# The Nature of Time: Should It Be Considered a 4th Dimension?

Because i received an infraction for posting my explanation for why time shouldn't be considered a 4th dimension, instead, i would like to address the issue as a question instead of an assertion.
Why should time be considered as a 4th dimension?

cheers,

The simple explanation is that relativity theory has the concept built into it. For example when changing inertial frames in special relativity, time and space change together.

i would add that time is still the "funky" 4th dimension. it has a different sign attached to it in that $\eta$ metric. there is also no "arrow of space" that i am aware of (except i guess in black hole). t is not qualitatively identical to x, y, or z, the latter 3 which are qualitatively identical.

Why should time be considered as a 4th dimension?
In Galilean space-time you could consider time the fourth dimension, but in relativity time is not the fourth dimension!

In relativity, the relative measure of time between any two observers is related to their relative orientations in space-time.

The only difference between an Euclidean 4-dimensional Galilean space-time and a Minkowski space-time is that the rotations work differently.

Both the Galilean E4 and the Lorentz O(1,3) make a 10-dimensional symmetry group.

So actually there is not that much of a difference!

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In Galilean space-time you could consider time the fourth dimension, but in relativity time is not the fourth dimension!

In relativity, the relative measure of time between any two observers is related to their relative orientations in space-time.

The only difference between an Euclidean 4-dimensional Galilean space-time and a Minkowski space-time is that the rotations work differently.

Both the Galilean E4 and the Lorentz O(1,3) make a 10-dimensional symmetry group.

So actually there is not that much of a difference!

The Lorentz metric is invertible, whereas the Galilean metric is not invertible.
The Lorentz metric yields three classes of vectors [spacelike, timelike, and null], whereas the Galilean metric yields only two classes...with spacelike and null coinciding.
The Lorentz group and the Galilean group have different sets of eigenvectors.

Time has the property that when scaled by c, (i.e., multiplied by the velocity of light) there results the dimension of distance. The other three dimensions (X, Y and Z) are also distances. When Minkowski unified time and space he scaled the temporal distance as "ct" This isn't something arbitrary - If you can think of it metaphorically as all objects always moving at a constant rate equal to c - then you traveled 3 x 10^8 meters in one second w/o even moving from your chair. This becomes a useful tool that leads to the invariance of the interval - all objects move at c, therefore while you are sitting at rest, you are traveling in time a distance ct whereas an object in motion (with uniform relative velocity v) would travel a distance
[(ct')^2 + (vt)^2]^1/2 where vt is the spatial component and ct' is the temporal component, both of which have units of distance. Rather than thinking of c as local light speed, think of it as the rate of expansion of the Hubble sphere - you like everything else are being effortlessly carried along a distance ct each second

If you can think of it metaphorically as all objects always moving at a constant rate equal to c - then you traveled 3 x 10^8 meters in one second w/o even moving from your chair. This becomes a useful tool that leads to the invariance of the interval - all objects move at c, ...

The above (e.g. a statement like "all objects move at c") would only apply to objects with nonzero rest-mass. That is, the above does not apply to light.

Why should time be considered as a 4th dimension?

Because the position of an event cannot be described except at a specific time and the time of an event cannot be described except at a specific position. (Albert Einstein)

Furthermore, using a space-time continuum also resolves certain problems in 19th Century physics, such as the invariance of the speed of light in the Michelson-Morley experiment and it makes Maxwell's equations invariant between different inertial frames of reference.

Garth

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It must be stressed here that time being the 4th dimension is coordinate time. This is very different from the time that clocks will measure (the so-called proper time): that is proportional to lengths of paths in spacetime and can involve as much space as they do time.

Given a 3D-Euclidean space, it does make sense to define a new, fourth dimension that can be defined as perpendicular to that space. That new dimension is associated with the "time" associated with that given 3D space. Mystical as this may sound at first, this construction is used in describing the evolution of 3D systems in Galilean physics... however, its interpretation as a spacetime geometry is not as familiar as Minkowski spacetime.

thanks all, that's quite a complicated variety of answers to chew on...
so far none of these answers tackles the fundamental nature of time as compared to that of space. Thats understandable, since time is something we thing with and only perilously about...

