# How Is The Relativity Of Simultaneity Consistent With Physics And Relativity

• hprog
In summary, the principle of relativity states that the laws of physics are the same in all inertial frames of reference. As a result, two events that are simultaneous in one frame are not necessarily simultaneous in other frames.
hprog
Hi.
The concept of relativity is that the laws of physics are the same in all inertial frames of reference.
As a result of relativity we have the relativity of simultaneity, which says that two events that are simultaneous in one inertial frame are not simultaneous in other frames.

Now consider we have a piece of wood and we put on a fire in exactly its middle, then according to the laws of physics (if the conditions are equal on both sides of the piece of wood) then the fire will arrive at the ends of the piece of wood simultaneously.
Or consider if we stand in the middle of a train and we trow two balls to the two ends of the train car with the same force, then according to physics the balls will arrive at the ends of the train simultaneously.

This is a result of the laws of physics and therefore according to relativity should be true regardless of the frame the situation is being observed.
However the relativity of simultaneity says that simultaneous events in one frame are not to be simultaneous according to another frame, so how is the situation to be explained?

(Note that every frame will agree on the middle point of objects since both halves are equaly contracted on frames that the object is in motion)

Hi, hprog,

In the two-balls version, there is nothing in the laws of physics that says that the balls have to have equal velocities. They just happen to have equal velocities in one frame.

In the fire version, there is nothing in the laws of physics that says that fire has to spread at equal speeds in both directions along a piece of wood that is moving with a certain velocity. The symmetry of the laws of physics only implies that the fire has to spread at equal speeds in both directions in a frame where nothing else breaks the symmetry, i.e., in a frame where the wood is at rest.

-Ben

hprog said:
However the relativity of simultaneity says that simultaneous events in one frame are not to be simultaneous according to another frame, so how is the situation to be explained?
Wait wait.

Relatviity does not say that cannot be the same; it simply says they may not be the same.

Nothing in your scenarios would cause any frame to see those events as asimultanoeus, so both frames might see them the same (excepting some things you might do to change them).

What exactly is the problem?

However the relativity of simultaneity says that simultaneous events in one frame are not to be simultaneous according to another frame, so how is the situation to be explained?

What situation??

oops, I just realized DaveC asked that...

So let me ask it this way: Are you asking if something would be non simultaneous from another inertial frame? from an accelerating frame?? Are your objects in an inertial frame or accelerating frame...say even rotating?

DaveC426913 said:
Nothing in your scenarios would cause any frame to see those events as asimultanoeus,
Did you perhaps mean "every" when you wrote "any"? The two events "fire reaches the left end of the piece" and "fire reaches the right end of the piece" aren't simultaneous to an observer with a non-zero (relative to the piece of wood) velocity component in the left-to-right direction.

DaveC426913 said:
What exactly is the problem?
The fire reaches both ends simultaneously in one frame, but not in another. So he has found something that's different in two frames, and is asking "hey wait a minute, wasn't everything supposed to be the same in both frames?"

A part of the answer is that the idea isn't that everything is the same. That wouldn't make much sense actually, because it would mean that if you get up from your chair and start walking towards the door, your speed relative to the chair would still be 0.

So what is the same? First of all, we need to understand that "the principle of relativity" is a rather loosely stated idea, so we shouldn't expect a complete answer to follow from it. The complete answer is contained in the theory that these ideas helped us find (i.e. special relativity). There are however lots of little details in the theory that could be interpreted as aspects of the principle of the relativity. For example, if A and B are two clocks, and the rate of change of B's clock in the frame comoving with A is slow by a factor of 25%, then the rate of change of A's clock in the frame comoving with B must also be slow by a factor of 25%. So I don't think it's possible to write down a complete answer.

One of the most important things that can be described as an aspect of the principle of relativity (the main thing really) is that equations that describe how properties of particles and fields change with time, can be stated in a coordinate-independent way. There's a slightly different version of each such equation associated with each inertial frame, but they will look more or less the same. (If a term that appears in the equation associated with one frame doesn't appear in the equation associated with another, it's just because its value is non-zero in the first frame and zero in the other). The coordinate-independent equations have coordinate-independent solutions, but there's a coordinate-dependent solution associated with each inertial frame. Example: The coordinate-dependent versions of Maxwell's equation associated with different inertial frames look the same, but the coordinate-dependent values of the E and B fields may be different in different frames.

