# Defining simultaneous as what you see now

1. Oct 1, 2014

### Erland

It can be argued that the definition of simulatneity in SR is artificial and only a matter of calculation and reconstruction. You can never now be aware of simultaneous events at other locations, according to the standard definition. Wouldn't it then be more natural to consider those events you can see now as simultaneous to your present?

For example, assume that the Earth and Mars are 5 light minutes apart from each other, and have synchronized clocks, according to Einstein's convention. We also assume that the Earth and Mars are effectively at rest relative to each other and that their common reference frame is an inertial frame.
Also, assume that:

At 8.00, I, on Earth, finish my breakfast.
At 8.05, I recieve a radio message from Mark on Mars saying "Hello Erland". I immediately reply and send the radio message "Hello Mark" to Mark.
At 8.10, Mark is beginning his breakfast.

Now, when I receive Mark's message at 8.05, I can make a calculation and infer that Mark sent his message at 8.00, just when I finished my breakfast. But I was totally unaware of that when it happened. I finished my breakfast without knowing that Mark at that very moment sent a message to me.
Isn't it quite artificial then, to say that these two events are simultaneous relative to me? Wouldn't it be more natural for me to say that the events when Mark sent the message and when I received it are simultaneous?

The idea is, then, to consider all events I can see now, or more general, can get knowledge about now from light speed signals, as simultaneous to my present.
In this case, we could for example follow the development of a supernova, say, 10 000 light years away, from eruption to extinction, and say that this development is going on now, and not during some weeks 10 000 ago, when noone of Earth could have the faintest idea of what was going on far away in the sky.

One might argue that if we define simultaneity in this way, it becomes dependent upon the observer. I can consider the two events when Mark sent the message and when I recieved it and sent the reply as simultaneous, but if Mark adopts the same definition of simultaneity, these two events are not simultaneous for him, but 10 minutes apart. He receives my reply at the moment when he begins his breakfast, and that is 10 minutes after he sent his message, according to him.

But in SR, simultaneity is dependent upon the observer even if we use the standard definition. Two observers who move relative to each other with a high constant velocity will not agree about which events are simultaneous, using the standard definition.
Also, if we use this alternative definition of simultaneity described here, it may be easier to find out what you actually see according to SR. For example, a length contracted object does not at all look contracted according to the standard formula, which in turn is connected to the standard simultaneity definition.

So, why not reformulate SR in terms of this alternative definition of simultaneity? Surely, someone must have got the same idea before. Does anyone know if there is anything written about this?
I tried to make some calculatons myself, but the math is more complicated than with the standard definition. With this alternative definition, simultaneity becomes dependent, not only upon the reference frame, but also upon every observer at every location. Mark and I have different opinions of which events are simultaneous, despite that we belong to the same reference frame (at rest relative to each other), while we agree about which events are simultaneous with the standard definition.
Maybe this is a decisive disadvantage compared to the standard definition.

2. Oct 1, 2014

### ghwellsjr

Yes, Einstein mentioned this in his 1905 paper:

"We might, of course, content ourselves with time values determined by an observer stationed together with the watch at the origin of the co-ordinates, and co-ordinating the corresponding positions of the hands with light signals, given out by every event to be timed, and reaching him through empty space. But this co-ordination has the disadvantage that it is not independent of the standpoint of the observer with the watch or clock, as we know from experience. We arrive at a much more practical determination along the following line of thought."

http://www.fourmilab.ch/etexts/einstein/specrel/www/

3. Oct 1, 2014

### Staff: Mentor

This approach is basically equivalent to saying that the speed of light is infinite so that the light travel time is zero. Consider, for example, the infra-red remote control that you use to control a television set from the comfort of your couch. There is a very small but measurable time lag between the signal leaving the remote control and reaching the controlled device; by declaring those two events to be simultaneous you are declaring that the elapsed time between them is zero for some observers. That's a good enough approximation for some problems (just about all of Newtonian mechanics; anything in day-to-day living) but it will fail calamitously for any system that is sensitive to such small timing differences. It would become impossible to design a working telecommunications system, GPS system, or for that matter any moderately interesting silicon chip, let alone a modern microcomputer.

4. Oct 1, 2014

### A.T.

It becomes dependent upon the observer's position, which is not the case in SR

No, it depends on the reference frame. Two observes at relative rest at different positions have the same simultaneity. It makes a huge difference if simultaneity depends on the position or just relative motion.

5. Oct 1, 2014

### Staff: Mentor

That is the main reason.

There are other reasons. For example, usually simultaneity is transitive, i.e. if event A is simultaneous with event B and event C is also simultaneous with B then A is simultaneous with C. That is not the case with the redefinition. Also, normally if two events are simultaneous then they cannot be causally related, whereas in your definition they could be causally related. But I think that the difficult math is the main reason.

Last edited: Oct 1, 2014
6. Oct 1, 2014

### pervect

Staff Emeritus
We already have a term, "null interval" , that covers the concept of two events connected by a lightlike signal. It would IMO be confusing to redefine the terms midstream.

7. Oct 2, 2014

Hi Erland –

I agree with the comments above that there’s no point in redefining “simultaneity”. The word strongly implies a two-way relation: if event A is simultaneous with event B, we assume B is also simultaneous with A. As you point out, that’s not the case if we take simultaneity as referring to each observer’s past light-cone.

I would rather emphasize that this word has no physical meaning, when applied to distant events. But I know of one paper that does do what you suggest, and the author claims some advantages to this approach: “Apparent Simultaneity”.

I think you’re making an important point here, though –
It’s not just that we couldn’t have known about the supernova earlier. If we think in terms of relativistic spacetime, the supernova is physically happening here "now” – since the invariant spacetime interval between that event far away and our seeing it here on Earth is “null”.

