You don’t have to go to general relativity to get space and time mixing. Under special relativity, if I am moving relative to you, my time direction is pointing a little bit in your space direction, and vice versa. It’s only in Newtonian mechanics that time entirely doesn’t mix with space.
There are problems with having real objects in 4-space too. The inverse square laws for gravitation and electromagnetism are that way because we’re in 3 dimensional space. If you have 4 dimensional space, then they become inverse cube laws. This causes a problem, because when you try to model stable orbits and atoms, there aren’t any - an object approaching another object either collides with it or is ejected from the system, and making stable atoms hits a similar snag. It’s pretty unlikely you could have an intelligence or even solid objects in a 4-space that has physical laws remotely similar to ours.
However, considering time as a bona-fide dimension (if one chooses to so consider), then you can certainly have a 4-dimensional cube. That is nothing more than a conventional 3-dimensional cube that exists for a certain duration of time, with a particular starting time and a particular ending time.
The measure of the time-dimension (that is, the length of the cube’s duration), to be perfectly cubical, needs to be just right. This is done by equating physical distances with time durations, using c as a conversion factor.
George Gamow illustrated this by showing an illustration of a conventional 3-cube with a little calendar attached to each vertex.
You are exactly correct, and I phrased my post poorly.
It’s fun to look at SR grids, where, for instance, my space-time is a “chessboard” grid – everything at 90 degrees – and your grid is made of lozenges or parallelograms.
(I searched for an example but couldn’t find one.)
“We”, Flesh-o-Pod? :dubious:
Strictly speaking the dimension of a space is just a number associated with that space and can loosely be thought of as the minimum number of parameters needed to describe that space. So it is something of a misnomer to talk about individual dimensions. That said physicists still talk about the time dimension and spatial dimensions and this does have some meaning.
To understand better it is best to go back to basics. Using Pythagoras’s theorem it is easy to express the displacement Δs on a plane in terms of displacement in Cartesian coordinates (x,y), i.e.:
Δs[sup]2[/sup] = Δx[sup]2[/sup] + Δy[sup]2[/sup]
Displacements cannot always be expressed using general coordinates so neatly as this, though so it is better to focus on infinitesimal displacements ds:
ds[sup]2[/sup] = dx[sup]2[/sup] + dy[sup]2[/sup]
This is known as a line element (of the metric), and it is basically a way of calculating distances when a space is parameterized in a certain way. The above is simply the line element of a plane parametrized by Cartesian coordinates.
In 3 dimensions, using Cartesian coordinates (x,y,z) the line element is:
ds[sup]2[/sup] = dx[sup]2[/sup] + dy[sup]2[/sup] + dz[sup]2[/sup]
It is important to realize at this point though that whilst the line element has the same form for Cartesian coordinates in Euclidean space, the choice of Cartesian coordinates is arbitrary and we can rotate or translate the coordinate system to create new Cartesian coordinates to express our line element.
It is also important to realize that the form of the line element depends not just on the space, but also on how the space is parameterized. For example the line element for the same space paramterized using spherical coordinates (r,θ,ϕ) is:
ds[sup]2[/sup] = dr[sup]2[/sup] + r[sup]2[/sup]dθ[sup]2[/sup] + r[sup]2[/sup] sin[sup]2[/sup]θdϕ[sup]2[/sup]
Now in relativity space and time are combined together into spacetime and the line element of (flat) Minkowski spacetime used in special relativity, in what are called Minkowski coordinates (t,x,y,z) (the SR equivalent of Cartesian coordinates), is:
ds[sup]2[/sup] = -dt[sup]2[/sup] + dx[sup]2[/sup] + dy[sup]2[/sup] + dz[sup]2[/sup]
(NB units are chosen such that the speed of light c, is equal to 1)
Now the most obvious thing to see is that the t-coordinate is different from the other coordinate as it has a minus sign before it. Displacements in spacetime can be divided into spacelike and timelike (and a 3rd category that sits between the other two called null or lightlike) and they are distinguished by their sign (ds[sup]2[/sup]>0 for spacelike displacements, ds[sup]2[/sup]<0 for timelike displacements and ds[sup]2[/sup]=0 for lightlike displacements).
Note that just as before we can rotate and translate the coordinate system to create new Minkowski coordinates, even though the form of the line element is unchanged. So for example if we perform a certain type of spacetime rotation known as a boost, we alter the axis of the t-coordinate and hence from the point of view of our original coordinates space and time become mixed.
Our justification for saying there are one dimension of time and three dimensions of space is because as long as we used special sets of parameteriziations (of which Minkowski coordinates are a subset) there will always be one parameter that is timelike and 3 that are spacelike.