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Fredrik said:
Did you perhaps mean "every" when you wrote "any"? The two events "fire reaches the left end of the piece" and "fire reaches the right end of the piece" aren't simultaneous to an observer with a non-zero (relative to the piece of wood) velocity component in the left-to-right direction.

The fire reaches both ends simultaneously in one frame, but not in another.
Ok. Sure.

Fredrik said:
So he has found something that's different in two frames, and is asking "hey wait a minute, wasn't everything supposed to be the same in both frames?"
And here I thought he was asking: "hey wait a minute, wasn't everythning supposed to be different in each frame?"

See here:
simultaneous events in one frame are not to be simultaneous according to another frame

He's arguing that the principle of relativity seems to say that everything is the same, and that relativity of simultaneity seems to contradict this by saying that this one thing isn't the same.

Fredrik said:
He's arguing that the principle of relativity seems to say that everything is the same, and that relativity of simultaneity seems to contradict this by saying that this one thing isn't the same.

Hm. OK.

To which my response would be:

Relativity does not say "everything is the same", it says "everyone follows the same rules , even though that might (and often does) lead them to completely different results. And everyone who does observe the rules has an equally valid viewpoint."

Forget a moment about the relativity of simultaneity, if fire spreads out in two directions with every thing being equal is there any reason for the fire going in one side faster than the other, or the ball flying in one direction faster than the other? this is physics and we need to have an answer and rule why this should happen.
Here is another such experiment, suppose we have a pool and we pure water directly in the middle of the pool and the water is spreading out to both sides, is there a reason why the water should arrive to one end of the pool before the other? farther more if the pool is an exact rectangle the water will arrive to the left and right sides (perpendicular to the direction of motion) simultaneously so why is this not happening in the direction of motion (and the same can be said for the ball and fire in which going sideways will yield simultaneous results).

The fire can be moving at different speeds with respect to different observers and be different distances from different observers. This doesn't even have anything to do with Einstein - Newton and Galileo would be fine with this as well.

hprog said:
Forget a moment about the relativity of simultaneity, if fire spreads out in two directions with every thing being equal is there any reason for the fire going in one side faster than the other,
There are many different ways to obtain this result. You could e.g. get it from the relativistic formula for addition of velocities. I think the best way to see it is by drawing a spacetime diagram, but you would have to work on understanding relativity of simultaneity first.

russ_watters said:
The fire can be moving at different speeds with respect to different observers and be different distances from different observers. This doesn't even have anything to do with Einstein - Newton and Galileo would be fine with this as well.
But in Galilean spacetime, a boost to a different inertial frame would change the velocities of the two "fronts" of the fire by the same amount, [strike]so that the two speeds would be the same in the new frame too[/strike] so that the two speeds in the new frame would differ from their values in the old frame by the same amount.

Edit: I corrected the last part after russ watters pointed out that I messed it up.

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Fredrik said:
There are many different ways to obtain this result. You could e.g. get it from the relativistic formula for addition of velocities. I think the best way to see it is by drawing a spacetime diagram, but you would have to work on understanding relativity of simultaneity first.

You are using the velocity addition formula which is based on the relativity of simultaneity, so it can not be considered a solution rather it is part of the problem itself.
Again my question every thing being equal why should things on one side be different than the other, and why should it be (1) only for different frames, (2) and only in the direction of motion.

russ_watters said:
The fire can be moving at different speeds with respect to different observers and be different distances from different observers. This doesn't even have anything to do with Einstein - Newton and Galileo would be fine with this as well.

No, in physics if every thing is being equal than we should get equal results.
In fact you have not disagreed that in the same frame every thing will be equal and that even in different frames it will be equal for the perpendicular direction, although the relativity of simultaneity does not require it...
This is clearly because of equal things will yield equal results, and the relativity of simultaneity cannot change this without breaking general physics, unless you come up with a better answer.

I'm not going to spend a lot of time on finding a solution that doesn't require you to understand relativity of simultaneity first. Maybe there is one, maybe there isn't. What's important here is what SR says about this situation. The Lorentz transformation is a part of the definition of SR, and the addition of velocities formula follow easily from that, so "what SR says" is precisely what you get from that formula.

