Why does observers agree on the spacetime interval?

[center o bass]

Since two observers with coinciding origins at t=t'=0 both measures a light wave to be moving at c they will both claim that r^2= (ct)^2 and thus that r^2-(ct)^2 = 0. Thus

r'^2 - (ct)^2 = r^2 - (ct)^2 = 0

which is really just stating that 0=0 for events following the spherical wave off light. I just wondered what the argument now is that

r'^2 - (ct)^2 = r^2 - (ct)^2

for any spacetime point (event) and not just in the trivial case following the spherical wave of light?

 

[HallsofIvy]

Since two observers with coinciding origins at t=t'=0 both measures a light wave to be moving at c they will both claim that r^2= (ct)^2 and thus that r^2-(ct)^2 = 0. Thus

r'^2 - (ct)^2 = r^2 - (ct)^2 = 0

which is really just stating that 0=0 for events following the spherical wave off light. I just wondered what the argument now is that

r'^2 - (ct)^2 = r^2 - (ct)^2

That's NOT true. I presume you mean r'^2- (ct')^2= r^2- (ct)^2. The simplest way to show that is to replace r' and c' with their expressions in terms of r and c using the Lorentz transformations.

for any spacetime point (event) and not just in the trivial case following the spherical wave of light?

 

[center o bass]

That's NOT true. I presume you mean r'^2- (ct')^2= r^2- (ct)^2. The simplest way to show that is to replace r' and c' with their expressions in terms of r and c using the Lorentz transformations.

Yes I ment that r'^2- (ct')^2= r^2- (ct)^2. Was there something else I said that was not true? Yes I know you could always go the lorentz transformations, however one could also use the spacetime interval to derive the properties of the correct group of transformations. It all depends where you start, doesn't it? On the other hand, in deriving the lorentz transformation one also considers a beam of light (or a light flash) as the event and then generalizes to an arbitrary event without further explenation.

I would like to explore SR directly from the space time interval, but need an argument why the fundamental equation holds true for any event, not just following spherical waves of light.

 

[Dale]

Since two observers with coinciding origins at t=t'=0 both measures a light wave to be moving at c they will both claim that r^2= (ct)^2 and thus that r^2-(ct)^2 = 0. Thus

r'^2 - (ct)^2 = r^2 - (ct)^2 = 0

which is really just stating that 0=0 for events following the spherical wave off light. I just wondered what the argument now is that

r'^2 - (ct)^2 = r^2 - (ct)^2

for any spacetime point (event) and not just in the trivial case following the spherical wave of light?

I actually prefer to go the other way. I.e. start out by defining the spacetime interval and then use that to derive the group of transformations that preserves the spacetime interval. Then we give that group a name, the Poincare group, and note that there is a sub-group that also preserves the origin. Then we give that group a name, the Lorentz group.

 

[TSny]

Here is an outline of an argument that is sometimes given (e.g., Pauli's Theory of Relativity page 9)

1.) A freely moving particle should travel along a straight line at constant speed in any inertial frame. This requires the transformation equations for (x, y, z, t) from one frame to another to be linear.

2.) If the transformation equations are linear and if (Δr)²-(cΔt)² = 0 in any frame for events connected by a light signal, then (Δr')²-(cΔt')² = k(v) [(Δr)²-(cΔt)²] for any two events. Here k[v] is a quantity that can depend only on the relative velocity of the two frames.

3.) By considering a transformation from one frame to another followed by a transformation back to the original frame, you can argue that k[v]*k[-v] = 1.

4.) k[v] can be shown to correspond to the factor by which the length of a rod is changed if the rod is held transverse to the relative motion of the two frames. By reasons of symmetry, this factor can't depend on the sign of v. So, k[v] = k[-v].

5.) Arguments (3) and (4) require k[v] = 1. (k[v] = -1 cannot be correct since k[v] must approach 1 as v approaches 0.)

So, we arrive at the invariance of the spacetime interval for any two events.

 

[Muphrid]

There's another way to go about this process. You start off with a Minkowski vector space, and you enforce the notion that, regardless of how space is stretched or distorted, the quantities that are physically meaningful are those that are invariant with respect to the distortion. Anything else is dependent on coordinates or on the particular frame and cannot be meaningful because no one else would necessarily agree on them.

 

[center o bass]

Here is an outline of an argument that is sometimes given (e.g., Pauli's Theory of Relativity page 9)

1.) A freely moving particle should travel along a straight line at constant speed in any inertial frame. This requires the transformation equations for (x, y, z, t) from one frame to another to be linear.

2.) If the transformation equations are linear and if (Δr)²-(cΔt)² = 0 in any frame for events connected by a light signal, then (Δr')²-(cΔt')² = k(v) [(Δr)²-(cΔt)²] for any two events. Here k[v] is a quantity that can depend only on the relative velocity of the two frames.

3.) By considering a transformation from one frame to another followed by a transformation back to the original frame, you can argue that k[v]*k[-v] = 1.

4.) k[v] can be shown to correspond to the factor by which the length of a rod is changed if the rod is held transverse to the relative motion of the two frames. By reasons of symmetry, this factor can't depend on the sign of v. So, k[v] = k[-v].

5.) Arguments (3) and (4) require k[v] = 1. (k[v] = -1 cannot be correct since k[v] must approach 1 as v approaches 0.)

So, we arrive at the invariance of the spacetime interval for any two events.

This is the kind of argument I am looking for. However looking it up in Pauli's book;
http://books.google.no/books?id=7xr...ir_esc=y#v=onepage&q=pauli relativity&f=false
when arguing about the k[v] proportionality constant, he still makes an explicit reference to the lorentz transformations themselves and he does not explain why the _same_ k appears both in the transformations and the proportionality between the intervals.

