different coordinates. And x and y can not have any arbitrary relationship but must be a diffeomorphism. I think this means that the Dirac delta function is diffeomorphism invariant, right? So if the dirac delta is required for some reason (TBD), then that requirement also specifies the existence of some properties of the space on which it resides, namely, that it admitts a diffeomorphic coordinate patches, right?

 Quote by strangerep The "infinite continuous class of reference frames in spacetime" is simply a consequence of the fact that the group of coordinate transformations which preserve the equation of inertial motion, i.e., $$\frac{d^2 {\mathbf x}}{dt^2} ~=~ 0$$ turns out to be a Lie group. This does not have to be "accepted on faith". An unaccelerated observer enclosed in an opaque box cannot tell whether he is stationary or moving relative to some external point.
All this does is render a mathematical description of the observations. It does not explain it. What I'm trying to understand is WHY space would have the properties it does that makes necessary these kinds of observations.

Even the author of the paper is trying to derive the Lorentz transformations from properties of spacetime. He starts with the a priori existence of diffeomorphic coordinate transformations on patches of an underlying manifold (paraphrased). I'd like to go even further back and explain the need of for this property of spacetime to begin with.

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 Quote by friend I don't think you can change x0 to y0 without specifying a function y(x). So I don't think you can go from the integral on the left to the middle integral.
Yes I can. Both steps are valid, provided both ##x_0## and ##y_0## are somewhere between ##-\infty## and ##+\infty##.

Anyway...

Although I'm happy to help you understand the content of published papers, I'm not going to follow you down a rabbit hole into crackpot land.

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 Quote by strangerep Yes I can. Both steps are valid, provided both ##x_0## and ##y_0## are somewhere between ##-\infty## and ##+\infty##. Anyway... Although I'm happy to help you understand the content of published papers, I'm not going to follow you down a rabbit hole into crackpot land.
I certainly hope it's not a "rabit hole". I was trying to find the simplest construction that relates a metric to a field. Then maybe that could be used in the derivation of both the fields of QFT and curvature of GR. And it occurs to me that fields consist of individual values at each point in a space. And the minimum section of space is a infinitesimally small flat portion at the point of interest. A metric is inherently needed to do integration. So what integration process uses the least portion of space and picks out individual field values? That would be the Dirac delta function. Intuitively that seems to me like a good place to start when trying to relate space and fields in terms of their smallest constituents. One would think that the integration of the dirac delta being one is a true statement independent of any physical reason to use it. So if it does prove useful in deriving physics, then we will have succeeded in deriving physics from inherent truth.

As I recall, the integration of the dirac delta is suppose to be one no matter how small the integration interval, as long as the interval contains the zero of the dirac delta's argument. And if the interval of integration is not infinite but is small instead, then my objection to your comments holds.

But even with an infinite interval of integration, I wonder if there must be a diffeomorphism between the x and y coordinates. For it seems that the field in both coordinates needs to be sufficiently well behaved in order to do the integration. Does that mean we should be able to construct a smooth and invertible function between x and y, a diffeomorphism?