When we say the supernova happened 10,000 years ago, that’s not completely wrong... but what we’re doing is transposing from the 3/1-dimensional spacetime of relativity to a 4-dimensional Euclidean spacetime, where time is just another dimension of space. This is the so-called “block universe” picture, where we imagine the whole universe existing all at once, over all its history. This is the picture we get by “calculating and reconstructing”, using the conventional definition of simultaneity that Einstein introduced in his 1905 paper.

We can reconstruct a somewhat consistent history of the universe this way – imagining the supernova as simultaneous with certain weeks in the Earth’s history 10,000 years ago. But as you say, such reconstructions are observer-specific; more importantly, this is not physically how spacetime is structured. There was a recent thread on “The Block Universe” where PeterDonis explained very well the kinds of conceptual errors we fall into when we tacitly assume this conventional definition of simultaneity is physically meaningful.

8. Oct 2, 2014

### Staff: Mentor

Spacetime is 4-d, yes, but it's not Euclidean. It's Minkowskian. A Euclidean spacetime would have a different metric. (Don't be misled by the fact that we multiply the time coordinate by $c$ to make its units the same as the space coordinates; that doesn't make the spacetime Euclidean, it just makes the units consistent.)

When we say the supernova happened 10,000 years ago, we're not actually changing the spacetime at all; we're just imposing a particular coordinate chart on it. It would be more correct to say that we're going from 4-d Minkowskian spacetime, with no coordinates at all, to a particular 3+1 split of that spacetime, imposed by choosing a particular set of coordinates (the one that says that 10,000 years of coordinate time elapsed between two particular events). You're correct that this 3+1 split is arbitrary and observer-specific, and has no invariant physical meaning.

Thanks for the kudos on the Block Universe thread, btw! :)

9. Oct 2, 2014

### ghwellsjr

I think you missed the point. In the coordinate system of an observer located with event A, B is simultaneous with A (and vice versa). A different observer located with event B defines a different coordinate system in which A is simultaneous with B (and vice versa).

10. Oct 5, 2014

### Erland

Thanks all for your replies, Especially, the paper linked by Conrad is very interesting. I am reading it now, and it is precicely what I wanted. The author points out some problems with the standard simultaneity definition that I didn't think of before.

The math is more complicated, yes, but it is not true that this backward light cone definition of simultaneity gives a simulatneity which is not transitive. For an observer at a specific position, two events are simultaneous if and only if they lie at the same backward light cone at that position, and this is clearly an equivalence relation, hence transitive. What is true is that it is not meaningful to talk about simultaneity for observers at different positions. On the other hand, once a position has been specified, simulatneity for observers ar that position is independent of the motion of the observer: two observers meeting at position will agree about which events are simultaneous to their meeting even if they move w.r.t. each other.

So you gain motion independence, but lose position independence, compared to the standardd approach.

11. Oct 5, 2014

### Staff: Mentor

Ah, ok, that is a different definition than I thought based on the previous discussion. I thought we were discussing this definition:

$\forall A,B \in M$
$A$ is simultaneous with $B$ iff $A\in \text{pastLightCone}(B)$

Where $M$ is a given spacetime manifold and $\text{pastLightCone}(B)$ is the set of all events connected to B by past-directed null geodesics.

That definition is not transitive. In fact, it is not even reflexive.

It sounds like the definition you are actually intending is:

$\forall A,B \in M$
$A$ is simultaneous with $B$ iff $\exists C \in M$ such that $A,B\in \text{pastLightCone}(C)$

That definition is transitive and reflexive, but it is overly broad. It makes all spacelike separated events simultaneous. Perhaps that is what you intended? But I don't think that is what anyone will understand by the term "simultaneous".

12. Oct 5, 2014

### Erland

No, it is not meaningful to say that two events are simultaneous in such a sense without specifying an observer.

Let us define an observer as a trajectory $T$ in spacetime such that the speed is less than $c$ at every point $C \in T$.

I would then make the following definition:

$\forall A,B$ in spacetime
$A$ is simultaneous with $B$ relative to $T$ iff $\exists C \in T$ such that $A,B\in \text{pastLightCone}(C)$.

This is indeed dependent upon the observer $T$, but it coincides for two observers at the same position at the same time, in the sense that for an event $C\in T\cap T'$, two events $A,B\in \text{pastLightCone}(C)$ are simultaneous relative with both $T$ and $T'$.

13. Oct 5, 2014

### Staff: Mentor

OK, thanks for clarifying. That seems reasonable. It is a lightlike foliation of the spacetime instead of the usual space like foliation, but it is both transitive and reflexive.

Of course, you should not call this foliation "simultaneity" since nobody will know what you are referring to if you use that label. But the foliation itself seems legitimate to me, and the resulting surfaces could certainly be used as a coordinate.

14. Oct 5, 2014

### robphy

The standard definition of simultaneity has events A and B spacelike-related.
These definitions allow the case when events A, B, and C lie on the same lightlike segment [implying that A and B are causally-lightlike related], otherwise A and B are spacelike-related.

(By the way, the practical definition of simultaneity is the observer-dependent assignment of the same value of t-coordinate to the two events.)

It seems.. (but admittedly I have not read the thread in detail)... that the OP is trying to assign the same coordinate [not t] to lightlike related events. One could do that [along a null segment] with one of the [Dirac] lightcone coordinates (u or v, possibly depending on sign conventions).

By the way, Ellis & Williams uses the term "world-picture" as a "view of objects in space-time on the past light cone of the point of observation"...for example, a photograph.