# Physical meaning of null spacetime interval

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• Jakz
Jakz
TL;DR Summary
Physical meaning of null spacetime interval
Hi all,

I've been trying to wrap my mind around the physical meaning of null spacetime intervals. I understand the metric of Minkowski geometry. And I understand that photons have no proper time and no reference frame: the spacetime interval between the emission and absorption of a photon is zero. They are the same event.

My question is: What does this mean? Is the fabric of spacetime warped to infinity by every photon passing through it? If I look up at Polaris at night, from my reference frame it appears to be 433 light years away in space, and 433 years away in time. But in spacetime, the emission of that photon and its absorption by my eye are the same event. Is this accommodated by a weird spacetime topology in which the worldlines of massive particles that are light-years apart in space intersect in spacetime for the exchange of energy? With "light" just being the way the interaction looks in our 3-dimensional perspective? Or am I off-track here? Thanks for any pointers you all can provide.

Jakz said:
the spacetime interval between the emission and absorption of a photon is zero. They are the same event.
The spacetime interval is zero, but they are not the same event. The spacetime interval is degenerate in that sense.

Along a null path, even though the proper time is not defined, you can still use an affine parameter to distinguish the different events.

cianfa72, David Lewis, vanhees71 and 4 others
Jakz said:
the spacetime interval between the emission and absorption of a photon is zero. They are the same event.
This is your Euclidean-based intuition talking, and it's wrong in Minkowski geometry. In Euclidean geometry, two points with zero distance between them must be actually the same point; not so with "distance" in Minkowski geometry. The emission and reception of a light pulse are not the same event despite their null separation (free advice: forget you ever heard the word photon until you actually study QFT and you'll have a lot fewer misconceptions to unlearn).

The obvious test of this is to ask somebody to stand ten meters away from you and throw a punch toward you. You see the punch thrown so your eye is null separated from the punch. Did it hurt? If not, your retina and the other guy's fist can't be at the same event.

The topology of Minkowski spacetime is the same as Euclidean space. It just has a strange "distance" measure that means some distinct events have zero distance between them. Bonus strangeness: you can connect any pair of events using at most two null curves. So it's always possible to find a zero "length" path between two events.

cianfa72, David Lewis, vanhees71 and 3 others
In a light-clock, the spacetime paths of the light-signals reflecting off mirrors are all null (lightlike)… but those reflection events are distinct. That’s why we can use this mechanism as a clock.

dextercioby, vanhees71, Peter Strohmayer and 2 others
Ibix said:
Bonus strangeness: you can connect any pair of events using at most two null curves. So it's always possible to find a zero "length" path between two events.
In the (1+1)-Minkowski case, this is essentially expressing a spacetime displacement vector in light-cone coordinates.

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vanhees71, Dale and Ibix
Peter Strohmayer said:
Space-time distance is a (constant) difference between time and space squared.
This is only true in flat spacetime. In curved spacetime the formula is more complicated and depends on the specific curved spacetime geometry.

vanhees71
Also worth noting that the spacetime "distance" is invariant rather than constant. Invariant means that all frames agree on the value, while constant means that the value doesn't change over time, or as you vary some physical condition. The latter doesn't really make sense for interval between events.

David Lewis, vanhees71 and Peter Strohmayer
Peter Strohmayer said:
Space-time distance is a (constant) difference between time and space squared.
For consistency, "space-time distance" should be "space-time squared-distance", akin to "squared-interval".
For clarity, the definition should read "difference between time-squared and space-squared".

PeterDonis
Ibix said:
This is your Euclidean-based intuition talking, and it's wrong in Minkowski geometry. In Euclidean geometry, two points with zero distance between them must be actually the same point; not so with "distance" in Minkowski geometry. The emission and reception of a light pulse are not the same event despite their null separation (free advice: forget you ever heard the word photon until you actually study QFT and you'll have a lot fewer misconceptions to unlearn).

The obvious test of this is to ask somebody to stand ten meters away from you and throw a punch toward you. You see the punch thrown so your eye is null separated from the punch. Did it hurt? If not, your retina and the other guy's fist can't be at the same event.

The topology of Minkowski spacetime is the same as Euclidean space. It just has a strange "distance" measure that means some distinct events have zero distance between them. Bonus strangeness: you can connect any pair of events using at most two null curves. So it's always possible to find a zero "length" path between two events.

