# Does invariance of proper time lead to the Lorentz transformation, or vice versa?

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## Main Question or Discussion Point

Hi, I've seen several explanations for sr on youtube. But they all start off explaining from a different perspective. I was wondering how the fundamental postulates of sr lead to the invariance of proper time between frames, and also what "order" everything is derived in. For example, does the invariance of proper time lead to the lorentz transformation, or vice versa?

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Proper time is time measured by the objects own clock, so it doesnt matter in what frame you are in. You are just checking his clocks, you dont care about yours.
It is invariant by definition.

A.T.
For example, does the invariance of proper time lead to the lorentz transformation, or vice versa?
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.

Ibix
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 realise 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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Dale
Mentor
does the invariance of proper time lead to the lorentz transformation, or vice versa?
As @Ibix said, it is perfectly valid to go either way.

From a strictly practical viewpoint, I would say vice versa, since you can easily demonstrate the invariant interval from the LT by a couple of lines of algebra. I think going the other way requires much more ingenuity than that.

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

TeethWhitener
Gold Member
A third way is to start from the invariant interval and show that the Lorentz transforms preserve it.
This may or may not be what OP was after, but I thought the invariance of proper time holds under all coordinate 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?.

Dale
Mentor
How can you derive LT from invarience of proper time? Proper time is invariant even in newtonian physics
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.

This may or may not be what OP was after, but I thought the invariance of proper time holds under all coordinate transformations, not just Lorentz transformations.
I assume that the OP was referring to what I just wrote above

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TeethWhitener
Gold Member
I assume that the OP was referring to what I just wrote above
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?

pervect
Staff Emeritus
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.

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

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?
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.

PeterDonis
Mentor
2019 Award
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.

PeterDonis
Mentor
2019 Award
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 not a symmetry group on the tangent space in the same sense, since the tangent space is a space of vectors, not points, and there is no obvious meaning to "translating" a vector in a vector space. Or, to put it another way, "translating" a vector would mean changing the point to which it is "attached", and there is no way to do that within a single tangent space, since that tangent space by definition is "attached" to a single point in the manifold--the spacetime. "Translating" would mean moving to some other point, and you can't do that within a single tangent space; you need a connection, as noted above.)

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.
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 dont use it.

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 doesnt 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 dont 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 dont 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.

Ibix
This may or may not be what OP was after, but I thought the invariance of proper time holds under all coordinate 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?.
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.

TeethWhitener
Gold Member
Ok, thanks everyone.
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.
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.

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.
Spacetime is still flat in Rindler coordinates. It is a chart on Minkowski spacetime.

A.T.
How can you derive LT from invarience of proper time?
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.

PeterDonis
Mentor
2019 Award
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.

Mister T
Gold Member
I dont 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.

Mister T
Gold Member
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

stevendaryl
Staff Emeritus
I dont 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:
1. Rotations (for example, $x \Rightarrow x cos(\theta) + y sin(\theta), y \Rightarrow y cos(\theta) - x sin(\theta)$)
2. Translations (for example, $x \Rightarrow x+a$)
3. Lorentz boosts (for example, $x \Rightarrow \gamma (x - vt), t \Rightarrow \gamma(t - \frac{vx}{c^2})$)
4. Reflections (for example, $x \Rightarrow -x$)
5. Time reversal ($t \Rightarrow -t$)

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Dale
Mentor
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 dont use it.
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.