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In summary, the invariance of proper time leads to the Lorentz transformation, and the Lorentz transformation preserves the invariance of the proper time interval.f

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It is invariant by definition.

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The invariance of proper time intervals is a fundamental assumption, otherwise you would get conflicting physical outcomes. The Lorentz transformation follows from the SR postulates, and such basic consistency requirements.For example, does the invariance of proper time lead to the lorentz transformation, or vice versa?

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You can start from a lot of different places. There isn't a Right One, there's just the one you like.

Einstein started from the principle of relativity and the invariance of the speed of light. From that he derived the Lorentz transforms and the invariance of the interval.

Another way is to start from the principle of relativity and show that this is only consistent with Newtonian physics and relativity, and then simply observe that experimental evidence is consistent with relativity, not Newton.

A third way is to start from the invariant interval and show that the Lorentz transforms preserve it.

A fourth way is to assert the Lorentz transforms and show that the constancy of the speed of light follows from them (this is almost what happened - Lorentz deduced the transforms, but didn't realize they applied beyond electromagnetism).

When you have a self-consistent system, it doesn't really matter where you start your deduction. Since scientists are interested in the real world, we often start backwards - with physical measurements that we hope to explain as outputs of a physical theory. Then we teach the theory we eventually came up with, and show logically and mathematically that the theory predicts the measurements. And we make new predictions for experiments we haven't tried yet and go out and do those to test the theory.

Einstein started from the principle of relativity and the invariance of the speed of light. From that he derived the Lorentz transforms and the invariance of the interval.

Another way is to start from the principle of relativity and show that this is only consistent with Newtonian physics and relativity, and then simply observe that experimental evidence is consistent with relativity, not Newton.

A third way is to start from the invariant interval and show that the Lorentz transforms preserve it.

A fourth way is to assert the Lorentz transforms and show that the constancy of the speed of light follows from them (this is almost what happened - Lorentz deduced the transforms, but didn't realize they applied beyond electromagnetism).

When you have a self-consistent system, it doesn't really matter where you start your deduction. Since scientists are interested in the real world, we often start backwards - with physical measurements that we hope to explain as outputs of a physical theory. Then we teach the theory we eventually came up with, and show logically and mathematically that the theory predicts the measurements. And we make new predictions for experiments we haven't tried yet and go out and do those to test the theory.

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As @Ibix said, it is perfectly valid to go either way.

I don't get this. How can you derive LT from invarience of proper time? Proper time is invariant even in Newtonian physics or in any other imaginable physics that allows definition of proper time.

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This may or may not be what OP was after, but I thought the invariance of proper time holds underA third way is to start from the invariant interval and show that the Lorentz transforms preserve it.

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The quantity ##d\tau^2=dt^2-dx^2-dy^2-dz^2## (in units where c=1) is not invariant in Newtonian physics. You can derive the Lorentz transform as the transform that leaves this expression unchanged in form.How can you derive LT from invarience of proper time? Proper time is invariant even in Newtonian physics

I assume that the OP was referring to what I just wrote aboveThis may or may not be what OP was after, but I thought the invariance of proper time holds underallcoordinate transformations, not just Lorentz transformations.

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Right, but that definition of proper time assumes Minkowski spacetime. The general definitionI assume that the OP was referring to what I just wrote above

$$\tau=\int{\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}}$$

gives a much larger symmetry group under which ##\tau## is invariant, right? I’m just trying to clear this up in my head. To get the Lorentz group of transformations specifically, if you start with invariance of proper time, you also have to postulate flat spacetime. Is that right?

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Right, but that definition of proper time assumes Minkowski spacetime. The general definition

$$\tau=\int{\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}}$$

gives a much larger symmetry group under which ##\tau## is invariant, right? I’m just trying to clear this up in my head. To get the Lorentz group of transformations specifically, if you start with invariance of proper time, you also have to postulate flat spacetime. Is that right?

My understanding is that in flat space-time , the symmetry associated with the invariance of the Lorentz interval is the Poincare group, and the Lorentz group is a subgroup of the Poincare group. The Poincare group can be regarded as a kind of product (the semi-direct product) of the translational symmetries and the Lorentz symmetries. I'm not terribly familiar with semi-direct products, alas.

See for instance https://en.wikipedia.org/w/index.php?title=Poincaré_group&oldid=842331217

In the context of general geometries, I believe it's standard to say that "the geometry of space-time is locally Lorentzian". MTW says this, for instance. General space-times are not (AFAIK) regarded as being translation invariant, for instance the curvature can change from one point to the next. I don't have a reference on this point though.

A key word here is "locally". The Lorentz group isn't a global symmetry group in non-flat space-times, it's only a local symmetry group.

I'm not sure how to write all this in precise mathematics, which is worrying. It could be that there are some hidden mental errors here because of that lack of rigor. For instance, I'm not sure how to rigorously define "locally". But take it for what it's worth, and maybe someone else will have more comments.

