Physics Query - the 'Relativity of Simultaneity'

[Refus de Sejour]

I’ve recently started reading a kind of physics-for-dummies, Clifford A. Pickover’s 'Time, a Travellers Guide' . It's fascinating stuff, to say the least, but I’ve been puzzled by one apparent contradiction which I'd welcome any attempts to clarify.

The first topic covered is that of the 'relativity of simultaneity'. This states that the temporal relation of events (whether they exists simultaneously, or in each other’s past or future) depends on their relative states of motion. Over great distances the implications of this are dramatic. My ‘now’ may seemingly occur simultaneous with event B in a distant galaxy, but if I move 6 feet across the room I will cause B to jump a day into my past or future, depending on whether I walk towards or away from it.

However Pickover states there is one restriction on this; the two events must be so close in time that there can be no causal connection between them; i.e., there is not enough time between event A and event B for light (or any other signal) to travel between them. While I can see the need for this restriction – i.e., preventing the reversal of cause and effect – this seems to contradict the initial hypothesis. The theory initially implies that there is no ‘universal standard time’. However, the above restriction seems to require just such a standard time – how else can events be thought of as being ‘close in time’ unless they are though of as sharing a common simultaneity as opposed to a relative one?

Any suggestions/speculations on this would be appreciated - preferably in relatively non-technical terms.

 

[Zombified]

You can still define the time interval between two events in a chosen frame of reference. Now, how close is close enough in time for this to work? It turns out to depend on distance. Basically, it has to far enough away from you that light can't travel the distance in the amount of time allowed in that reference frame.

Now, the time and distance intervals change when you change frames of reference. But they don't change arbitrarily, there is a relationship between the change in the time interval and the change in the distance interval. And because the speed of light is the same in all frames of reference, if the events are far enough apart in one frame, they are far enough apart in all frames.

The rule in a formula is,

x² - c²t² > 0

where x is the distance between the two events and t is the time difference, in a particular chosen frame, and c is our old friend the speed of light.

The expression on the left turns out to be "invariant" under the sorts of coordination transformations you do in special relativity (Lorentz transformations), and so this interval serves as a well-defined relativistic distance between two events in space-time (hinting at why you'd consider time, or time-times-speed-of-light, to be another dimension). Mathematically,

x² - c²t² = x'² - c²t'²

where primes (') denote measurements in a different frame of reference.

If the interval is positive, the events have a spacelike seperation, and neither can cause the other. If the interval is negative, they have a timelike seperation, and although the interval may change, they will always appear in a consistent order, and the earlier event could cause something to happen at the later event.

 

[epepke]

The theory initially implies that there is no ‘universal standard time’. However, the above restriction seems to require just such a standard time – how else can events be thought of as being ‘close in time’ unless they are though of as sharing a common simultaneity as opposed to a relative one?

Your intuition is pretty much correct, even if your application of it isn't.

Differently moving observers will measure space differently and time differently, so they'll measure distance and [/i]elapsed time[/i] differently. However, if you combine distance and elapsed time into a single quantity called the interval, then you have something that all intertial observers will measure as the same.

There is no unversal standard time or unversal standard space, but there is universal standard spacetime (well, at least in SR), where the time and the space are related exactly by the speed of light.

Calculating the interval is easy. First of all, pick a system where the speed of light is 1. For example, take a nanosecond as the unit of time. In that time, light travels a little bit more than a foot. (Call it a bigfoot.) Then envision the universe as a 4-dimensional box, except for one trick: multiply the time by i, the square root of -1. Then use the Pythagorean theorem the same way that you would to calculate distance to calculate the interval instead.

You'll get three kinds of intervals: real valued, zero, and imaginary-valued. Imaginary-valued intervals are the ones we're used to (timelike); every observer will agree upon the order of events. Zero-valued intervals represent light (lightlike). Real-valued intervals (spacelike) represent the kind of simultaneity problems that the book is talking about.

Another way to think about it is this: spacelike=more distance than time, timelike=more time than distance, lightlike=exactly as much time as distance.

If you think about it, those are the only kind of events that can be simultaneous. Simultaneous means that the change in time as measured by somebody is zero. Unless they are in the same place, there has to be a spacelike interval between them, because any distance is greater than zero.