The simple explanation is that relativity theory has the concept built into it. For example when changing inertial frames in special relativity, time and space change together.
Perhaps I'm misinterpreting this explanation, but it seems to me you are saying: "Time should be treated as a 4th dimension because we treat it as a 4th dimension." That seems a bit circular to my question. I don't see your example as an explanation either, since it leaves room to assume the reason time and space change together in inertial frames could be due to the way they are arranged in an equation, not what's indicated by nature.
thanks,

i would add that time is still the "funky" 4th dimension. it has a different sign attached to it in that $\eta$ metric. there is also no "arrow of space" that i am aware of (except i guess in black hole). t is not qualitatively identical to x, y, or z, the latter 3 which are qualitatively identical.
what does the sign indicate? You have pointed out how different time is to a spatial dimension, why then should it be treated as a dimension? Perhaps this is a question of semantics, but if it is called a "4th dimension", that implies it is built upon the former 3 dimensions, just as the Y-dimension is only thus in reference to the X- dimension, etc.
I am interested in hearing more about the arrow of time in a black hole.
thanks,

I would be interested in hearing what you think the word "dimension" means.

what does the sign indicate? You have pointed out how different time is to a spatial dimension, why then should it be treated as a dimension? Perhaps this is a question of semantics, but if it is called a "4th dimension", that implies it is built upon the former 3 dimensions, just as the Y-dimension is only thus in reference to the X- dimension, etc.
The question is: "How are events connected up in space and time?"

Take an infinitesimal interval ds separating two events:

Now the separation in 2 dimensions is given by Pythagoras' theorem:

dx2 + dy2 = ds2

now expand it to 3 dimensions:

dx2 + dy2 + dz2 = ds2

now expand it to 4 dimensions where the fourth dimension is time; do we get

dx2 + dy2 + dz2 + dt2= ds2?

Unfortunately this isn't correct, there are two things wrong with it.

First there is a question of units, we have added the squares of 'apples' and oranges'! We need a conversion factor to convert time into distance, such a factor has the dimension of velocity, so call it c, we have to multiply dt2 by c2.

Secondly in SR we do not add the time2 but subtract it. This changes the 4D space we are constructing from Euclidean space to Minkowski space. You may ask why do we do this, the first answer is because that is the way the world works, and this approach has been verified in all the experiments that verify SR.

In Minkowski space the maximum velocity is c, massive objects can only approach c asymptotically and massless objects such as photons can only travel in vacuo at c, so c is the speed of light in vacuo.

We now have:

dx2 + dy2 + dz2 - c2 dt2= ds2.

This is called the Minkowskian metric and accurately describes the behaviour of objects with clocks and rulers moving relatively to each other at high speed.

The result of this construction of a 'space-time' continuum out of space and time is that time is seen to be a dimension like the other three but with a difference. It bears the same mathematical relationship to them that the Imaginary numbers do the the Real. The fact that if time is a dimension then it is not exactly the same as the others is intuitively self-evident.

I hope this helps.

Garth

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It's too bad that there are so many 'definitions' of what 'time' 'means' to fit each theory---and how it is incorporated at 'what' level of those theories.

maybe time will tell which times are tales

That really isn't true, rewebster. To scientists and engineers, this issue is unambiguous.

We now have:

dx2 + dy2 + dz2 - c2 dt2= ds2.

This is called the Minkowskian metric and accurately describes the behaviour of objects with clocks and rulers moving relatively to each other at high speed.

The result of this construction of a 'space-time' continuum out of space and time is that time is seen to be a dimension like the other three but with a difference. It bears the same mathematical relationship to them that the Imaginary numbers do the the Real. The fact that if time is a dimension then it is not exactly the same as the others is intuitively self-evident.
Out of curiosity Garth. why do you call this fourth dimension time?

To me time is measured by the metric, not by the t coordinate.

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Out of curiosity Garth. why do you call this fourth dimension time?

To me time is measured by the metric, not by the t coordinate.
If there is no motion when moving between the two events then dx, dy and dz are all zero. Then
ds2 = -c2dt2 and the space-like interval is i x time. One could write everything in terms of the time-like interval d$\tau$:

dt2 - c-2dx2 - c-2dy2 - c-2 dz2 = d$\tau$ 2

in which case if dx, dy and dz are all zero then d$\tau$ is dt, the interval is time in the rest frame.

Using this form of the metric $\tau$ is the proper time or time-like space-time interval between the two events.

Garth

If there is no motion when moving between the two events then dx, dy and dz are all zero. Then
ds2 = -c2dt2 and the space-like interval is i x time. One could write everything in terms of the time-like interval d$\tau$:

dt2 - c-2dx2 - c-2dy2 - c-2 dz2 = d$\tau$ 2

in which case if dx, dy and dz are all zero then d$\tau$ is dt, the interval is time in the rest frame.

Using this form of the metric $\tau$ is the proper time or time-like space-time interval between the two events.

Garth
I fully understand what you are saying but I don't look at it that way at all. To me space-time is a frame independent representation of reality.

In the case you mention, the proper time vector happens to be pointing in the same direction as the t axis. This is the trivial case.

But clearly, in a more general case, where we have several objects that are in relative motion with each other, we immediate see that not all the directions of the proper time vectors point towards the t axis.

Then if we analyze this situation a bit further we can see that the t axis does not measure time but instead is a means to describe relative angles between the individual proper time vectors.