Thanks. It's actually due to that last point that I was wondering about the physical significance of null intervals. Can information be transmitted at the intersection of null cones? I've seen this put forward as a potential explanation for entanglement: the particles are connected along the null surfaces, it's just our retrieval of this information that inevitably involves lags in space and time. Might point to holography too? I definitely have a lot more to learn, just thought it was an interesting idea.

Jakz said:
Can information be transmitted at the intersection of null cones?
Classically, information can certainly be transmitted by electromagnetic radiation, so that would basically be "yes". However:

Jakz said:
I've seen this put forward as a potential explanation for entanglement
This is a whole separate question, discussion of which belongs in the QM interpretations subforum (since what you are referring to is the transactional interpretation of QM). Classical electromagnetism can't answer this question.

vanhees71
Jakz said:
I was wondering about the physical significance of null intervals
Null intervals are physically significant because they identify light cones. Or other massless particle worldlines. The invariance of null intervals is the second postulate.

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Jakz said:
I was wondering about the physical significance of null intervals.
Null intervals are causal boundaries: they are the spacetime geometry implementation, so to speak, of the physical law that "nothing can travel faster than light".

vanhees71
Maybe it's that vivid:

Historically, we have defined the units "metre" and "second" and measured the "speed of light" with them. Physically it is the other way round: we define the duration of a freely chosen constant physical process as a "second" and then determine how long a "metre" is by the propagation of a pulse of light in a given time.

It is in the pulse of light that the fusion of space and time takes place. A light pulse always covers in space what it needs in time.

Between the two events of the emission A and the arrival B of a light pulse, the temporal distance is always equal to the spatial distance ("light-like distance"). In the distance, space and time coincide, their geometric difference, the space-time interval, is zero.

If the first light pulse is immediately followed by a second one in an other direction, the event of its arrival C is always farther away from A in time than in space ("time-like distance"). Time exceeds space, the space-time interval is positive.

When two light pulses are emitted from some event, the arrival event D of one light pulse is always farther away from the arrival event E of the other light pulse in space than in time ("space-like distance"). Space exceeds time, the space-time interval is negative.

A is the cause of B and C. D cannot be the cause of E and E cannot be the cause of D. This causal relations or space-time intervals between two events are invariant in different reference frames.

Dale
PeterDonis said:
Null intervals are causal boundaries: they are the spacetime geometry implementation, so to speak, of the physical law that "nothing can travel faster than light".
I wonder if we should phrase that the other way around - "nothing can travel faster than light" is an imprecise way (or, perhaps better said, a flat spacetime special case way) of saying "causal influences cannot propagate outside future lightcones".

I appreciate that is probably an "A" level view more than "I", but it's in a similar vein to "geometry first" approaches to SR. It shortcuts the inevitable arguments about things like high redshift galaxies that are receding faster than light. It's fine that they are (in a loose sense) travelling faster than light because they aren't moving outside their future lightcones and "nothing travels faster than light" is a flat spacetime only rule.

Jakz, Dale and vanhees71
Keeping the focus on Minkowski spacetime, let's further restrict to the (1+1)-Minkowski case.

From my post last year
( https://www.physicsforums.com/threa...cle-go-through-spacetime.1048187/post-6832310 )

A way to visualize the square-interval is to draw the "causal diamond"
joining the endpoints of the spacetime-displacement vector.
For a timelike displacement, it's the intersection of the "future cone (and its interior) of the past event" with the "past cone (and its interior) of the future event".
Its area is equal to the square-interval $\Delta t^2-\Delta(x/c)^2$
In the figure below, the area is 64.
So, the magnitude of the diagonal of the diamond equals 8.

The intersection of the
"future cone (and its interior) of the past event (call it event O)" with the
"past cone (and its interior) of the future event (call it event Q)"
is the set of events that "can be signaled by O, then signal Q".
So, the squared-interval of timelike-OQ is a measure of the signed-area in spacetime of such events that can be signaled by O, then signal Q.
In the null case, this area is zero... (but this doesn't mean there are no such intermediate events... just that the area is zero).
Why? (It's based on light-cone coordinates from the eigenvectors of the boost in (1+1)-Minkowski.)
The lightlike sides have sizes
$\Delta u=\Delta t+\Delta(x/c)$
$\Delta v=\Delta t-\Delta(x/c)$
The area is $\Delta u\Delta v=\Delta t^2-\Delta(x/c)^2.$
Since the boost has determinant 1, the signed-area is invariant under boosts.

