B On the four dimensions of special relativity

[trees and plants]

Hello. Questions: How can special relativity describe four dimensions when we only see three spatial dimensions and we experience time?Why time is the fourth dimension and not another physical quantity? Is this only a generalisation that was needed to describe more physical phenomena and the three spatial dimensions alone could not do this? Thank you.

 

[jbriggs444]

If you want to meet with someone, you need to specify a place and a time. In general, that takes four coordinate values -- three for space (longitude, latitude and elevation, for instance) and one for time.

That's essentially all there is to it. A "dimension" is nothing more than a place where you can stick in a coordinate value. The formal mathematical details come under the heading of linear algebra and vector spaces.

 

[phyzguy]

You are basically asking, "Why is the universe the way it is and not some other way?" Nobody can answer this question. It has been noticed that by treating time as a fourth dimension which is slightly different from the three spatial dimenisons, that many things make sense that didn't before.

 

[robphy]

Hello. Questions: How can special relativity describe four dimensions when we only see three spatial dimensions and we experience time?Why time is the fourth dimension and not another physical quantity? Is this only a generalisation that was needed to describe more physical phenomena and the three spatial dimensions alone could not do this? Thank you.

The four-dimensionality is a mathematical construction which happens to allow us to write down physical laws and make predictions, which seem to have been successful in agreeing with experiment. One is certainly allowed to attempt a different formulation and make the same predictions... but, so far, it doesn't seem that there is a viable alternative that is as simple or as successful.

Note that this four-dimensionality is not restricted to special relativity.
Four-dimensionality appears in Galilean physics as well.
We draw position-vs-time diagrams for simplicity... but position, treated fully, is three-dimensional.

The principle of relativity suggest that space and time are more intertwined than other pairs of quantities.
The success of special and general relativity and their geometrical interpretations suggest that we should consider a four-dimensional geometrical object, where time and space are featured.

If the "reality" of 4-dimensional objects bother you,
say instead it is "as if" there were a 4-dimensional object... then carry on, do the math, make the predictions, etc... be marveled when it still agrees with experiment.

 

[Dale]

Why time is the fourth dimension and not another physical quantity?

The math works out and correctly predicts the outcome of experiments that way.

You can always make a higher dimensional vector simply by adding on additional independent quantities of interest. So you could make a 5 dimensional space by (latitude, longitude, altitude, temperature, humidity). That sort of thing is always mathematically possible.

What becomes interesting is the transformation properties of such objects and the mathematical operations that can be done. In the case I just described you can do something like a coordinate transformation where you change from using true north to using magnetic north. That will change the first three coordinates together, but will not change temperature or humidity. Similarly you can change from Celsius to Kelvin and that will change the fourth coordinate, but none of the others. Similarly, rulers measure differences in the first three coordinates only and thermometers measure the fourth coordinate only and both are independent of each other. So there is no sense in which the temperature really “belongs” with the first three. It is there as the “fourth dimension” only for convenience.

With time it is different. If we make a four vector as $(t,x,y,z)$ then we actually do find interesting transformation properties and operators. The time measured by a clock is not given by $\Delta t$ but by $\sqrt{\Delta t^2-(\Delta x^2 + \Delta y^2 + \Delta z^2)/c^2}$. And when you build inertial coordinates based on a moving reference frame you get a sort of rotation between space and time.

Because of those transformation properties and operations, it is clear that time belongs with space in a much stronger sense than temperature did. Putting them together is both mathematically convenient, but also makes physical sense based on the real world behavior of actual clocks etc.

 

[robphy]

quite a flurry of responses in a short interval!

 

[trees and plants]

Note that this four-dimensionality is not restricted to special relativity.
Four-dimensionality appears in Galilean physics as well.

Can you describe a few things about the four-dimensionality of Galilean physics and its difference with that of special relativity?