 

[Muphrid]

That Lorentz transformations keep the spacetime interval constant is a mathematical consequence of the geometry. They're useful, but trying to derive that the spacetime interval is invariant under them gets things backwards. It is a physical statement that observers agree on the interval. LTs come from that, not vice versa.

 

[center o bass]

That Lorentz transformations keep the spacetime interval constant is a mathematical consequence of the geometry. They're useful, but trying to derive that the spacetime interval is invariant under them gets things backwards. It is a physical statement that observers agree on the interval. LTs come from that, not vice versa.

I agree with you that the invariance of the interval is the defining property of the LT's. That is the reason I want to start with this fact and then further explore the transformations. However if SR is based only on it's postulates then this fact should be derived from the postulates and the only thing I am able to derive is that (r')^2 - (ct')^2 = r^2 - (ct)^2 for a light ray.

 

[Muphrid]

Postulate: The spacetime interval is an invariant that all observers agree on.
Consequence: there are transformations of spacetime that convert between observers and obey this postulate. The Lorentz transformations are a subset of these.

Is the problem in verifying mathatically this is the case? Or is it in deriving the transforms from the invariance of the interval?

Or instead you don't consider invariance of the interval a postulate?

 

[center o bass]

Postulate: The spacetime interval is an invariant that all observers agree on.
Consequence: there are transformations of spacetime that convert between observers and obey this postulate. The Lorentz transformations are a subset of these.

Is the problem in verifying mathatically this is the case? Or is it in deriving the transforms from the invariance of the interval?

Or instead you don't consider invariance of the interval a postulate?

I do not consider the invariance of the interval a postulate. I'm trying to derive the invariance from the postulates of SR, namely

The postulate of relativity and the postulate of the constancy of the velocity of light.

 

[TSny]

This is the kind of argument I am looking for. However looking it up in Pauli's book...
when arguing about the k[v] proportionality constant, he still makes an explicit reference to the lorentz transformations themselves and he does not explain why the _same_ k appears both in the transformations and the proportionality between the intervals.

Pauli does not assume the Lorentz transformation. Note the comment at the bottom of page 9 where he states, "If one also bears in mind that any motion parallel to the x-axis must remain so after the transformation, formulae (1) of section 1 will be seen to follow immediately." So, these formulas are a consequence of the linearity of the transformations and the fact that the interval in the primed frame must be k times the interval in the unprimed frame. This also explains why the k is the same in the interval relation and in formulas (1).

 

[Muphrid]

Then probably the most direct path is to say that the principle of relativity only allows a few simple geometries, and the only one of those that admits an invariant speed is the Minkowski geometry. LTs and invariant intervals follow from that.

So I would try to prove that only a few geometries obey relativity in general first.

 

[Dale]

I do not consider the invariance of the interval a postulate. I'm trying to derive the invariance from the postulates of SR, namely

The postulate of relativity and the postulate of the constancy of the velocity of light.

Well, as far as I know "invariance of the interval" means "invariance of the interval under the Lorentz transform", so the question itself seems to assume the Lorentz transform. I.e. you cannot prove that something is invariant under a transform without deriving the transform.

So I would just derive the Lorentz transform from the postulates in the usual way, and then show that the interval is invariant under it.

 

[center o bass]

So, these formulas are a consequence of the linearity of the transformations and the fact that the interval in the primed frame must be k times the interval in the unprimed frame. This also explains why the k is the same in the interval relation and in formulas (1).

Do you have any suggestion how one could go about proving that this is the case?
Parallel before and after suggests y'=y, z'=z while linearity suggests x' = A*x - B*vt, but if i subsitute that back into the relation between the intervals there does not seem like much can be learned from it.

Well, as far as I know "invariance of the interval" means "invariance of the interval under the Lorentz transform", so the question itself seems to assume the Lorentz transform. I.e. you cannot prove that something is invariant under a transform without deriving the transform.

So I would just derive the Lorentz transform from the postulates in the usual way, and then show that the interval is invariant under it.

Since the postulates imply the transformations and the transformations imply the invariance I would think that one could get to the invariance more directly? It seems to be what Pauli is trying to do at page 9 in the book i posted above.
The reason I find it attractive is because you then can at once consider the more general transformations.

 

[center o bass]

I.e. you cannot prove that something is invariant under a transform without deriving the transform.

I found a satisfactory derivation in Landau and Lifhitz 'The classical theory of fields' which makes no reference to the transformations themselves.

 

[Dale]

I found a satisfactory derivation in Landau and Lifhitz 'The classical theory of fields' which makes no reference to the transformations themselves.

Then how can it derive the invariance? Invariant wrt what if not the transform?

 

[center o bass]

Then how can it derive the invariance? Invariant wrt what if not the transform?

Ofcourse it is invariant with respect to the group of transformations satisfying the invariance. But the invariance is just a quantity which different observers agree on and the transformations are derivable from that quantity. I guess a similar approach in euclidian space would be to derive the invariance of a distance by considering pythagoras and using that invariance to derive the properties of the rotation group.

 

[TSny]

I think what "center o bass" is saying is that L&L make no explicit use of the Lorentz transformation equations in deriving the invariance of the interval. You can read their derivation at http://archive.org/details/TheClassicalTheoryOfFields

The derivation is in section 2 starting on page 3. The Lorentz Transformation equations are derived later in section 4, page 9.

 

[Ich]

Thanks for the reference. L&L are brilliant. Of course they use all four postulates (1,2,homogeneity, isotropy), but not the transforms. I didn't know this derivation.

 

[TSny]

I was surprised to find that most of the L&L volumes are now available at archive.org.

They are definitely amazing pieces of work.