From https://www.desmos.com/calculator/4jg0ipstya

You can interact with it.
Move the events A and Z.

Since the signed-area is $\Delta u\Delta v=\Delta t^2-\Delta(x/c)^2.$,
• when area > 0, then AZ is a timelike vector
• when area <0. then AZ is a spacelike vector
• when area =0, the AZ is a lightlike vector
https://www.physicsforums.com/insights/relativity-rotated-graph-paper/

You could try to generalize this to higher dimensions..
but then you'll have a more complicated expression involving the dimensionality of spacetime (in a coefficient and in a power of the spacetime-volume)

In the (2+1)-case for a timelike displacement, you'd have this 3-D volume between two cones...
but then you'd have to raise that to the 2/3-power and multiply by the appropriate coefficient to get the squared-interval.

https://www.geogebra.org/m/pr63mk3j

For a spacelike displacement in higher dimensions, one probably has to work harder to come up with an appropriate construction.

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Peter Strohmayer, Dale and vanhees71
All this confusion could be avoided, if the physicists (including myself) were a bit more careful in inventing the jargon. The fundamental form of Minkowski space is a non-degenerate bilinear form, but it's not positive definite, and that's why it, in distinction to the positive definite scalar product of a Euclidean space, does NOT induce a metric or a distance measure. Unfortunately many physicists call it a "distance" and a "metric" also it's just a fundamental form with signature (1,3) (or (3,1), depending on the convention used).

Dale
Ibix said:
I wonder if we should phrase that the other way around - "nothing can travel faster than light" is an imprecise way (or, perhaps better said, a flat spacetime special case way) of saying "causal influences cannot propagate outside future lightcones".
Yes, in many textbooks this is indeed how it is phrased. The light cone statement is more precise because it refers to a definite mathematical object (and it also has the advantage of generalizing to curved spacetimes, where concepts like "speed" become problematic).

Dale and vanhees71
Thanks everyone for the responses.
vanhees71 said:
All this confusion could be avoided, if the physicists (including myself) were a bit more careful in inventing the jargon. The fundamental form of Minkowski space is a non-degenerate bilinear form, but it's not positive definite, and that's why it, in distinction to the positive definite scalar product of a Euclidean space, does NOT induce a metric or a distance measure. Unfortunately many physicists call it a "distance" and a "metric" also it's just a fundamental form with signature (1,3) (or (3,1), depending on the convention used).
I feel I'm on the cusp of understanding this, but not quite there. Distance is exactly what I don't get about Minkowski space. What is it, if it collapses to zero for light? And it's not just at the limit that this matters: distance goes towards zero as massive objects approach the speed of light. For example a spaceship can go from the Milky Way to Andromeda in (proper time) seconds if energy and acceleration are disregarded (suppose an alien spaceship is just doing a flyby).

Can you expound on the physical meaning of "not positive definite"?

Jakz said:
What is it, if it collapses to zero for light?
A useful concept, just like it is in Euclidean space. It's something that everyone will agree on, whatever coordinate system they choose. And it has physical significance - for spacelike intervals it's the Euclidean distance a person would measure and for timelike ones it's the elapsed time along the path.

Again, your Euclidean intuition of how you think a distance "ought to behave" is misleading you. Why shouldn't its square be zero or negative?

@vanhees71's argument is, I think, that we should reserve "distance" for the concept in Euclidean space (and locally-Euclidean curved spaces in general) and use something else ("interval" is the correct term) for the equivalent concept in Minkowski spacetime (and locally-Minkowski curved spacetimes in general) because using the same term confuses students because they import Euclidean intuitions of how distance ought to behave, and it just doesn't work that way. And they confuse the interval along a path with the Euclidean distance along its spatial projection (which I suspect is what's getting you in your Andromeda example). My observation is that he's correct in your case.

So if it is not helpful to you to think of the interval as a distance, don't think of it that way. Euclidean distance and Minkowski interval are analogous concepts, not exactly identical ones. I think most of us do find it a helpful analogy, but not everyone is alike in their thinking.

jbriggs444, Jakz, Mister T and 3 others
Jakz said:
Distance is exactly what I don't get about Minkowski space. What is it, if it collapses to zero for light?
It's the interval, not the distance. The interval is analogous to the distance in Euclidean space. When the interval is zero you are looking at the margin between timelike intervals and spacelike intervals. When an interval between two events is timelike it means you can be at the two locations by moving from the first event to the second event. If the interval is lightlike (has a value of zero) it means only a beam of light can pull that off. If the interval is spacelike, nothing can.