Fredrik said:
But in Galilean spacetime, a boost to a different inertial frame would change the velocities of the two "fronts" of the fire by the same amount, so that the two speeds would be the same in the new frame too.
No, one might go up while the other goes down, say, if you're in a frame traveling with one flame front that one is motionless while the other is moving at twice the speed it moves wrt to the object that's burning.

hprog said:
No, in physics if every thing is being equal than we should get equal results.
No, that's not what relativity means at all. Relativity says they obey the same laws of nature, not that you get the same results when measuring the same phenomena from different reference frames.

russ_watters said:
No, one might go up while the other goes down, say, if you're in a frame traveling with one flame front that one is motionless while the other is moving at twice the speed it moves wrt to the object that's burning.
Yes, I messed up what I was trying to say.

Quick guess: It's not sufficient to apply equal force. Equal force must be applied for equal time. While the force is being applied, the balls will move apart, so they are not released from exactly the same point.

OK, I give up. It's a mess. And they don't even have to learn about relativistic forces at http://ocw.mit.edu/courses/physics/8-033-relativity-fall-2006/lecture-notes/lecture11_dyn2.pdf" I'll stick with relativity of simultaneity.

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All three scenarios fire/balls/water are governed by the same thing: relativistic velocity addition. If you know the velocity in one frame you use relativistic velocity addition to get the velocity in another frame. This ensures that if they arrive simultaneously in one frame then they do not in any other frame.

DaveC426913 said:
Relatviity does not say that cannot be the same; it simply says they may not be the same.

That's an excellent point! In thinking about the meaning and use of the Lorentz transformation recently I've been led to the conclusion that the clearest and most indisputably valid use of the LT is to consider that it first of all provides the initial conditions for an invariant form of the wave equation for EM propagation. Physically meaningful results seem to occur when you restrict the choice of transformed values to t = t', both for setting the initial conditions for the solution of the wave equation and thereafter in evaluating it for other values of t.

I imagine that last post raised a few eyebrows so I'll try to explain. If I'm not mistaken, from the 2 Lorentz transformation equations we can eliminate the mutual variables to arrive at 8 total equations:

$$x = \gamma (x' + vt') \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t = \gamma (t' + \frac{vx'}{c^2})$$
$$x' = \gamma (x - vt) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t' = \gamma (t - \frac{vx}{c^2})$$
$$x = \frac{c^2}{v}(t - \frac{1}{\gamma} t') \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t = \frac{1}{v}(x - \frac{1}{\gamma}x')$$
$$x' = \frac{c^2}{v}(\frac{1}{\gamma}t - t') \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t' = \frac{1}{v}(\frac{1}{\gamma}x - x')$$

This provides the inverse transformation without having to specify the change of direction of the velocity. That gives the Lorentz transformation special status in that it contains its own inverse transformation. While that seems to give it a basis for covariance, one might wonder if it, at the same time, destroys its value as a transformation in some way. At any rate, exactly 2 values of the variables x, x', t and t' may be chosen and the other 2 follow from the transformation.

If we choose to set t = t' = 0 (because we know the value of the wave equation for some parameter at that time) then x and x' are forced to be equal to zero. We force the origin of the coordinate systems both to be co-located. But for all other values where t = t', x and x' diverge at a linear rate proportional to v.

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russ_watters said:
No, that's not what relativity means at all. Relativity says they obey the same laws of nature, not that you get the same results when measuring the same phenomena from different reference frames.

ok, I can see how we can measure an outcome differently. However, there can only be one outcome.

If I set fire to a piece of wood, the file will spread depending on what ever laws of physics are in effect.

However, there can only be one outcome, the fire will only spread in one way, despite what we measure. There is a causal relationship in effect here.

So my understanding of SR in the case mentioned is that different observers can measure different outcomes, however there is only one proper outcome.

