# About the definition of simultaneity in SR.

1. Nov 27, 2014

### Coffee_

"I've looked this up and came up to similar questions but haven't seen it been explained very clear yet. Excuse me if such questions were already posted here.

There are two different kinds of simultaneity it seems to me:

Let's take the event of two lightning bolts striking earth. Observer A is positioned exactly in between the lightning strikes and sees the light of both bolts reaching him at the same time. Observer B, closer to bolt 1, will see bolt 1 happening before bolt 2. Observer C, closer to bolt 2, will see bolt 2 happening before bolt 1. So here we have people seeing a different order of events using a very classical reasoning. So no SR involved here.

(According to SR though, these observers should all see the same simultaneity for the bolts, since $\beta=0$ and $\gamma=1$. This would mean $\delta{t}=\delta{t'}$ for any of those observers.)

The only explanation I came up with is that there exists some ''absolute simultaneity'' for all observers at rest relative to eachother. They can define for the simultaneity to be: ''all observers at rest to eachother should calculate how the person exactly in between the two events would percieve them, this is then the meaning of simultaneity in the context''.

So basically there is:

1) Simultaneity as in percieving the order of events without any calculations

2) Simultaneity as in, doing calculations based on how far you are from the two events and how long the time difference of arrival of lightf from the events is. Based on this you can conclude if they would have been simultaneous in the frame of a person excatly in between the events.

The question is now:

Please tell me if my reasoning is somewhat correct, and where it can be more accurate. Do I just forget about the simultaneity as defined in 1) and only think about it in terms of 2? This would mean that the if I drop a ball right now, and then in 8 minutes I look up and see a sunspot appearing on the sun, the dropping of the ball and the appearing of the sunspot would be simultaneous.

2. Nov 27, 2014

### stevendaryl

Staff Emeritus
The relativity of simultaneity is not about the order in which you see events. The time that you see an event depends, as you know, on when the event took place, and also how far away it is. So for your three observers, for them to figure out the times for the two lightning strikes, they have to do some computation:

$t_{event} = t_{observed} - \frac{D}{c}$

where $t_{event}$ is the time the event occurred, $t_{observed}$ is the time that it is observed, $D$ is the distance to the event, and $c$ is the speed of light.

So if your three observers that are at rest relative to each other compute the time of the event, they'll get the same answer.

But if the observers are in motion relative to one another, they will get different answers.
Right.

Yeah, the notion of simultaneity based on the time you observed the events is not too relevant to SR.

3. Nov 27, 2014

### Staff: Mentor

Neither #1 nor #2 are quite right, and if you fix them the apparent incompatibility between them goes away.

In #1, when you talk about "perceiving the order of events", exactly which events are you talking about? The event that you are perceiving is not the lightning strike, it is light from the lightning strike landing on the retina of your eye. There's never any question about whether these light-reaches-your-eyes events are or are not simultaneous - all observers, regardless of their state of motion, will agree about whether two flashes reach your eyes at the same time or not. However, that doesn't tell us anyone about whether the two lightning strikes that created the flashes happened at the same time. To determine that, we have to do the calculation you describe in #2.

In #2, you just need to strike out the entire second sentence. The person "in the middle" (who would also have to be at rest relative to you for the light to reach his eyes at the same time) also has to make the distinction between the lightning strike event and the light-reaches-his-eyes event. All observers who are at rest relative to you, regardless of their position, will calculate the same result for the simultaneity or not of all events.

4. Nov 27, 2014

### Coffee_

Again assuming no one is moving realtive to eachother. For a person exactly in between the events the time difference between the events happening is exactly the same time difference that he will see the light arriving at him. This is what I meant. To calculate the time difference between events it's equally as good to calculate the time difference at which light arrives at an observer in between the events. I said this because this is basically what is done in some thought experiments with the train.

5. Nov 27, 2014

### Staff: Mentor

Yes. Just forget about 1), it is not what is referred to by "simultaneity".

SR always assumes that all observers are intelligent and recognize that they need to account for the finite speed of light. They also all recognize that the event of receiving a flash of light is a different event from the one where that flash was emitted. So, if a given observer assigns $t_r$ and $x_r$ to the time and position of the receiving event and $t_e$ and $x_e$ to the time and position of the emitting event then, they know that they are related by: $c=\frac{x_r-x_e}{t_r-t_e}$.

The simultaneity of receiving a pair of flashes of light never automatically implies the simultaneity of emitting the pair of flashes. It always has to be calculated to determine that (although certain configurations make the calculation easier). If you know $x_r-x_e$ and $t_r$ then you can use the above formula to calculate $t_e$.

6. Nov 27, 2014

### Staff: Mentor

This is NOT the definition of simultaneity. It is just a particular scenario where the actual definition is easy to calculate since some of the terms cancel. Suppose that we have two emission and two reception events. Using the labeling above we can write:

$$\frac{x_{r1}-x_{e1}}{t_{r1}-t_{e1}}=c=\frac{x_{r2}-x_{e2}}{t_{r2}-t_{e2}}$$

Now, in the specific case where someone is exactly in between the events we have $x_{r1}-x_{e1}=x_{r2}-x_{e2}$ so $t_{r1}-t_{r2}=t_{e1}-t_{e2}$, as you said. This is not a different calculation from what I posted above, but just the same calculation simplified.

7. Nov 27, 2014

### Coffee_

I see, so the definition of simultaneity in SR can be best described as follows.

I take an xyz coordinate frame which represents me as an observer. Two events happen at positions (x1y1z1) at an unknow $t_{1}$ and (x2y2z2) at an unknown $t_{2}$. I can see the positions of those events and I can measure the difference in time between the light arrival of those events. From this information I can find this unknown $t_{1}$ and $t_{2}$. I can then consider $t_{2}-t_{1}$ as a representative for simultaneity for me.

Any observere which has a translated and or rotated frame relative to me so that our $t_{me}=t_{observer}$ will measure different positions for the events but can still calculate the same $t_{1}$ and $t_{2}$ and thus, the same measure for simultaneity $t_{2}-t_{1}$ ?

More or less correct?