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I like this statement on wiki:

Thus, similar to definitions of other fundamental quantities (like space and mass), only the units of time measurement are defined in science, not time itself.

http://en.wikipedia.org/wiki/Time

I think that when most people wonder about a fourth dimension, they're thinking of a fourth spatial dimension. Time is of course a temporal dimension so naturally time as the fourth dimension comes as an unexpected answer.

But what, really is the significance of this? No one is really claiming that time is a fourth spatial dimension. But it so happens that the math works out quite well if we treat it in a similar way to the spatial dimensions. I think the bottom line is that time is being treated as a fourth dimension only because of the similarity between separate relativistic transformations of space and time.

So even though we treat time and space similarly in the mathematics, they are indeed inherently different quantities.

I like this statement on wiki:

Thus, similar to definitions of other fundamental quantities (like space and mass), only the units of time measurement are defined in science, not time itself.

http://en.wikipedia.org/wiki/Time

Although we all have a psychological experience of time and its 'passing', as physicists we can only talk about time as something we measure with a clock.

The clock measures units of time defined by some regular physical process such as the vibration of a caesium atom in an atomic clock.

MeJennifer Everything is relative.

The space-time diagram is drawn in the instantaneous rest frame of the 'first' observer, other observers are on time-like world-lines that are inclined to the 'first' time axis, that is in the rest frame of the first observer's measuring its proper time.

These other moving observers' instantaneous 4-velocities make different angles to each other and the diagram's time axis, however the same situation can be drawn in the rest frame of another observer. In this case the first observer's 4-velocity is now inclined.

Garth

The space-time diagram is drawn in the instantaneous rest frame of the 'first' observer, other observers are on time-like world-lines that are inclined to the 'first' time axis, that is in the rest frame of the first observer's measuring its proper time.

These other moving observers' instantaneous 4-velocities make different angles to each other and the diagram's time axis, however the same situation can be drawn in the rest frame of another observer. In this case the first observer's 4-velocity is now inclined.
I presume you are not saying that each object has his own space-time.

If one has a collection of objects in relative motion in space-time, we readily can see that the t dimension does not represent time. Instead, the metric describes time.

Do you understand my point?

Anyone that has to use 'time' , 'time as/in measurement' , 'measurement of movement' , and 'measurement using/as time' , or any variation of them HAS to have a concrete 'definition' or standard of those as it applies to the field/theory/experiment they are doing (an applied application, such as a researcher, experimenter, etc.) as it requires them to do so.

From the same wiki 'time' page:

"Time has long been a major subject of science, philosophy, and art. The measurement of time has occupied scientists and technologists, and was a prime motivation in astronomy."

If someone was 'into' a Newton aspect, they'd hopefully use some definition of time that Newton ascribed to; relativity uses 'spacetime'; etc.

When thinking about the fundamental aspects of time (what is time? , Is spacetime the correct 'thinking' about 'time'? and Can anyone come up with a 'definition' for 'time' to work with all theories?), I think is coming in under the idea of the 'The Philosophy of the Physics of Time'.

To me, this is where some discussions run into problems:

Philosophy of the ideas of Physics and the various theories (theorists) vs. the ideas of the 'accepted' principles of the various Physics Theories (applied physics)

It seems to me there's even a subset under the Theorists:

Challenging/questioning a theory (philosophy of the theory) vs. Defending/explaining the theory (knowledge of the theory)

Time to an applied physicist may be accepted as 'the measurement of time'; but, Time (to me) is still something that is still not completely explained/ doesn't have a complete explanation.

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I agree with Stainsor in some ways that whoever first attributed the word 'dimension' with the word 'time', altered the 'definition' of 'time'.

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Help me out - who said: " I know what 'time' is until someone asks me, and then I don't know. Was it Eddington, Feynman ...

I think that time is a dimension because it has a relationship with the "other" 3 dimensions that we know, like the 3rd dimension has a relationship with the 2 previous ones. I can imagine also that we are in a "free fall" in this 4th dimension, probably a C. Moving fast in the 3D dimension world (relatively) affect time (relatively), does the time affect the 3D dimension ? i believe so, and i believe that the "expansion" of the universe is the result of the "time effect" on the 3D dimension world, and finaly, i picture (but this is my thinking only) that MASS is the friction to the time flow...

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I presume you are not saying that each object has his own space-time.
Each observer makes their own foliation of space-time into space and time.
If one has a collection of objects in relative motion in space-time, we readily can see that the t dimension does not represent time. Instead, the metric describes time.
The metric is written in a specific coordinate system of one inertial observer, or if that observer is accelerating, it is written in the osculating coordinate system of that observer at one particular event. The t-dimension of a space-time diagram represents the proper time of that observer as it is drawn in their (momentary) rest frame. The interval integrated along another object's world-line between two events measures the proper time between them, it is frame-independent.