PhilDSP said:
I imagine that last post raised a few eyebrows so I'll try to explain. If I'm not mistaken, from the 2 Lorentz transformation equations we can eliminate the mutual variables to arrive at 8 total equations:
...
This provides the inverse transformation without having to specify the change of direction of the velocity. That gives the Lorentz transformation special status in that it contains its own inverse transformation.
The inverse transformation isn't among the equations you posted. The inverse would express t and x as functions of t' and x'. These equations are much prettier in matrix notation by the way, especially when we use units such that c=1: $$\begin{pmatrix}t'\\ x'\end{pmatrix}=\gamma\begin{pmatrix}1 & -v\\ -v & 1\end{pmatrix}\begin{pmatrix}t\\ x\end{pmatrix},\qquad \begin{pmatrix}t\\ x\end{pmatrix}=\gamma\begin{pmatrix}1 & v\\ v & 1\end{pmatrix}\begin{pmatrix}t'\\ x'\end{pmatrix}$$I also wouldn't say that there's something "special" about "containing its own inverse". Every square matrix with non-zero determinant does that. And the word "special" often refers specifically to determinant =1.

rede96 said:
So my understanding of SR in the case mentioned is that different observers can measure different outcomes, however there is only one proper outcome.
Yes, the results of measurements will be different, but this isn't exclusive to SR and GR. Even in pre-relativistic physics (Newtonian mechanics in Galilean spacetime), different observers will have different result. For example, if you measure the velocity of the chair you're sitting on right now, you will get the result 0. If you get up and start walking in your positive x direction, and then measure the velocity again, the result will be something like -1 m/s.

I wouldn't describe this as "different outcomes", because that sounds too much like "one observer finds that Mike crashed his spaceship into an asteroid and died at the age of 30, while another finds that he missed the asteroid and lived to be a hundred". (I understand that you didn't mean that).

Fredrik said:
The inverse transformation isn't among the equations you posted. The inverse would express t and x as functions of t' and x'.
I think I see what you're saying. I believe the 8 equations above express the Lorentz transformation, with all its permutations, in the most general unrestricted form. You've chosen to retrieve solutions given the pre-existence of values for x and t as specified by your beautiful and useful matrix formulation. I've chosen to search for solutions given the pre-existence of values for t and t'. So the related matrix formulation would be the, not as pretty, but maybe not lacking in usefulness form here:

$$\begin{pmatrix}x\\ x'\end{pmatrix}=\frac{1}{v\gamma}\begin{pmatrix} \gamma & -1\\ 1 & -\gamma\end{pmatrix}\begin{pmatrix}t\\ t'\end{pmatrix},\qquad \begin{pmatrix}t\\ t'\end{pmatrix}=\frac{1}{v\gamma}\begin{pmatrix} \gamma & -1\\ 1 & -\gamma\end{pmatrix}\begin{pmatrix}x\\ x'\end{pmatrix}$$

Here the matrix is identical for both the forward and inverse cases.

I had in mind when using the term special, the more general concept of transforms which can't necessarily be expressed in matrix form, for instance, the Fourier, Laplace and Hilbert transforms. The Fourier transform is especially interesting since the transform and its inverse are the same except for a change in the sign from positive to negative. But that sign change lies within the exponent. Has anyone attempted to express the Fourier transform using a matrix formulation? That would be extremely useful.

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Right fredrick: I would say there is one event that different observers can measure differently. I don't like the usage of the word "outcome" either: 'the fire reaches both sides simultaneously' is two events, not one.

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Fredrik said:
The inverse transformation isn't among the equations you posted.
I said a little too much in this sentence. The second line of equations is of course the inverse transformation.

russ_watters said:
Right fredrick: I would say there is one event that different observers can measure differently. I don't like the usage of the word "outcome" either:

Yes, I can see what you are getting at. 'Outcome' may not be the best description but I was struggling to find the right terminology.

russ_watters said:
'the fire reaches both sides simultaneously' is two events, not one.

This is where I need some help with the terminology. I don't see this as two 'events'. The fire can reach both sides simultaneously in one frame and be measured to reach the left (or right) first in another. However only one 'event' could have occurred. (Or whatever the right word is.)

So if two observers were to get together after the fire to compare notes, they would both argue their respective observation were true. But there is only one correct result.

The question is how to tell which one?

I would suggest that they would have to take their relative speed and distance to the burning wood into account. If they did that, they would both come up with the same answer for when the fire reached the end of the piece of wood.