Jakz said:
Thanks everyone for the responses.

I feel I'm on the cusp of understanding this, but not quite there. Distance is exactly what I don't get about Minkowski space. What is it, if it collapses to zero for light? And it's not just at the limit that this matters: distance goes towards zero as massive objects approach the speed of light. For example a spaceship can go from the Milky Way to Andromeda in (proper time) seconds if energy and acceleration are disregarded (suppose an alien spaceship is just doing a flyby).

Can you expound on the physical meaning of "not positive definite"?
In Euclidean space, when we say "##A=(x,y,z)## is a distance ##s## away from the origin", what we mean is that if you draw a sphere with radius ##s## centered on the origin, that sphere includes ##A##. The sphere includes a lot of other points that are not ##A## but all of them are a distance ##s## away from the origin. The equation for the sphere is ##s^2=x^2+y^2+z^2##.

In Minkowski spacetime, when we say "##A=(c t,x,y,z)## is a spacetime interval squared ##s^2## away from the origin", what we mean is that if you draw a right hyperboloid with semimajor axis ##s^2## centered on the origin, that hyperboloid includes ##A##. The hyperboloid includes a lot of other events that are not ##A## but all of them are an interval squared ##s^2## away from the origin. The equation for the hyperboloid is ##s^2=-c^2 t^2 + x^2 + y^2 + z^2##.

Notice that unlike the Euclidean case, the Minkowski case can have ##s=0## for events other than the origin. This set of events is called a light cone. Also it is possible for ##s^2<0## in which case the hyperboloid is a hyperboloid of two sheets and physically represents the time on a clock or ##s^2>0## in which case the hyperboloid is a hyperboloid of one sheet and physically represents the length on a ruler.

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Jakz, vanhees71 and PeterDonis
I will elaborate on what others, esp. Ibix, have said ("distance" is here used as an analogy, not an identity, so do not simply automatically rely on your Euclidean understanding of the word), but take my ideas with a grain of salt, as I am just a student of special relativity.

Whenever you say that there is an analogy between two structures, it means that you can use three languages: the two specific to each situation and a general one embracing the other two.

In this case, for me it is illuminating that the general language is "solving a problem".

How does this translate into the specific language of each situation?

In the ordinary physical space, which is Euclidean, the problem is for example how much you must walk to get from point A to point B. So the solution is a distance, that is to say, something that is a challenge, a difficulty to be overcome.

In Minkowskian space, as I think that PeterDonis also pointed out, the problem is related to causality, i.e. whether you are still able to send a signal from event A which can cause or prevent event B. Here you may see a distance in the geometric representation of the problem, but don't take that literally, since such representation is only a metaphor. Call it better an "interval", as vanhees71 dictated, so as to avoid the Euclidean intuition. But what is necessary for you to gain a better intuition is that you realize that the interval is still the number of units that solve the problem, but it is not anymore a challenge o a difficulty to be overcome. On the contrary, the bigger the Minkowskian interval is, the easier it will be that there is causal influence between events A and B. If A and B are timelike events, it means that a massive object can witness both events and the bigger the proper time (i.e. ticks of its wristwatch) that it experiences, the easier that the causal influence is, since that means that you could also send a faster messenger (experiencing less proper time) which would be arriving arriving well in advance. If instead only a light signal can witness both events, it means that you are, in PerterDonis' words, at the boundary of causal influence. If finally the events are spacelike, you runt into impossibility of any influence.

To sum up, whenever you make an analogy, understand that there is a generic spirit that is shared by both poles, but that this may translate into each specific case in a different manner, "mutatis mutandi" (changing what must be changed). Thus in the Euclidean case the interval is a "difficulty" and the smaller it is, the easier it is for you to achieve your objective, i.e. solve your (translational) problem. In the Minkowskian case, the interval is an "assistance" and the smaller it is, the tougher it is for you to solve your (causal) problem. Hence null interval means, in the Euclidean context, the limit where your difficulty and your problem vanish, whereas in the Minkowskian context, it means the limit after which, for lack of assistance, your problem becomes unsolvable.