 

[Zombified]

Re: Re: Physics Query - the 'Relativity of Simultaneity'

Calculating the interval is easy. First of all, pick a system where the speed of light is 1. For example, take a nanosecond as the unit of time. In that time, light travels a little bit more than a foot. (Call it a bigfoot.) Then envision the universe as a 4-dimensional box, except for one trick: multiply the time by i, the square root of -1. Then use the Pythagorean theorem the same way that you would to calculate distance to calculate the interval instead.

In case it isn't clear, epepke has given pretty much the same interval as I did above, but changed units and took a square root:

s = sqrt(x² + (i t)²)
= sqrt(x² - t²)

s² = x² - t²

which is what I gave above if you set c=1 as epepke did. s² can be negative, which is where imaginary values of s come from.

Multiplying time by i to use Pythagoras' theorem is a neat trick to remember the sign. In fact, thinking about this in terms of Pythagoras' theorem is kind of interesting. Another way to get that sign is to look at it this way. Think about the position of a point in space or an event in spacetime as a vector. You can get the distance between two points (vectors) by taking their difference and then taking the dot product of that difference with itself:

s² = Δx · Δx = sum Δx_i²

We want to introduce something to get that negative sign into one of the components. epepke's suggestion is to multiply a component by i. Here's another way: multiply one of the vectors in the dot product by a special matrix. Call it g and take its components as

g₁₁ = g₂₂ = g₃₃ = 1,

g₀₀ = -1

g_ij = 0 for i != j

Then the formula for the interval becomes,

s² = Δx · Δ(g x) = sum g_ij Δx_i Δx_j

This is the same formula in more complicated notation. Why do this? The matrix g is called the metric. If you set it to an identity matrix, then you just get Pythagoras and the metric is the metric for Euclidean space. Set it to the metric above, and you get special relativity in four-dimensional space time.

General relativity is all about generalizing that metric so that it can represent curved spacetime. Then the metric isn't as simple as shown above, and you have to find your intervals by integrating along a curved path, but the general idea is the same.

I know this is a lot more detail than was originally asked for, but physics doesn't get enough air time hereabouts...

 

[epepke]

Re: Re: Re: Physics Query - the 'Relativity of Simultaneity'

In case it isn't clear, epepke has given pretty much the same interval as I did above, but changed units and took a square root

Yeah. It's exactly the same. I just did it in words, because math makes some people sleepy.

Taking the square root is, I think, justified, as people normally think in terms of distances rather than distances squared. But you can do the same thing without taking the square root, which gives you positive, zero, or negative.

Incidentally, this is one reason I prefer to use quaternions over 4-vectors in Minkowsky space or 4-vectors with imaginary time or 4-vecors with munged operators. Because it's nice to have the interval be real with the more mundane timelike intervals and the interval be imaginary outside the light cone. However, for some reason, quaternions never caught on for relativity.

Also, quaternions are actual numbers, and the right-hand rule falls out of them automatically.

 

[Zombified]

Incidentally, this is one reason I prefer to use quaternions over 4-vectors in Minkowsky space or 4-vectors with imaginary time or 4-vecors with munged operators. Because it's nice to have the interval be real with the more mundane timelike intervals and the interval be imaginary outside the light cone. However, for some reason, quaternions never caught on for relativity.

I never learned the quaternion method. I understand they're useful for dealing with rotations (especially in computer graphics) as well.

I prefer the metric method. That's because the metric is going to appear in GR anyway, and one may as well be prepared for it.

Some books use a metric with opposite sign to what I showed above; then timelike intervals are real and spacelike intervals are imaginary. You get to pick, but you have to be careful which convention a given author uses.

Rereading my second post I see I forgot to mention that Δx₀ = c Δt and Δx_1,2,3 = Δx,Δy,Δz (spatial coordinates.)

Edit to add: I know math makes some people sleepy, but I can't help myself. I love math!

 

[epepke]

I never learned the quaternion method. I understand they're useful for dealing with rotations (especially in computer graphics) as well.

It isn't really taught anywhere that I know of, but it works great.

The quaternions that are useful for rotations (actually, orientations) are unit quaternions. They're nice because they have no gimbal lock and correctly differentiate between a single and double turn. Also, interpolation between quaternions is easy, because you can pick any method you like that is constrained to a 3-sphere. Given two quaternions, it's easy to find an axis and an amount of rotation.