This is what I would call the 'proper' result.

rede96 said:
However, there can only be one outcome, the fire will only spread in one way, despite what we measure. There is a causal relationship in effect here.
There is a causal relatinship between setting the fire and the fire reaching the right end. There is also a causal relationship between setting the fire and the fire reaching the left end. There is no causal relationship between the fire reaching the left end and the fire reaching the right end.

DaleSpam said:
There is a causal relatinship between setting the fire and the fire reaching the right end. There is also a causal relationship between setting the fire and the fire reaching the left end. There is no causal relationship between the fire reaching the left end and the fire reaching the right end.

Not quite sure I understand that. Isn't fair to say that depending on where the fire is first set on the piece of wood will determine how the fire will spread?

So when the fire reaches the left and right side is determined by the initial ignition, hence why they are causally related.

rede96 said:
Not quite sure I understand that. Isn't fair to say that depending on where the fire is first set on the piece of wood will determine how the fire will spread?

So when the fire reaches the left and right side is determined by the initial ignition, hence why they are causally related.
They are both causally related to the initial ignition, not to each other. I.e. The fire reaching the left does NOT cause the fire to reach the right, and vice versa.

OK, here is an attempt to do it with forces. The question is most easily addressed by the Lorentz tranformations for each of the two events, or by the addition of velocities. However, the OP asked, what, in detail, happens with a force that is symmetric in both frames? Let's set up everything in the frame on the ground, and never switch frames.

At t=0 the front and back walls of the train are at x=±L. They move to the right with constant velocity U, so x=±L+Ut (Eq 1a,b). Each wall has constant charge density, and the walls are oppositely charged, generating a constant E field between them.

At t=0, two charges q of opposite sign and rest mass m are released from x=0. They experience constant forces in opposite directions f=±qE.

Starting with Newton's second law f=dp/dt, gives p=pu±ft.

Substituting p=mvV, where mv is the relativistic mass, gives mvV=muU±ft

Solving for V, and http://integrals.wolfram.com/index.jsp" over time to get displacement gives x(t)=±{sqrt[(muU)±ft)^2+m^2]∓sqrt[(muU))^2+m^2]}/f (Eq 2a,b).

Choosing specific values for all constants (U<1 since c=1; mu=m/sqrt(1-U^2)), and http://fooplot.com/" Eq 1 and 2 to obtain the t values where they intersect, seems to indicate that the backward charge intersects the back wall before the forward charge intersects the front wall, consistent with answers obtained by changing frames. Corrections for conceptual* and algebraic errors are of course welcome

*I neglected the self force

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DaleSpam said:
They are both causally related to the initial ignition, not to each other. I.e. The fire reaching the left does NOT cause the fire to reach the right, and vice versa.

Ah, ok. I see. Thanks

DaleSpam said:
They are both causally related to the initial ignition, not to each other. I.e. The fire reaching the left does NOT cause the fire to reach the right, and vice versa.

After thinking about this for a while, I was wondering, there is some dependency between the fire spreading to the left and the fire spreading to the right.

Using a different example:

I get into my car, turn my engine on, put it in gear and put my foot down on the accelerator. There is causal relationship between the front of my car moving off and the accelerator being depressed and there is a causal relationship between the back of my car moving off and the accelerator being depressed.

Making this analogous to the wood and fire, you would say that the front of my car moving off does not cause the back of the car to move off.

However, I would say that that the physics of the car means that it is not possible for the front to move without the back moving. So they are dependant events. (If that is the right terminology)

In my frame the front and back of the car move simultaneously. As they are dependant, this is the proper sequence that all other observers must agree, even though certain frames may measure the front and back movements differently.

I see this being the same as the fire burning on the wood. The physics of the wood and where the fire was lit will determine how the fire will spread. So the movement of the flame is ‘predetermined’ (without going into quantum physics!)

Therefore, if it just so happens that the fire reaches the ends of the wood simultaneously, this must the sequence agreed by all observers. Even though some might measure a different result.

Anyway, the point being that in the case of the wood on fire or my car moving off, there is absolute simultaneity. E.g. both ends of my car must move off together and flame must spread to each end equally.

Any observations that show a different result do not accurately describe the events.

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