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vanhees71, Jakz, Dale and 1 other person
Saw said:
On the contrary, the bigger the Minkowskian interval is, the easier it will be that there is causal influence between events A and B.
This is not correct. First, causes can travel over null intervals, which have zero Minkowski "length". Second, spacelike intervals are nonzero, but causes cannot travel over them.

vanhees71
Picking up one loose end:

Jakz said:
Is the fabric of spacetime warped to infinity by every photon passing through it?
No. The "fabric of spacetime" is just not Euclidean, so a zero interval between two events does not mean they are at the same point in spacetime.

vanhees71
After some cleanup the thread is reopened

Thanks Dale, Saw, and Ibix for your clarifying comments. I definitely was being sloppy in letting myself think of "interval" as a distance. But, if I may, two follow-up questions:

1. I've always understood "invariant" to mean something like "that which really exists". That is, invariant objects/networks exist in a 4D block universe; and space and time are just coordinate systems used to describe any one worldline from the perspective of another. In a Euclidian 3D space, a stick can be rotated to any orientation, and from some perspectives it will appear to be shorter than it actually is. But there is an invariant length that corresponds to something real. And it seems the same is true in spacetime. That's why I find the null interval puzzling. Yes, the two points have different coordinates, but isn't their separation a matter of our perspective as massive entities? And it's not as if the null interval is just an odd fact of geometry: there's something real (a transfer of energy) that happens along it. Why, in this case, do we treat the space and time intervals as "real", and the invariant interval as inconsequential?

2. In continued reading on the subject, I found the description below that comes closer to describing to what happens along the null interval. Is this an accurate description? (note: no intention to go off topic into quantum mechanics here, just wondering if this is a reasonable description of the physical energy exchange).

"The physical significance of a null spacetime interval is that the quantum state of any system is constant along that interval. In a sense, the interval represents a single quantum state of the system, so (for example) the emission and absorption of a photon can be regarded as a single quantum act." https://mathpages.com/rr/s2-01/2-01.htm

"A completely free massless particle – if such a thing existed – might just be represented by a monochromatic plane wave, but a real photon is necessarily emitted and absorbed as a directed quantum of action, so it corresponds to a bounded null interval in spacetime. (Note that the quantum phase of a photon does not advance while in transit between its emission and absorption, unlike massive particles. The oscillatory nature of electromagnetic waves arises from the advancing phase of the source, rather than from any phase activity of a photon “in flight”.)" https://mathpages.com/rr/s9-10/9-10.htm

(btw, an interesting historical note in the second article: "It’s interesting that some of the ideas of Feynman’s talk that day, such as the need for a complete absorber in the future, were already familiar to Einstein. After the talk was over, Pauli raised some objection to the theory, and asked if Einstein agreed. “No”, Einstein replied, “I find only that it would be very difficult to make a corresponding theory for gravitational interaction”.")

Jakz said:
Yes, the two points have different coordinates, but isn't their separation a matter of our perspective as massive entities?
Their spatial separation is a matter of perspective (or coordinate choice, more precisely). The interval is not.
Jakz said:
Why, in this case, do we treat the space and time intervals as "real", and the invariant interval as inconsequential?
We don't. The invariant interval is utterly fundamental to all of relativistic physics, and the spatial and temporal distances are merely a convenience for calculation and comparison to measurement.

Jakz said:
1. I've always understood "invariant" to mean something like "that which really exists".
Invariant simply means that the value of some quantity does not vary for different reference frames. If you want to think of that as meaning "real" then that is a philosophical issue because the definition of real is a philosophical issue.

Jakz said:
In a Euclidian 3D space, a stick can be rotated to any orientation, and from some perspectives it will appear to be shorter than it actually is. But there is an invariant length that corresponds to something real.
You can project the shadow of the stick onto a wall. As you rotate the stick the length of the shadow changes. Is the shadow real?

Jakz said:
That's why I find the null interval puzzling. Yes, the two points have different coordinates, but isn't their separation a matter of our perspective as massive entities?
What do you mean by "separation"? If you are referring to the spatial separation, then yes, that's a quantity that varies depending on the reference frame. But the interval has a value of zero, and that value is invariant. The value really is zero!

Jakz said:
(btw, an interesting historical note in the second article: "It’s interesting that some of the ideas of Feynman’s talk that day, such as the need for a complete absorber in the future, were already familiar to Einstein. After the talk was over, Pauli raised some objection to the theory, and asked if Einstein agreed. “No”, Einstein replied, “I find only that it would be very difficult to make a corresponding theory for gravitational interaction”.")
There's a more thorough discussion of that talk in Surely Your Joking Mr. Feynman. I first read that superbly entertaining book in the early 1990's and just read it again recently. I was surprised when I realized how much it had influenced me. He applies the rational scientific way of thinking to everything and what I realized is that I had been doing that throughout my career. Always arguing with with college administrators when they rationalized things in an unscientific way, or tried to apply scientific methods incorrectly or in an inappropriate way.