However, I've always been fond of them and wished they could be used more. Quaternions (and octinions, and sedenions, and whatever it is they call that thing with 32 numbers) are algebraically satisfying in a way that vectors aren't.

 

[Diamond]

A great non-mathematical book that explains the RoS and other parts of Relativity is: Relativity Visualized by Lewis Epstein

 

[Refus de Sejour]

Thanks guys. Though I won't pretend to follow all the math, I get the general idea. I'll seriously study the equations to try and see exactly whats going on - it's not going to be easy, considering I pretty much abandoned math when I encountered long division at the age of 14, but I'll give it my best shot.

Re the speed of light being the same in all frames of reference; damn that's weird. I accept and understand that it's the case, it just seems such so unusual - the idea that no matter what speed you're going, c is always going to be the same speed faster than you. Hmmm.

 

[epepke]

Thanks guys. Though I won't pretend to follow all the math, I get the general idea. I'll seriously study the equations to try and see exactly whats going on - it's not going to be easy, considering I pretty much abandoned math when I encountered long division at the age of 14, but I'll give it my best shot.

If I were you, then I'd try to understand it without using the equations. If I were me, which I am, I'd do the same thing. My training was in math, and when I first was taught about SR in high school, it was equation-intensive, but I never really grokked it until I looked at it from a geometric perspective.

First of all, all of the equations in SR come out of a simple right triangle and the Pythagorean theorem, but it's not so obvious when you look at the Lorentz transformations as usually presented, with the t and the t' and all that.

There's also an unfortunate historical accident. The equations were empirically derived quite a long time before Einstein, but the people who derived them were disturbed by how ad hoc they looked. Einstein came up with the beautiful and elegant picture that showed why they had to be that way.

Re the speed of light being the same in all frames of reference; damn that's weird. I accept and understand that it's the case, it just seems such so unusual - the idea that no matter what speed you're going, c is always going to be the same speed faster than you. Hmmm.

This, of course, was the puzzle that made it necessary to figure out a new kind of relativity (as opposed to the old kind that Galileo came up with and Newton took for granted). But that's what the experiments showed.

 

[Diamond]

This, of course, was the puzzle that made it necessary to figure out a new kind of relativity (as opposed to the old kind that Galileo came up with and Newton took for granted). But that's what the experiments showed.

Not just the experiments. Einstein wrote that it was the form of Maxwell's equations of Electromagnetism were the real starting point since the velocity of light is derived from two constants, neither of which are altered by relative velocity.

 

[Zombified]

Yes, the incompatibility between Maxwell and simple Galilean transformations was an important factor. In fact, the ad hoc derivation of the Lorentz transformation that epepke refers to was an exercise in finding a coordinate transformation that would leave Maxwell's equations invariant. Lorentz found the transformation, but had no physical justification for it.

Einstein's peculiarly clever bit was to figure out how to derive Lorentz's transformation from just the assumption of the constancy of the speed of light and geometry, without reference to Maxwell's equations. The argument has purely to do with the operational definitions of length and time interval measurement.

As a result, Maxwell's equations have special relativity built into them for free. In fact, they get a lot simpler written down using four-dimensional space-time vectors instead of the traditional notation.

 

[epepke]

Not just the experiments. Einstein wrote that it was the form of Maxwell's equations of Electromagnetism were the real starting point since the velocity of light is derived from two constants, neither of which are altered by relative velocity.

Maxwell's equations showed that the speed of light did not depend on the speed of the source. It wasn't so clear at the time that it did not depend on the speed of the observer. A luminiferous ether theory seemed that it might have been able to maintain the Galilean xforms.

But experiments were done, the most famous of which was Michelson-Morley, which gave a null result, suggesting that nor did it depend on the speed of the observer.

One of the two Ms, I forget which, never accepted the null result. And there are ether afficionados today. But the Lorentz transforms have been so overwhelmingly supported that this is perverse.

 

[TillEulenspiegel]

Well I always personally liked the Pythagorean approach as the sum of the squares seemed most elegant ( see location line).

MMX MAY have indeed been flawed. the two reasons being that the experiment was done on a 2-axis plane not in 3 dimensions, the other is that the distance traversed by the light beam may have been too small. even tho the maths in place were ~ correct. With new observations it appears that Lambda and the Aether may be worthy of reexamination regardless of what label we give it (DE...). Perhaps Einstein second guessing himself was the wrong thing for him to do .