Geometric objects, such as such intervals, are frame-independent, however each observer has to make measurements of them from one particular frame of reference or another, i.e. in the coordinate system of their own rest frame.

Does that help?

yogi
What then is time? I know well enough what it is, provided that nobody asks me; but if I am asked what it is and try to explain, I am baffled.
St. Augustine of Hippo (He was a bishop in N.Africa). Confessions XI 14 (AD 354-430)

In that same chapter he also said , as a prayer to God, amongst other sayings:
It is therefore true to say that when you had not made anything, there was not time, because time itself was of your making.
( i.e. time 'began' when the universe did - not bad for the fifth century!)
and
How can the past and future be when the past no longer is and the future is not yet? As for the present, if it were always present and never moved on to become the past, it would not be time but eternity.

But I won't go on otherwise this Thread will be moved to Philosophy.

Garth

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MeJennifer said:
I presume you are not saying that each object has his own space-time.
Each observer makes their own foliation of space-time into space and time.
Clearly I am not talking about foliations here Garth. I am talking abour space-time.

I asked you why you think that the t dimension in space-time represents time, then I provided arguments as to why that view is generally incorrect except for the trivial case.

It seems, however, that you are not interested in this discussion.

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Clearly I am not talking about foliations here Garth. I am talking abour space-time.

I asked you why you think that the t dimension in space-time represents time, then I provided arguments as to why that view is incorrect except for the trivial case.

It seems, however, that you are not interested in this discussion.
I could say "likewise", but I would not be so rude.

I too am talking about space-time.

When we talk about 'time' we are talking about something that is measured by a clock carried by a particular observer, i.e. in a particular foliation of 'space-time'.

When that observer makes measurements of objects moving between two events we recognise that the time between them as measured by the first observer will be different to the time as measured by the 'moving' observer.

The two measurements of time are related by the Lorentz time transformation.

Garth

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When that observer makes measurements of objects moving between two events we recognise that the time between them as measured by the first observer will be different to the time as measured by the 'moving' observer.

The two measurements of time are related by the Lorentz time transformation.
Obviously. But that is not what we are discussing.
The discussion is about what time is.

You claim that the t dimension in space-time (and I am obviously not talking about things like diagrams, but I am talking about the phenomenological space-time) is time in relativity, I claim that time in relativity is proper time.

Frankly, I don't understand how one could possibly say that the t dimension is time in relativity. In Galilean relativity one can say that the t dimension is time, even in the Newton/Cartan formulation, but certainly not in special or general relativity.

If we take it to the next level and include curved space-times the invalidity of the statement that the t dimension represents time becomes even more obvious. And one step beyond, there are even metrics that are not diagonizable.

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The discussion is about what time is.
Yes, and I thought we agreed that although there are several ways of talking about time, such a psychological time, in physics we are talking about something we can measure with a clock, did we not?
You claim that the t dimension in space-time (and I am obviously not talking about things like diagrams, but I am talking about the phenomenological space-time) is time in relativity, I claim that time in relativity is proper time.
Frankly, I don't understand how one could possibly say that the t dimension is time in relativity. In Galilean relativity one can say that the t dimension is time but certainly not in special or general relativity.
I am saying that time for a particular observer is their proper time, and it is the t-axis on that particular space-time diagram drawn from their perspective, that is, in their own rest frame.
If we take it to the next level and include curved space-times the invalidity of the statement that the t dimension represents time becomes even more obvious. And one step beyond, there are even metrics that are not diagonizable.
But even so, observers in such a space-time will have a clock and ruler and are able to make local measurements of time and space in their own frame of reference. What they cannot do is then extend that frame of measurement elsewhere in a way consistent to other observers.

Garth

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Obviously. But that is not what we are discussing.
The discussion is about what time is.

You claim that the t dimension in space-time (and I am obviously not talking about things like diagrams, but I am talking about the phenomenological space-time) is time in relativity, I claim that time in relativity is proper time.

Frankly, I don't understand how one could possibly say that the t dimension is time in relativity. In Galilean relativity one can say that the t dimension is time, even in the Newton/Cartan formulation, but certainly not in special or general relativity.

If we take it to the next level and include curved space-times the invalidity of the statement that the t dimension represents time becomes even more obvious. And one step beyond, there are even metrics that are not diagonizable.

It's not clear to me what your position on this topic is. (I get hints of what is not your position... but it's murky to me what your position is.) For the benefit of the readers of this thread, in a short paragraph, can you precisely define "dimension"? "t"? "time"? "proper time"?

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It's not clear to me what your position on this topic is. (I get hints of what is not your position... but it's murky to me what your position is.) For the benefit of the readers of this thread, in a short paragraph, can you precisely define "dimension"? "t"? "time"? "proper time"?
Thanks! I've been plowing through endless wikipedia articles trying to keep up...
-said,