Understanding the Interval Metric

[FayeKane]

Okay, tlet me know where I trip up:

The everyday concept of "distance" allows an event to occur in only one place at one time and nowhere else.

But because of the interval metric's signature (it has both positive and negative terms), the definition of four-dimensional distance implies that there is zero distance between a point in space-time and an an infinite number of other points called the "null cone."

My understanding is that the four-distance is called a "pseudo metric" because it is not transitive. That is, if the 3-distance distance A to B is nonzero and the 3-distance B to C is nonzero, the four-distance between A and C may be zero even if A and C are separated in both time and space (in equal amounts).

To clarify, I mean that an event at (3-dimensional) location A at time t also exists at a different spatial location B at time t+n, because if you add the same number to both the space and time coordinates of A, the value of the metric (the distance AB) remains unchanged.

Is that correct?

If so, it would seem to imply that a point mass which winks into existence for an instant at time t, continues to exist in future times, but in multiple locations. Specifically, as time unfolds, the event, (which happened in the past) continues to exist in the future as a sphere expanding outwards at c, and in fact, this is what the four-dimensional null cone looks like in three dimensions.

...and is THAT correct?

So when the null cone of this mass event intersects (passes through) a different event (say, a human at exactly 5 pm), the absolute distance between the instantaneous mass event and that human is zero (momentarily).

Is that correct too?

If so, then isn't the passage of the null cone through the human identical to (i.e., just another name for) the gravity wave produced by the instantaneous mass event?

To give a concrete example, this appears to mean that when you feel the tug of the Moon's gravity, what you're feeling is the physical moon itself touching you, but in the past.

If all this is true, I think it has very interesting and possibly important implications which involve my other question (in another message thread) about mass being a condition of space.

But I want to know if I made a mistake in my understanding before I point them out.

Thank you,

-- faye

 

[javierR]

t
^
| \      /
|  \    /
|   \  /
|    \/  --------> x

The diagram above shows the light cone on a t vs. x plot. The V is the cone itself. Take the point of the cone to be x=0 and t=0 and let some event occur there, like a light flashes. Along the lines of the cone, the 4-separation $$s^{2}=(ct)^{2}-x^{2}=0$$, which represent the light of the flash event traveling away (at light speed c). Anything inside the cone has $$s^{2}>0$$...things of this sort must be moving at a speed v less than light since then x=vt and $$(ct)^{2}>(vt)^{2}$$. Such objects have a non-zero mass like the moon. Outside the cone represents stuff going faster than light (and there's no evidence of any such thing in our model of things). I left out the flipped cone expanding downward from the vertex, which represents things in the past.

The light cone of an event is not the actual event or object expanding out...it's just a coordinate system for spacetime and a way of cutting into different significant regions, from the point of view of a particular observer (an inertial reference frame). Anything that has a path along the cone lines must be traveling at light speed.

Consider this example. We stand next to a lightbulb that we turn on at t=0, and we say we and the lightbulb are at the same x-position "x=0". Let's say the light from the bulb moves away from us toward someone else at x=5 feet. Then the light would follow the line on the right side of the cone in the diagram. But at time t=anything after 0, our position and the light bulb are still at x=0. So we and the bulb have moved directly up the middle of the diagram from the vertex, inside the cone. Each object, the light and each event has a specific *point* in the diagram.

 

[FayeKane]

> The light cone of an event is not the actual event or object expanding out...it's just a coordinate system for spacetime

Yes, but it's more than that. The phrase "null cone" refers to a set of points in spacetime, yes. But the nexus of those points is also the physical location, in four dimensions, of a single event that occurred at time t and location x,y,z relative to some arbitrary coordinate definition.

It seems that the metric implies, indeed, mandates, that the gravitation produced by the object in the present IS the object in the past. This may sound like mere renaming of things, but I think it's more than that. It means that for any two events happening at different places and times, there exists a third place and time at which both events exist at the same absolute location (interval from an arbitrary 4D origin).

Why might this be important? Well, to give just one example, it MAY shed light on quantum entanglement when you see things this way. A quantum event occurs with an indeterminate spin at time T at location L, which you can call (T,L) where L= (x,y,z) in some arbitrary coordinate system. The event continues to exist as a light cone at all points (T+n, L+m) for which n=m, that is, when the space distance and the time distance are the same (i.e. on the null cone).

It would seem that all events occurs at all possible places at some time, and at some time in all possible places, and that we call an event a "particle" only at the place where the space and time distances from the event are are both zero. However, at every other place and time equally distant from the event, we call the same event a "wave".

The quantum event interacts with the nearest object in 4D space, where the spin becomes determined. But since all null cones intersect, the now-defined spin can also be detected at larger 3D distances at later times, because all those places and times are literally at zero distance from the original event.

Entanglement only seems to be "spooky action at a distance" when you pay attention to distance in space and ignore

We see certain interactions as mysterious and confusing because everything with mass local to us moves through time at the same rate as us, and so we (perhaps arrogantly) assume:

-- that we are at rest when we are not (relative to a 4D coordinate system)

-- that events we interact with occur only at the place and time where we just happen to be

-- that light sometimes acts like a particle and sometimes like a wave.

-- that "waves" are generated by an event and move at c (like gravitation and radiation), when in fact they are actually particles sitting still in spacetime (relative to a 4D coordinate system) and it is we who are moving through them at c.

Stationary phenomena only become "waves" when we pretend we are at rest. But seeing things as they are makes understanding them WAY simpler.

The thing is, none of this should be news. It's not even opinion; it 's all, every word of it, merely observations of the consequences of the signature of the interval metric-- the main observation being that distances from an event in time and space cancel if they are equal.

You may grudgingly acknowledge that, yes, in some formal sense, events occur at multiple places and times, but that this is merely a mathematical formalism/curiosity that doesn't help us understand anything, which physicists once believed about the advanced solutions to Maxwell's equations.

But I assure you that it isn't--particularly when you toss in Wheeler and Feynman's explanation of why we can interact with with events in our future but not our past (absorber theory), as well as the well-known observation that time is imaginary space (though it's usually phrased the other way around). That is, a line extending into the time direction is no different than a line extending into a spatial direction, except that it has a negative length. That is also a consequence of the signature if the interval metric.

I'll stop here, but before calling me a crackpot, please at least note that I haven't asserted anything new, and in fact, I haven't asserted anything at all; I've only just pointed out what we already know but just don't usually think about because we like to pretend that we're stationary. I want to emphasize that there is no new "theory" here and that, since it is so simple, I'm certain thoughtful people noticed it long ago (like the 1920's).

The ONLY thing I'm saying is that this seems to be more significant than is commonly realized because I've never seen anyone talk about it--and I have only barely touched on the implications for simplifying the understanding of interactions between mass and things moving at c.

Okay, tear me to shreds. (If you can, then I actually want you to--seriously). In particular, I suspect that I'm very likely misunderstanding entanglement. But what am I forgetting? What am I believing that isn't true?

Or is all this, while true, merely insignificant?

I will happily prove that last assertion false, even if none of this really sheds any light on entanglement.

-faye