 

[etotheipi]

Can you describe a few things about the four-dimensionality of Galilean physics and its difference with that of special relativity?

my two pennies, in the classical mechanics, galilean spacetime is a four dimensional affine space, and the points in this spacetime are events. The galilean transformations preserve the time interval between two events, and also the spatial distance between any two simultaneous events [and for any $a,b \in A^4$, there is an absolute notion of whether $a$ and $b$ are simultaneous, i.e. there is a mapping $t$ from the space of parallel displacements $\mathbb{R}^4$ to $\mathbb{R}$, and we can check that $t(a-b) = 0$]. The space and time dimensions are pretty much kept separate, they don't "transform into one another".

But now in the special theory, minkowski spacetime is also a four dimensional affine space, except now for instance the lorentz transformations do not preserve either the time interval between two events, nor the spatial distance between two events. Whether two events $a,b \in A^4$ are simultaneous also depends on the 4-velocity of the observer [they each project the difference vector between the events onto their differing 4-velocities]! When you change your basis $\{ \mathbf{e}_{\mu} \}$ describing the spacetime, loosely, the transformation of the temporal components also depends on the spatial components, and vice versa. In that way, the space and time dimensions are treated more on a level footing.

 

[trees and plants]

my two pennies, in the classical mechanics, galilean spacetime is a four dimensional affine space, and the points in this spacetime are events. The galilean transformations preserve the time interval between two events, and also the spatial distance between any two simultaneous events [and for any $a,b \in A^4$, there is an absolute notion of whether $a$ and $b$ are simultaneous, i.e. there is a mapping $t$ from the space of parallel displacements $\mathbb{R}^4$ to $\mathbb{R}$, and we can check that $t(a-b) = 0$]. The space and time dimensions are pretty much kept separate, they don't "transform into one another".

But now in the special theory, minkowski spacetime is also a four dimensional affine space, except now for instance the lorentz transformations do not preserve either the time interval between two events, nor or spatial distance between two events. Whether two events $a,b \in A^4$ are simultaneous also depends on the 4-velocity of the observer [they each project the difference vector between the events onto their differing 4-velocities]!

So, the fourth dimension which is time was considered in Galilean physics also.

A person in Galilean physics could also add more dimensions in a manifold?We are interested in time as the fourth dimension in special relativity because of the Lorentz transformations about it? Because it transforms?

 

[trees and plants]

It is a way to find coordinates like in two dimensions where we have one dimension of space and one dimension of time?Like the orthogonal lines that form the two dimensional coordinate system we know but in this case the four dimensional coordinate system does not have a graphical representation?

 

[jbriggs444]

It is a way to find coordinates like in two dimensions where we have one dimension of space and one dimension of time?Like the orthogonal lines that form the two dimensional coordinate system we know but in this case the four dimensional coordinate system does not have a graphical representation?

A "space-time diagram" is a way to present one dimension of space and one dimension of time on a piece of flat paper. However the notion of "orthogonal" that is used looks weird because it is hyperbolic geometry. One must not expect lines of simultaneity to always be at 90 degree angles to lines of constant position. And one must not expect to read interval lengths directly from the paper.

 

[Janus]

Can you describe a few things about the four-dimensionality of Galilean physics and its difference with that of special relativity?

I'll try by analogy:
Prior to SR, time and space were treated as absolute measures that everyone could agree on. The analogy is treating time and space like the directions of North-South(Time) and East-West(space). No matter what direction someone is facing, they agree on these directions.
However, SR replaced the N-S, E-W idea of time and space with a Front-back and Left-right convention. Thus your measurement of an event's location in time and space relative to you, is determined by your own perception of front-back and Left-right. They have become "frame-dependent". Changing the direction you are "facing"( changing inertial frames of reference) also changes those events' location relative to you in time and space.
In this analogy, I reduced things to 2 dimensions, 1 time and 1 space, but adding 2 more spatial dimensions isn't an issue.

 

[trees and plants]

So, as i see special relativity is about three dimensional physical bodies and about time and this forms the four-dimensional Minkowski space. If we had only a point with one spatial dimension and one of time, it would be a two-dimensional flat spacetime in special relativity?

