Does SR = invariance of dτ

In summary: Perhaps if we have a more fundamental requirement like\int_{{\tau _0}}^\tau {f(\tau - {\tau _0})d\tau } = athis will require the length of τ-τ0 to be invariant wrt to coordinate changes in (t,x). For example, maybe {f(\tau - {\tau _0})} might be a probability distribution along a path so that its integral along the path must be 1 in any coordinate system.Yep, that's correct. If you know that \tau is a direct observable, you can see this just by considering how the integration variable transforms under a change of coordinates. The
  • #36
friend said:
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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  • #37
strangerep said:
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?
 
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<h2>1. What is SR and how does it relate to invariance of dτ?</h2><p>SR stands for Special Relativity, a theory developed by Albert Einstein to explain the relationship between space and time. Invariance of dτ refers to the concept that the spacetime interval, dτ, between two events is the same for all observers regardless of their relative motion.</p><h2>2. Why is invariance of dτ important in the theory of Special Relativity?</h2><p>Invariance of dτ is important because it is a fundamental principle of Special Relativity. It allows for the consistent measurement of time and space between events, regardless of the observer's frame of reference. Without this invariance, the laws of physics would be different for observers in different frames of reference, leading to inconsistencies and contradictions.</p><h2>3. How is the invariance of dτ demonstrated in experiments?</h2><p>The invariance of dτ has been demonstrated in numerous experiments, such as the Michelson-Morley experiment and the Kennedy-Thorndike experiment. These experiments use precise measurements of the speed of light and show that it remains constant for all observers, regardless of their relative motion. This supports the concept of invariance of dτ in Special Relativity.</p><h2>4. Can the invariance of dτ be violated?</h2><p>No, the invariance of dτ is a fundamental principle in Special Relativity and has been repeatedly confirmed by experiments. It is a foundational concept in our understanding of space and time and has not been found to be violated in any scenario.</p><h2>5. How does the invariance of dτ impact our understanding of the universe?</h2><p>The invariance of dτ, along with other principles of Special Relativity, has greatly impacted our understanding of the universe. It has led to the development of theories such as time dilation and length contraction, which have been confirmed through experiments and have revolutionized our understanding of space and time. It also plays a crucial role in modern physics, including the theories of general relativity and quantum mechanics.</p>

1. What is SR and how does it relate to invariance of dτ?

SR stands for Special Relativity, a theory developed by Albert Einstein to explain the relationship between space and time. Invariance of dτ refers to the concept that the spacetime interval, dτ, between two events is the same for all observers regardless of their relative motion.

2. Why is invariance of dτ important in the theory of Special Relativity?

Invariance of dτ is important because it is a fundamental principle of Special Relativity. It allows for the consistent measurement of time and space between events, regardless of the observer's frame of reference. Without this invariance, the laws of physics would be different for observers in different frames of reference, leading to inconsistencies and contradictions.

3. How is the invariance of dτ demonstrated in experiments?

The invariance of dτ has been demonstrated in numerous experiments, such as the Michelson-Morley experiment and the Kennedy-Thorndike experiment. These experiments use precise measurements of the speed of light and show that it remains constant for all observers, regardless of their relative motion. This supports the concept of invariance of dτ in Special Relativity.

4. Can the invariance of dτ be violated?

No, the invariance of dτ is a fundamental principle in Special Relativity and has been repeatedly confirmed by experiments. It is a foundational concept in our understanding of space and time and has not been found to be violated in any scenario.

5. How does the invariance of dτ impact our understanding of the universe?

The invariance of dτ, along with other principles of Special Relativity, has greatly impacted our understanding of the universe. It has led to the development of theories such as time dilation and length contraction, which have been confirmed through experiments and have revolutionized our understanding of space and time. It also plays a crucial role in modern physics, including the theories of general relativity and quantum mechanics.

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