However it is worth noting that there is more than one way to cook a goose and for example in so-called light-cone coordinates (u,v,y,z) two of the parameters are lightlike and two are spacelike and the line element is:
ds[sup]2[/sup] = -2dudv + dy[sup]2[/sup] + dz[sup]2[/sup]
Also parameters can be both timelike and spacelike, depending on the location in spacetime. For example in the Schwarzschild coordinates (t,r,θ,ϕ) for the Schwarzschild metric (representing a black hole), the line element is:
ds[sup]2[/sup] = -(1 - r[sub]s[/sub]/r)dt[sup]2[/sup] + (1- r[sub]s[/sub]/r)[sup]-1[/sup]dr[sup]2[/sup] + r[sup]2[/sup]dθ[sup]2[/sup] + r[sup]2[/sup] sin[sup]2[/sup]θdϕ[sup]2[/sup]
Where r[sub]s[/sub] is the Schwarzschild radius. When r<r[sub]s[/sub] the signs of the r and t coordinates flip in the line element so that the r coordinate becomes timelike and the t coordinate becomes spacelike. Note this is an artifact of the coordinates chosen though and it doesn’t mean that time and space physically swap.
So from the point of view of spacetime, it is still a misnomer to call one dimension as temporal or timelike and 3 dimensions as spatial or spacelike, but we do it anyway as our particular way of looking at the World is to choose 4 parameters to describe events: one to describe the time of the event and 3 to describe the spatial location.
I’ve been wanting to ask for about 50 years now, ever since I first read it in Gamow’s book One, Two, Three . . . Infinity when I was in 7th grade: What is the rationale for the minus-sign before the dt[sup]2[/sup]? Can that be explained in lay terms, or can it only be explained by diving into an intricate mathematical excursion? Is there some more-or-less “intuitive” way to look at it, or is that “just the way it is”?
It is kind of “just the way it is”. Think of it this way: Consider a stick in ordinary (Newtonian) space, with bunch of people looking at it from different directions. Each of these people imagines a coordinate system where the x-direction is “to the right”, the y-direction is “straight ahead”, and the z-direction is “up”, with the origin centred between their eyeballs. Since they are all facing and oriented in different directions, none of their x, y, and z directions match any of the other people’s x, y, and z directions, but that’s OK for now.
Using their own coordinate system, each person can find the location of the end points of the stick. Coordinates (x1, y1, z1) for one end, and coordinates (x2, y2, z2) for the other end. Since every person has their own coordinate system, each person has their own values of x1, y1, z1, x2, y2, and z2, but that’s fine as well.
But then if you ask each person to compute L = sqrt((z2-z1)^2 + (y2-y1)^2 + (x2-x1)^2), despite the fact that everyone had their own sets of individual numbers, they suddenly all agree on what L is. Basically they have each computed the length of the stick in their own coordinate system, and even though they don’t agree on the individual components that went into it, they come up with the same answer. Basically it is a statement of the fact that looking at a stick from a different direction doesn’t change its length.
In special relativity, for two events at (x1,y1,z1,t1) and (x2,y2,z2,t2), the quantity T = (z2-z1)^2 + (y2-y1)^2 + (x2-x1)^2 - c^2*(t2-t1)^2 is a similar quantity. Different observers will disagree about all of the individual coordinates, but necessarily will agree about this combined quantity.
As to “why that is”, that’s just a property of the universe. We could imagine (mathematically, that is) universes with no minus sign or two minus signs, but we just happen to live in a universe with one minus sign.
Yes there is a sort of intuitive way to look at it. It comes from the constancy of the speed of light in inertial frames
Lets call an infinitesimal spatial displacement dr, so that:
dr[sup]2[/sup] = dx[sup]2[/sup] + dy[sup]2[/sup] + dz[sup]2[/sup]
and
ds[sup]2[/sup] = -dt[sup]2[/sup] + dr[sup]2[/sup]
In order for a speed to be constant in regardless of inertial frame of reference, there must be a value of |dr/dt| that is independent of the Minkowski coordinates.
Looking at the case when:
ds[sup]2[/sup] = 0
implies that
dt[sup]2[/sup] = dr[sup]2[/sup]
for all dt and dr in Minkowski coordinates (as the value of ds[sup]2[/sup] is independent of how the space is parameterized)
and therefore
|dt| = |dr|
and
|dr/dt| = 1
Which must be true independently of the choice of Minkowski coordinates. In my first post I said the units were chosen so that c=1, so |dr/dt| = c.