Dale
Jakz said:
Yes, the two points have different coordinates, but isn't their separation a matter of our perspective as massive entities?
There is no frame where two null separated events are in either the same location or the same position. There is also no frame where the time order of two null separated events swaps.

Jakz said:
Why, in this case, do we treat the space and time intervals as "real", and the invariant interval as inconsequential?
We don’t. It is just that the a spacetime interval being 0 doesn’t mean two events are the same event. A null interval is real and consequential, but it is not the same as a 0 Euclidean distance.

A Euclidean distance indicates a sphere. A Euclidean distance of 0 is a degenerate sphere, which is a single point.

A Minkowski interval indicates a hyperboloid. A spacetime interval of 0 is a degenerate hyperboloid, which is a cone (called the light cone).

Bandersnatch, vanhees71, Mister T and 2 others
Dale said:
There is no frame where two null separated events are in either the same location or the same position. There is also no frame where the time order of two null separated events swaps.
If I interpret this correctly, you are saying that boosts and rotations preserve time orientation [and chirality] and that you are restricting your attention to future-directed, right-handed frames?

Dale
jbriggs444 said:
boosts and rotations preserve time orientation [and chirality]
This is the usual definition, yes. The more technical term is that boosts and rotations are elements of the proper orthochronous Lorentz group. This is the component of the full Lorentz group that is connected to the identity.

Dale
jbriggs444 said:
If I interpret this correctly, you are saying that boosts and rotations preserve time orientation [and chirality] and that you are restricting your attention to future-directed, right-handed frames?
Yes, as @PeterDonis confirmed.

This goes back to the idea that surfaces of constant spacetime interval from some origin form hyperboloids. For timelike intervals it is a hyperboloid of two sheets, one future and one past. There is no way to smoothly move from one sheet to the other, they are disconnected.

This is in contrast to spacelike intervals which form hyperboloid a of one sheet. So any fixed spacelike imterval can be smoothly connected.

So spacetime intervals split spacetime into four distinct regions about any event: future timelike, past timelike, lightlike, and spacelike. There is no arrow of time, but there is a clear separation between future and past.

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jbriggs444 said:
If I interpret this correctly, you are saying that boosts and rotations preserve time orientation [and chirality] and that you are restricting your attention to future-directed, right-handed frames?
The orthochonous Lorentz group preserves time-ordering between timelike or light-like separated events. That's why only such events can be causally connected, because this group doesn't preserve the time-ordering of space-like separated events, i.e., you can always transform to another frame, where the temporal order is switched or to a frame, where the events are simultaneous.

Jakz said:
TL;DR Summary: Physical meaning of null spacetime interval

Hi all,

I've been trying to wrap my mind around the physical meaning of null spacetime intervals. I understand the metric of Minkowski geometry. And I understand that photons have no proper time and no reference frame: the spacetime interval between the emission and absorption of a photon is zero. They are the same event.

My question is: What does this mean? Is the fabric of spacetime warped to infinity by every photon passing through it? If I look up at Polaris at night, from my reference frame it appears to be 433 light years away in space, and 433 years away in time. But in spacetime, the emission of that photon and its absorption by my eye are the same event.

This is a bit of a necro post, but I have a different take. As others have mentioned, the emission event and reception events are different events, it's incorrect to assume that because the space-time interval is zero, that that means that the events are the same.

What I think one needs to know about null intervals is that mathematically, they have an affine geometry, not a Euclidean geometry.

wiki said:
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting"[1][2]) the metric notions of distance and angle.

What can do with an affine geometry? Affine geometries have a concept of "in the same direction", which allows one to talk about geodesics. These would be "null geodoesics". And affine geometry have a concept of marking regular intervals along any particular null geodesic, which is an "affine parameterization" of the null geodesic.

But there isn't any meaningful notion of "distance" along a null interval, even though there is a notion of "equal" intervals along any one particular geodesic. A distance would require the ability to compare regular intervals from one null geodesic to another - but there's no unique way of doing this.

vanhees71, Ibix and Dale

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