Nature of the time dimension in 4D space-time

[Mr Peanut]

I read on a website (http://math.ucr.edu/home/baez/physics/Relativity/GR/gravity.html):

"The world we live in consists of four dimensions, the three space dimensions and one that is not exactly time but is related to time (it is in fact time multiplied by the square root of -1)."

In my layman's world, as I think about multidimensional spaces, I envision basis vectors that have the same unit quantities. They all describe direction and distance (in inches say). Is this too narrow? Do we really have three dimensions that are length and another dimension that is some other beast altogether?

Or, is the nature of this fourth dimension indistinguishable from the other three? Is sqrt(-1)T a unit of distance?

 

[espen180]

Maybe they are referring to the metric equation

$$c^2d\tau^2=c^2dt^2-dx^2-dy^2-dz^2$$

The nature of the time dimension is different than the nature of the spatial dimensions in that we are free to move in the spatial dimentions, while our movement in the temporal dimention is dependent on our movement in the spatial ones.

The dimensions all have the same units of meters or seconds, depending on preference.

 

[yuiop]

In my layman's world, as I think about multidimensional spaces, I envision basis vectors that have the same unit quantities. They all describe direction and distance (in inches say). Is this too narrow? Do we really have three dimensions that are length and another dimension that is some other beast altogether?

Or, is the nature of this fourth dimension indistinguishable from the other three? Is sqrt(-1)T a unit of distance?

Personally, I believe time is some other beast altogether.

Let us say we have an event with coords (x1,y1,z1,t1) and another event (x2,y2,x2,t2) then the invariant interval $\Delta S$ between those two events is $$\sqrt{(c^2 \Delta t^2-\Delta x^2- \Delta y^2- \Delta z^2}$$ and it is immediately obvious that the time dimension is treated differently to the space dimensions and therefore is not the same beast. The time dimension is multiplied by c to convert it from units of time to units of length. We could just as easily divide both sides of the equation by c^2 to obtain units of time throughout.

 

[Trenton]

yuiop - you are supposed to have x,y,z and their deltas in light-seconds (if you are measuring t in seconds) hence th c^2 term only applying to the time delta. You can convert distances to time but you would still have time delta being the odd one out as it would then be the only term not divided by c^2.

And most physicists would struggle to conceptualize it including myself.

I accept that is strange as many motorists, particularly in America, think of journeys only in terms of time!

 

[HallsofIvy]

I read on a website (http://math.ucr.edu/home/baez/physics/Relativity/GR/gravity.html):

"The world we live in consists of four dimensions, the three space dimensions and one that is not exactly time but is related to time (it is in fact time multiplied by the square root of -1)."

In my layman's world, as I think about multidimensional spaces, I envision basis vectors that have the same unit quantities. They all describe direction and distance (in inches say). Is this too narrow?

Yes, much too narrow. For one thing, it is perfectly plausible to have a "space" vector in which basis vectors in the "x" direction are measured in feet, in the "y" direction in meters, and in the "z" direction in yards! But when you assert "they all describe direction and distance" you are simply refusing to consider "four dimensions, the three space dimensions and ...". Mathematically, of course, a vector can have any number of dimensions with any units you like. Of course, you have to have some way of "merging" one kind of unit into the other between axes. Many of the peculiarities of relativity can be interpreted in terms of "rotating" the "four dimensional space-time" system.

Do we really have three dimensions that are length and another dimension that is some other beast altogether?

I wouldn't put it that way. We experience that fourth dimension as "time" and I can
see no reason not to call it time. Of course, if we do that, we have to measure "distance", on a "world line" in that "four dimensional space-time continuum" as
$$\sqrt{x^2+ y^2+ z^2- t^2}$$
having a "signature" (in Riemann space terms) of "+, +, +, -" rather than just "+,+,+,+".

Or, is the nature of this fourth dimension indistinguishable from the other three? Is sqrt(-1)T a unit of distance?

No, its not "indistinguishable from the other three"- we do distinguish time from space! And, yes, sqrt(-1)T is a "unit of distance"- it is that sqrt(-1) that distinguishes it.