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Which is the usual assumption for special relativity.that definition of proper time assumes Minkowski spacetime.

To get the Lorentz transforms you always have to assume flat spacetime, regardless of which direction you are going. It is for SR so spacetime is flat by definition.To get the Lorentz group of transformations specifically, if you start with invariance of proper time, you also have to postulate flat spacetime. Is that right?

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The general definition

$$

\tau=\int{\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}}

$$

gives a much larger symmetry group under which ##\tau## is invariant, right?

Yes, although calling it a "symmetry group" might be misleading. It's the group of diffeomorphisms (with a bunch of technicalities that I don't think we need to delve into here).

Note also that interpreting ##\tau## as a "proper time" assumes that the curve along which the integral is being done is everywhere timelike. Obviously not all curves will meet this requirement. But the arc length along a curve obtained by the integral you give will be invariant under diffeomorphisms regardless of the type of curve it is.

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I'm not sure how to write all this in precise mathematics

The rigorous way to do it is to define a flat tangent space at each point of the manifold, and write all local tensor equations in that tangent space. Then, to do integrals like the one @TeethWhitener wrote down, where you have to integrate along a curve, you need a connection, i.e., a way of mapping vectors/tensors in the tangent space at one point on the curve to vectors/tensors at other points. GR uses the Levi-Civita metric compatible connection for reasons which are well explained in textbooks like MTW (IIRC Wald goes into this in some detail as well).

I'm not sure how to rigorously define "locally"

Per the above, the rigorous definition is "in the tangent space at some chosen point". The Lorentz group is obviously a symmetry group in the tangent space at any point, since that tangent space is just flat Minkowski spacetime. (Note that, at least as I understand it, the Poincare group, which includes translations, is

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Well yes, but this expression leads to proper time only in minkowsky spacetime and for timelike separations. Proper time, to the best of my knowledge, is defined as time measured by the clocks in objects rest frame. This definition is as good in Newtonian as in minkowski spacetime, even though it is redundant concept in Newtonian physics and therefore we don't use it.The quantity ##d\tau^2=dt^2-dx^2-dy^2-dz^2## (in units where c=1) is not invariant in Newtonian physics. You can derive the Lorentz transform as the transform that leaves this expression unchanged in form.

Perhaps what you wrote is what OP meant, but if i was OP i would be utterly confused (which i was):)

The rigorous way to do it is to define a flat tangent space at each point of the manifold, and write all local tensor equations in that tangent space. Then, to do integrals like the one @TeethWhitener wrote down, where you have to integrate along a curve, you need a connection, i.e., a way of mapping vectors/tensors in the tangent space at one point on the curve to vectors/tensors at other points. GR uses the Levi-Civita metric compatible connection for reasons which are well explained in textbooks like MTW (IIRC Wald goes into this in some detail as well).

This seems that geometrical proper time defined by given equation doesn't even need spacetime metric, all you need is connection. Or am i reading it wrong? This interests me, since in Newtonian gravity that is exactly the case - we don't have spacetime metric, but we do have connection (accoding to MTW, if i remember correctly). Do you have some idea how would the integral looked in this case without metric? I am little at loss, because i don't see connection present in the integral. You do scalar product of tangent vector to the curve, which gives scalar. You do this at every point along the curve and you get function. Now you integrated this function. In no place i see connection comming in.

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I assumed that we were only considering global inertial frames on flat spacetime (i.e. plain-vanilla SR). If you want to consider more general cases (even transforms to non-inertial frames on flat spacetime), then yes, I believe you are correct.This may or may not be what OP was after, but I thought the invariance of proper time holds underallcoordinate transformations, not just Lorentz transformations. Wouldn't this be a much larger group than the Lorentz group (something like the general linear group)? You'd have to further assume flat spacetime to get the Lorentz group, right?.

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Yes, it just occurred to me that assuming flat spacetime by itself isn't even enough, because proper time is invariant in Rindler coordinates as well.I assumed that we were only considering global inertial frames on flat spacetime (i.e. plain-vanilla SR). If you want to consider more general cases (even transforms to non-inertial frames on flat spacetime), then yes, I believe you are correct.

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Spacetime is still flat in Rindler coordinates. It is a chart on Minkowski spacetime.Ok, thanks everyone.

Yes, it just occurred to me that assuming flat spacetime by itself isn't even enough, because proper time is invariant in Rindler coordinates as well.

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Not from proper time invariance alone, but from the 2 SR postulates plus some additional assumptions, one of which is that frames must agree on directly observable quantities like proper time intervals.How can you derive LT from invarience of proper time?

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Proper time, to the best of my knowledge, is defined as time measured by the clocks in objects rest frame.

Not quite. Proper time is time measured by a clock that is carried along with the observer or object. Or, to put it another way, it is arc length along a timelike worldline. It is independent of any choice of frames, and does not even require defining a frame.

