How is the notion of relative simultaneity to be understood?

Question

I have been reading a bit on Einstein’s special theory of relativity. It seems clear to me from the theory that two observers do not necessarily agree on what events occur simultaneously. However I have difficulty understanding how that relates with the concept of now.

Let’s consider that in my reference frame I am holding my phone typing this question. Suppose that in my reference frame there is someone approaching me from a far distance at a speed close to that of light. Then that person would find my future self to be simultaneous with his/her now (let’s say it’s a future self that is asleep because it’s night time).

I do not know how to interpret this. I am not yet in that state. It seems that I will eventually become that future self but that is not what I am yet. Should I interpret this as simply saying that experiments carried out by that person will reveal my future self to be simultaneous with the person BUT NOT to mean that the future self exists yet (experiments such as obtaining the distance between him/herself and my future self and dividing it by the speed of light)? In other words should I interpret this as saying that the laws of physics only give you what is simultaneous IN your reference frame but not necessarily what is happening in the moment (in the way that it is the case that I am typing on my phone at this moment and not sleeping). And if I bite that bullet it seems I’m forced to consider that the person I say is approaching me might only be doing that in my reference frame and not in his.

The question itself is a bit confusing as well because according to relativity the moment depends on the observer. I guess the question might be a bit more philosophical than scientific but if there is any light you can shed on the problem it would be very appreciated.

Answer 1

It seems clear to me from the theory that two observers do not necessarily agree on what events occur simultaneously. However I have difficulty understanding how that relates with the concept of now.

The concept of “now” is not a physical concept, it is a philosophical concept typically associated with the metaphysical ontology known as presentism. From a physics perspective the concept of now is neither theoretically necessary nor experimentally measurable.

If you have some particular affinity for the concept then you are free to use it. You can identify any spacelike surface you choose and call it “now” without fear that some experiment could disagree. You could choose a surface of simultaneity for some specific observer, but it is not mandatory.

It is also possible to say that “now” is a purely local concept so that “now” has meaning for some observer but there is no sense in which a given observer’s “now” coordinates with other locations.

Because it is a philosophical concept you will find a lot of disagreement on it, and none of it is important for physics. If it bothers you, then discard it, or if you like it then keep it. Just recognize that it is not physical.

Answer 2

There is something you don't mention explicitly in your question, but your unease may be connected with it: a question whether relativity of simultaneity challenges our notion of causality.

The interesting thing is: it does not. If an event B is triggered by an event A then there is no transformation that would lead to a presentation where that happens out of order.

For instance, let's say there is a setup where a light source B will emit a pulse of light at the point in time that a photosensitive cell strapped to B registers a pulse of light coming from source A.

You can transform to a coordinate system such that the distance between the sources A and B is length contracted in terms of that coordinate system, so that the time for the light to traverse the distance is correspondingly shorter, but there is a limit to that. There are only transformations that make that duration shorter, no transformation is available that would reverse the order.

Relativity of simultaneity presents a degree of freedom that leaves causality unaffected.

Here is another comparison.

The following is not a complete description of relativity of simultaneity (something essential is left out), but it may help to open a window.

Let's say that you are in the 18th century, exchanging letters with someone on another continent. For the sake of the thought demonstration: the ships on which the letters make the journey do not report how many days it takes them to travel from west to east, and from east to west.

It could be the same amount of days, but maybe there are predominent east-to-west blowing winds which would result in the east-to-west journey taking less days than the west-to-east journey.

Let's say the arrival time of the letters is the only information you have to try and synchronize calenders with the person you are exchanging letters with.

The simplest approach is that you take the round trip time, and you divide that by 2. If the round trip time is 60 days the a single journey could be just half that, 30, and on that assumption you could decide on calender synchronization.

But: you don't know.

It could be that the actual distribution of the round trip time is 20-40. The interesting thing is: the fact that you don't know does not present an obstacle. As long as everybody doing business with each other is using the same convention no discrepancy will be encountered.

Relativity of simultaneity in terms of special relativity goes a step further than the above. In the story above there is the detail: "the ships on which the letters make the journey do not report how many days it takes them"

In the special relativity flavor of relativity of simultaneity there is intrinsically no way of measuring how the round trip time breaks down.

Answer 3

What it means is that our common sense notion of simultaneity is rather meaningless when we consider events that are spacelike separated.

For you, your local 'now' is a particular point in spacetime at which you exist, and its time coordinate is continually increasing at a fixed rate in your frame- if you like, you can think of it as a series of successive events at the spatial location you occupy.

If someone elsewhere is travelling towards you, they will also have a local 'now' which is a point in spacetime travelling forward through time at a fixed rate in their frame.

Each of you has a plane of simultaneity, but that simply means a slice through spacetime with common t coordinate. In the frame of a person moving relative to you, your horizontal slice through spacetime is a sloping one, with a t' coordinate that varies all along your direction of travel.

As a consequence, you and the traveller will label distant events with different time coordinates, but each event has only one location in spacetime- it is just that you and the traveller use different coordinates to label it.

To make that more concrete, imagine you are standing still and you have two friends, one walking away from you to the right and one walking towards you from the right. At a suitably great distance, the time corresponding to 'now' for each of you might vary by a year, so you might consider it to be October 2022 at that distance point, while one friend will consider it to be October 2023 and the other October 2021.

Now suppose there is a planet at that distant point, on which there are three people, one stationary relative to you, one walking towards you and one walking away, so the three people on the distant planet share the same relative movements, and hence the same three frames as you and the two people here on Earth. All three of the distant people share the same 'now'. What they disagree about is the date of 'now'. If the person who is stationary relative to you considers the date 'now' to be October 2022 (ie simultaneous with your current 'now') because their time is synched with yours, then the other two people will consult their calendars, and one will say it is October 2021 while the other will say it is October 2023.

You will see some philosophers argue that presentism (which is the idea that things only exist in the present) cannot be true, owing to the relativity of simultaneity, but that is a specious argument. The three people on the distant planet can all exist at a common 'now'- they just label it differently.