 

[Passionflower]

The time an observer takes between two events is a curve in spacetime not a dimension of spacetime.

In other words the accumulated time on a clock is total length of the path traveled through spacetime

 

[A.T.]

The time an observer takes between two events is a curve in spacetime not a dimension of spacetime.

In other words the accumulated time on a clock is total length of the path traveled through spacetime

You should specify that you mean the Minkowski Spacetime, where the coordinate time is a dimension.

There is nothing wrong with interpreting proper time as a dimension, so that the coordinate time becomes the path integral. In this geometrical interpretation the nature of the (proper)time dimension is very much like of the spatial dimensions, and the metric is Euclidean "+,+,+,+" (and not pseudo-Euclidean like in the Minkowski Spacetime)

 

[Passionflower]

There is nothing wrong with interpreting proper time as a dimension, so that the coordinate time becomes the path integral. In this geometrical interpretation the nature of the (proper)time dimension is very much like of the spatial dimensions, and the metric is Euclidean "+,+,+,+" (and not pseudo-Euclidean like in the Minkowski Spacetime)

And you can succeed doing that in curved spacetime?

 

[DrGreg]

There is nothing wrong with interpreting proper time as a dimension, so that the coordinate time becomes the path integral.

Actually, in my humble opinion, there's rather a lot wrong with that, except in the special case of a universe containing of a single particle. In the absence of gravity, space-time is a vector space, but space-propertime isn't. There is no one-to-one mapping between events in space-time and the "events" in space-propertime (due to the twins paradox).

 

[A.T.]

And you can succeed doing that in curved spacetime?

Sure, the curved space-(proper)time is also curved.

See chapter 6 of this:
http://www.relativitet.se/Webtheses/lic.pdf
for a derivation of the Schwarzschild metric in space-(proper)time.

Here a visualization of an 1+1 embedding:
http://www.adamtoons.de/physics/gravitation.swf

 

[Passionflower]

Sure, the curved space-(proper)time is also curved.

See chapter 6 of this:
http://www.relativitet.se/Webtheses/lic.pdf
for a derivation of the Schwarzschild metric in space-(proper)time.

Looks like an interesting document, if I can find the time I will read it.

 

[A.T.]

Actually, in my humble opinion, there's rather a lot wrong with that, except in the special case of a universe containing of a single particle.

It works fine with multiple particles. But it's a different type of diagram, which has to be interpreted differently than the Minkowski diagrams.

Here the twins in both diagrams:
http://www.adamtoons.de/physics/twins.swf
Unlike in the Minkowski diagram, you can see the age difference directly as a coordinate offset in the space-propertime diagram.

What you don't see well in the space-propertime diagram is the meeting between two objects, as an intersection of world-lines. But that is a logical consequence of having a time dimension on the same footing as the spatial dimensions: In a purely spatial diagram an intersection of two paths doesn't imply a meeting.

In the absence of gravity, space-time is a vector space, but space-propertime isn't.

Not sure what you mean. It seems to me that it is still a vector space, just with a vector component that corresponds to different observed quantity than in Minkowski spacetime.

There is no one-to-one mapping between events in space-time and the "events" in space-propertime

True, but that doesn't make one of the interpretations wrong.

 

[Hesh]

This may sound dumb but time doesn't exist time is not naturally occurring. Time is something we made up like for example a day 24 hours, hours don't exist its just the rotation of the Earth there is no natural unit of time like what is a year just a label we put on the Earth circling the sun its not time. Its hard to explain but time is non existent we only have a hard time beliving time doesn't exist because we invented it so long ago that time is just hard wired into our brains and makes it seem possible that time is a real thing but it isnt. Time is just a label we gave stuff to explain it easier. You dig?

 

[cosmik debris]

Those diagrams are terrific A.T. Did you make them?

 

[Passionflower]

AT, very interesting but I still have doubts wrt general applicability. I am hand waving here but if we cannot Wick rotate the spacetime I foresee problems, e.g. non-stationary spacetimes, does that make any sense?