 

[trees and plants]

Could i make another two questions in relation to those i made here but off topic i think and post them in this thread or not?

 

[jbriggs444]

So, as i see special relativity is about three dimensional physical bodies and about time and this forms the four-dimensional Minkowski space. If we had only a point with one spatial dimension and one of time, it would be a two-dimensional flat spacetime in special relativity?

Neither special relativity nor Galilean relativity have anything much to do with extended physical bodies in one, two or three dimensions. Instead, both are about setting up an abstract space within which we can describe things.

For instance, one might describe how an unaccelerated particle moves by writing down an equation: x = vt.

That equation works in Galilean relativity.
That equation works in Special relativity.

The difference comes in when you try to change coordinate systems and describe the same motion from the point of view of a piece of moving graph paper.

 

[trees and plants]

I see more clearly how things work in physics after this discussion i think.

Neither special relativity nor Galilean relativity have anything much to do with extended physical bodies in one, two or three dimensions. Instead, both are about setting up an abstract space within which we can describe things.

For instance, we can describe how particles move by writing down an equation: x = vt.

That equation works in Galilean relativity.
That equation works in Special relativity.

The difference comes in when you try to change coordinate systems and describe the same motion from the point of view of a piece of moving graph paper.

That change of coordinate systems is when coordinates transform for example from a Cartesian coordinate system to a curvilinear coordinate system or when we change bases?

 

[trees and plants]

Neither special relativity nor Galilean relativity have anything much to do with extended physical bodies in one, two or three dimensions. Instead, both are about setting up an abstract space within which we can describe things.

For instance, one might describe how an unaccelerated particle moves by writing down an equation: x = vt.

That equation works in Galilean relativity.
That equation works in Special relativity.

The example you said with the equation is it in a two dimensional flat manifold with one dimension of space x, and one dimension of time t in Galilean physics and special relativity? If we had $f(x,y,z,t)=0$ would it be in a four-dimensional manifold?

 

[jbriggs444]

That change of coordinate systems is when coordinates transform for example from a Cartesian coordinate system to a curvilinear coordinate system or when we change bases?

Simpler than that. One can consider just rotating the coordinate system. So instead of lining up the graph paper on an east-west axis, you line it up on a northeast-southwest axis.

Or a moving coordinate system. Instead of having the graph paper at rest, have it moving rightward at one meter per second.

As it turns out, a coordinate system transformation in Special Relativity turns out to be very much the same thing as a rotation.

 

[trees and plants]

Simpler than that. One can consider just rotating the coordinate system. So instead of lining up the graph paper on an east-west axis, you line it up on a northeast-southwest axis.

Or a moving coordinate system. Instead of having the graph paper at rest, have it moving rightward at one meter per second.

On a textbook i have i think it said about rotating frames of references. So, what is the motivation in physics to use moving and rotating coordinate systems?

 

[Nugatory]

If we had only a point with one spatial dimension and one of time, it would be a two-dimensional flat spacetime in special relativity?

One spatial dimension is a line not a point, but otherwise, yes.

 

[Nugatory]

Could i make another two questions in relation to those i made here but off topic i think and post them in this thread or not?

New questions go in new threads... but be sure that you understand the answers you’re getting in this one before you go spinning off in new directions.

 

[jbriggs444]

On a textbook i have i think it said about rotating frames of references. So, what is the motivation in physics to use moving and rotating coordinate systems?

For a continuously rotating coordinate system, consider weather systems on the surface of the Earth. It is convenient to use a coordinate system in which Chicago, for instance, stays in one place.

For a moving coordinate system, consider air travel where a craft has an air speed and a ground speed.

Exercises in your textbook may consider the rest frame of an elevator car or of a carousel. Boats on rivers are also popular.

 

[Dale]

On a textbook i have i think it said about rotating frames of references. So, what is the motivation in physics to use moving and rotating coordinate systems?

Please put new questions in new threads. Different questions may be better served by different forum members