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I don't get this. How can you derive LT from invarience of proper time? Proper time is invariant even in Newtonian physics or in any other imaginable physics that allows definition of proper time.

You need other things besides invariance of proper time. The reason this hasn't been addressed is likely because of the way the question was worded. If you want to know all of the things you'd need, in addition to the invariance of proper time, to derive the Lorentz transformation equations; or what's needed in addition to the Lorentz transformation equations to derive invariance of proper time, those are some good questions. They're just different questions from the one asked.

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Proper time, to the best of my knowledge, is defined as time measured by the clocks in objects rest frame.

It's defined as the time that elapses between two events that occur at the same location. So if you are always co-located with a particular clock, then that clock is measuring proper time. Being a relativistic invariant, proper time has the same value in all reference frames, so there is no need to refer to any particular reference frame when defining it

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I don't get this. How can you derive LT from invarience of proper time? Proper time is invariant even in Newtonian physics or in any other imaginable physics that allows definition of proper time.

I think people mean the invariance of the expression ##d\tau = \sqrt{dt^2 - \frac{1}{c^2} (dx^2 + dy^2 + dz^2)}##

The transformations that leave the right side invariant I think are combinations of:

- Rotations (for example, ##x \Rightarrow x cos(\theta) + y sin(\theta), y \Rightarrow y cos(\theta) - x sin(\theta)##)
- Translations (for example, ##x \Rightarrow x+a##)
- Lorentz boosts (for example, ##x \Rightarrow \gamma (x - vt), t \Rightarrow \gamma(t - \frac{vx}{c^2})##)
- Reflections (for example, ##x \Rightarrow -x##)

- Time reversal (##t \Rightarrow -t##)

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Fair enough. I assumed that the OP meant “spacetime interval” and answered accordingly. I certainly could have misinterpreted his question. I will wait for him to clarify.Proper time, to the best of my knowledge, is defined as time measured by the clocks in objects rest frame. This definition is as good in Newtonian as in minkowski spacetime, even though it is redundant concept in Newtonian physics and therefore we don't use it.

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The most easy way to derive Lorentz transformations is to use the invariance of space-time intervals. See my SRT FAQ article:

https://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

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We use tensor notation to write its components, just as we use tensor notation to write the componetns of Christoffel symbols. But neither is a tensor. The fact that the Lorentz transformation doesn't directly apply in Rindler coordinates is a practical illustration of the non-tensorial nature of the Lorentz transform.

Because the Lorentz transform isn't a tensor, one needs detailed assumptions about the coordinate system that's being used to know when it applies. Tensor expressions work in all coordinate systems, non-tensor expressions are not so general. There are various possible ways of specifying coordinates. Usually in physics, specifying a line element is regarded as the simplest way of communicating the choice of coordinates. See for instance Misner's "Precis of General Relativity" https://arxiv.org/abs/gr-qc/9508043

misner said:Equation (1) [[note: this is the aforementioned line elmenet]] defines not only the gravitational field that is assumed, but also the coordinate system in which it is presented. There is no other source of information about the coordinates apart from the expression for the met-ric. It is also not possible to define the coordinate system unambiguously in any way that does not require a unique expression for the metric. In most cases where the coordinates are chosen for computational convenience, the expression for the metric is the most efficient way to communicate clearly the choice of coordinates that is being made.

So from this point of view, when one choose one's coordinates in any space-time, one necessity define a line element that describes the geometry of that space-time by defining the invariant interval between nearby points in that space-time.

Note that in general, over large distances, the invariant interval between points along a geodesic connecting said two points will not necessarily be a quadratic form. Consider for instance the full expression for the distance between two points on a sphere for example. Furthermore, there won't necessarily be a unique goedesic between two points in a general space-time. However, if the points are close enough, there's always a unique geodesic between them, due to the existence of local convex neighborhoods in topology, and also some function that does give the interval along this unique geodesic. The low order series expansion of this function turns out to be a quadratic form in the differences between coordinates, which is the line element that defines the geometry.

When this line element is in the Minkowskii form ##dx^2 + dy^2 + dz^2 - c^2\,dt^2## the Lorentz transform applies. When it's not in that form, the Lorentz transform doesn't necessarilly apply.

Globally, when coordinates exist such that the line element is Minklwskii at all points, we have the flat space-time of special relativity, and the associated coordinates define by this line element (recall the previous remarks I cited from Misner on this point) define an inertial frames of reference.

While the Lorentz transformation is not a tensor, local Lorentz invariance is a coordiante-independent notion. If one considers a single point in space-time, there are multiple choices for an orthonormal set of basis vectors at that point. Smooth transformations between these choices of basis vectors exist, and these transformations will be members of the Lorentz group (which includes rotations as well a Lorentz boosts, and possibly reflections as well - I'd have to refresh my memory on the last point).

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