Perhaps I am wrong.

 

[bobc2]

I read on a website (http://math.ucr.edu/home/baez/physics/Relativity/GR/gravity.html):

"The world we live in consists of four dimensions, the three space dimensions and one that is not exactly time but is related to time (it is in fact time multiplied by the square root of -1)."

In my layman's world, as I think about multidimensional spaces, I envision basis vectors that have the same unit quantities. They all describe direction and distance (in inches say). Is this too narrow? Do we really have three dimensions that are length and another dimension that is some other beast altogether?

Or, is the nature of this fourth dimension indistinguishable from the other three? Is sqrt(-1)T a unit of distance?

There is a concept of four purely spatial dimensions that is fully compatable with special relativity (constant light speed, Lorentz transformations, etc.), and I'll see if I can describe it properly--if I slip someone can jump in and clarify the 4-D space picture. In that context some aspect of a given observer moves along his world line at the speed of light. Time is simply a parameter, exactly in the sense that we use parametric equations for y(t) and x(t) to describe the motion of a projectile in free flight. A typical mechanical clock keeping time along the 4th dimension is afterall just a periodic spiraling of the tip of the clock arms. That is, the clock is a four-dimensional structure with extremely small 3-D widths compared to the long path length along the clock's 4th dimension. The primary reason the 4th dimension is special is because the 4-dimensional structures populating the 4-D space have extremely long lengths (billions of miles long?) along the 4th dimension compared to sizes in X1, X2, and X3. But, that certainly is not any basis for calling it a different beast. We see very long strings in 3-D and don't ascribe anything unusual about the length of the string (it could be a mile long and 1mm diameter).

Time just passes with motion along the 4th dimension, exactly as it does driving along an interstate. You can put time markers along the highway and call it a time dimension if you wish, but that does not rob it of its spatial nature.

And you certainly do not need an ict. Just X4 = cT. It's c because the motion along the 4th dimension is always at the speed, c. If you drove at constant speed along the interstate then you could say that X1 = vt.

The term spacetime as some kind of mixture of space and time has been seriously mismanaged in this view.

 

[A.T.]

AT, very interesting but I still have doubts wrt general applicability. I am hand waving here but if we cannot Wick rotate the spacetime I foresee problems, e.g. non-stationary spacetimes, does that make any sense?

Not sure about the Wick rotation, but the space-propertime is already Euclidean, so you can apply Euclidean solutions directly, if you want to solve problems geometrically in it.

However, I'm not saying that it is generally (or at all) sensible to use space-propertime for calculations, and the OPs question is not really about simplifying math. Space-propertime is mainly a visualization tool, that shows some things better (and some things worse) than Minkowski-Diagrams.

I think it is good to introduce both diagrams to demonstrate that Minkowski-spacetime is just one possible geometrical interpretation of the Relativity math, and not some unique truth. You will find many threads here like this one, where people try to interpret some deeper philosophical meaning into the opposite sign of dt^2 in the Minkowski-metric, which is just a convention.

 

[Trenton]

Not sure the minus is just a convention in Minkowski spacetime. I thought it was the whole point of it. If you consider the oft quoted return trip to Alpha Centuri example, the astronaut ages less than those on Earth who didn't take the trip. The x,y and z are the same (as it is a return trip) so the t coord is the only factor. If it were positive the astronaut would age more.

A much more vexing convention is the minus in gravitational potential. I wonder if there might be a deeper meaning here.

Take a simple system of a single planet in an eliptical but stable orbit around a star. What is the total mass/energy of this system? Obviously this would include the masses but also, correct me if I am wrong, the energy of the orbit. The orbit is constantly converting potential energy to kinetic energy and back again. As kinetic energy is positive, so must be potential energy - otherwise the total energy of the system would vary.

That said, gravity seems to be an energy creator or maybe it facilitates energy borrowing from a vacuum causing space to expand somehow! If we do consider grav pot as part of the total mass/energy of a system, how does one assess the total mass energy of a system consisting of a black hole eating a star? If there really is a zero sized singularity at the center, infalling matter would realize 'infinite' kinetic energy. Even with a finite but very small entity at the center of the hole, the KE would outweigh the total mass of the system with ease.

I put infinite in single quotes because I see this as a word that paints 1000 pictures, mostly Salvador Dalis. It seems to me there is no such value. We can say that the path of an electron around a nucleus is an infinite path just as we can say there are an infinite number of integers - but there never comes a time when it can be said the electron has completed infinity orbits or one has counted to infinity.

The designers of computers were right when they decided that a divide by zero was an error. I am wondering if perhaps it is better to follow suit and treat all solutions to equations in which terms go to infinity as invalid. That would get rid of the singularity and allow only an asymtope to a singularity - which is what I strongly suspect is the case anyway.

 

[Trenton]

Just as a 'corrective' footnote to the above. I am aware that with infinity being disallowed on the grounds that it is unreachable and a non-value, this does invalidate the idea of history having no origin and implies that there must have been a point of creation, a dawn of time.

An origin is something I am very uncomfortable with for the following reasons. As a point of faith, the laws of math and physics are timeless. For example 1 + 1 = 2 would be true regardless of time and regardless of wether or not there were any entities in existence to be added. The speed of light in vacuo would be as it is today even if there were no photons and no vacuum for them to move through. And as another point of faith, the universe (not limited to that which can be observed), is a consequence of the laws of math and physics and therefore also be timeless - an endless series of big bangs and crunches (or inversions) stretching infinitely in the past.

I believe this is a fairly well known conundrum - except that I have no idea if a name has been given to it. If anyone knows what it is called so I can google it, I'd be most grateful

Best regards - Trenton Maiers

 

[Dale]

Not sure about the Wick rotation, but the space-propertime is already Euclidean, so you can apply Euclidean solutions directly, if you want to solve problems geometrically in it.

No, it is not Euclidean. Euclidean implies that it is a metric space, but it is not even a topological space, let alone a metric space. That was Dr. Greg's point above. The fact that a single event maps to multiple "events" in space-propertime also means that it is not diffeomorphic to R4. Your drawings and animations are great, as always, but you cannot take them too far. Specifically, you cannot take them as a 4D Euclidean metric space.

 

[A.T.]

No, it is not Euclidean. Euclidean implies that it is a metric space,

The metric in space-propertime is given by:
dt^2 = dx^2 + dy^2 + dz^2 + dtau^2

but it is not even a topological space,

I'm not familiar with the criteria there, but would like to learn why not?

The fact that a single event maps to multiple "events" in space-propertime also means that it is not diffeomorphic to R4.

The correspondence of physical events to single points in R4 is a concept specific to Minkowski spacetime. Space-propertime has a different correspondence. Neither one is wrong or right.

 

[Dale]

The metric in space-propertime is given by:
dt^2 = dx^2 + dy^2 + dz^2 + dtau^2

No, it is not a metric because it is not a metric space. The same point can have more than one proper time so one point has a non-zero and non-unique distance to itself. For example, what is the space-propertime interval between some decay event and itself? You cannot answer it because the space is not a metric space.

Space-propertime has a different correspondence. Neither one is wrong or right.

I never said it was wrong or right, it is just not a metric space so it is not Euclidean. I don't see anything wrong with your diagrams, in fact, I enjoy them, but you shouldn't say that they are something they are not.

 

[A.T.]

The metric in space-propertime is given by:
dt^2 = dx^2 + dy^2 + dz^2 + dtau^2

No, it is not a metric because it is not a metric space.

I think you have it backwards here. It is the metric that makes the metric space a metric space, not the other way around.
From http://en.wikipedia.org/wiki/Metric_space :

In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.

I just defined the distance dt in space-propertime above. That makes it a metric space.

The same point can have more than one proper time

No, a point in space-propertime has a unique proper time coordinate.

For example, what is the space-propertime interval between some decay event and itself?

The idea of physcial events corresponding to unique points in some space is a Minkowski concept. The points and intervals in space-propertime have a different physical meaning. Your question doesn't make sense, because it applies the Minkowski idea of events to a different concept.

In this trivial case however the space-propertime interval, being dt, is zero.

You cannot answer it because the space is not a metric space.

No, that is not the reason. Metric space is a purely mathematical concept that simply requires some distance measure for the elements of the set. The interpretation of elements/distances as physical events/quantities is physics, and not relevant for the mathematical property of being a metric space.

 

[Dale]

In this trivial case however the space-propertime interval, being dt, is zero.

No, it is not. Suppose the decay event occurs at the reunion of the twins in your diagram here: http://www.adamtoons.de/physics/twins.swf The decay event then has one x coordinate but two different t coordinates. Because you can make any number of twins meet at the same event, all with different proper times, a single physical event can have an infinite number of proper-time coordinates.

You quoted Wikipedia re: metric spaces. You missed the important part of what you quoted "a notion of distance (called a metric) between elements of the set is defined". Assuming that you want to do physics then the elements of the set should be physical, and there is no notion of distance between physical occurences here as shown in your own drawings.

Of course, if you are not interested in physics you can make an elf-unicorn space and write a Euclidean metric on that space which is entirely disconnected from physics. If you wish to insist that your space propertime has physical meaning and relates to physics then I don't see how you can claim it is a metric space. If I were you, I would rather give up the metric than the connection to physics. As a connection to physics it is pedagogically interesting, as a metric space it is not.

 

[A.T.]

No, it is not. Suppose the decay event occurs at the reunion of the twins in your diagram here: http://www.adamtoons.de/physics/twins.swf The decay event then has one x coordinate but two different t coordinates.

No, the decay event has a unique tau coordinate: the proper time interval of the decayed object, which is independent of the tau coordinates of the twins.

But I agree with your general point you are tying to make here which is:

a single physical event can have an infinite number of proper-time coordinates.

You quoted Wikipedia re: metric spaces. You missed the important part of what you quoted "a notion of distance (called a metric) between elements of the set is defined". Assuming that you want to do physics then the elements of the set should be physical,

I guess that is the main point of disagreement. As I stated before, I don't see how the connection to physics is relevant for purely mathematical properties like being a metric space.

and there is no notion of distance between physical occurences here as shown in your own drawings.

The distance in space-propertime is the coordinate-time interval which is a physical quantity, which could be interpreted as a distance.

Of course, if you are not interested in physics you can make an elf-unicorn space and write a Euclidean metric on that space which is entirely disconnected from physics.

It is not disconnected from physics, it just has a different correspondence to physical concepts.

If you wish to insist that your space propertime has physical meaning and relates to physics then I don't see how you can claim it is a metric space.

Again, I don't see how the connection to physics is relevant for purely mathematical properties like being a metric space.

If I were you, I would rather give up the metric than the connection to physics.

I don't think I have to choose between the two.

As a connection to physics it is pedagogically interesting, as a metric space it is not.

That might be true. It might not be pedagogically interesting to use the metric space property of space-propertime, because it doesn't fit with the intuitive understanding of physical distances. But that doesn't mean that space-propertime is not a metric space.

If propertime can be interpreted as a distance "traveled" by individual objects (as is the case in Minkowski space time), then why cannot coordinate time be interpreted as a distance "traveled" by all objects (as is the case in space-propertime)? I don't see how the later is fundamentally different or less intuitive than the former.

 

[nada1]

If this is old hat it fits. The simplest answer to time's nature is to define it exactly as its three fellows, ie distance, unit the meter. Feynman suggested this but did so only for temporary convenience. But if you do say that the unit of time is the time required for light to travel one meter, then the consequences for a unit system are lovely. Energy and mass identical, momentum segues into energy, fruit fairly flies off the tree. Back to some of the responses, I read a few and wholeheartedly agree with Feynman's answer to somebody asking him to explain time. He said it's way too complicated for me to do that for you. Or words to that effect. Anyway, basing a unit system on the presumption that time is, well, "metric" has so many obvious advantates for the student of electromagnetism that mention here surely is old hat. It fits because I'm old.