I am starting this thread to discuss the 4th dimension. Do we really know what it is? There appears to be two explanations, one from the mathematic point of view and the other from the physics point of view. In mathematics, it is treated like a vector not much differently then the preceding or following vectors, as Alphanumeric described. In physics it is said to be time, though it has never been proven, it is generally regarded as such and taken for granted. This thread is opened for discussing this dimension and seeking to better understand what is known about it and what is not. While explanations and ideas are welcome, proof of any sort is more welcome. Is mathematics mixing up the words vector with dimension? Is physics 100% certain the 4th dimension is time?
What do you mean "proven"? Has it been proven that the "first dimension" is length? Oh wait, maybe the first is width. Or height.
It's pretty clear that time is the fourth dimension. This is proved by the fact that GR is the correct theory of gravity at low energies, insofar as we've been able to test it. There is no alternative explanation that is supported by experiment.
No, it has not been proven that the 1st dimension is length, width, or height, depending on which you prefer to call it. I recall one person calling it distance, while another saying it was linear. None of the dimensions have been proven to be what they are. The first three as we generally describe them can be interchangable, as your quote shows. Take a cube, it has height, width, and length. X, Y, and Z. Independent, any of these coordinates would be considered a dimension, though only when they combine do they become 2 or three dimensional. What if we take only 1 coordinate and move it, what becomes of it. If this is a matter of logical interpretation and meaning, then it can be reasonably deduced unless I am mistaken. These coordinate add up spatially when combined. By adding the 4th coordinate we say time is reached, though wouldn't it be an extra spacial region when that 4th is added where the area or object in question is dispersed. Would or does space turn into something dynamic like time.. How does GR prove this and clear it up. I would like to understand the mechanics by which this is, otheriwise I won't understand it enough to accept it fully, even if I do at word value.
That was my point: the dimensions are assigned "numerical values" as and when required, depending upon what is being considered. Therefore time is the fourth when using space-time equations. No, it doesn't work like that. Time is not a spacial dimension, it's just a dimension. You could "add" colour as a dimension if that was part of the remit.
I follow. So these definitions are used for equations and assigned as needed. This applies in physics I'll assume, based on the above written discussion. In math, they operate differently, unless I am mistaken. Sorted. Ok then, if we are building on space itself, taken this spatial object with three dimensions to it, and then add a forth spacial dimension, what would happen to the object.. Also, considering, objects aren't solid phantoms, they are made of billions of atoms, which break down to glouns, partons, and wee bits. These can hardly be described by 3 dimensions. The object has a bond holding these atoms together, which also could not be described by 3 dimensions. When I consider this, I am left wondering how many does the object actually contain outside of time...
Let the object in question be Newton's famous apple! Mathematically speaking, how many dimensions does it contain... Physically speaking, same question...
I'm not sure what you mean by that bit. What fourth spacial dimension? It depends on what sort of description you're looking for. When it comes down to it what, if anything, can you describe completely using only three parameters? What do you mean by "outside of time"?
are Maths and Physics not merged into one another for this question?, you need them both at the same time.
The notion that time is the fourth dimension is misleading. There is a significant amount of mathematics dealing with n-dimensional space, where n can be any integer. The modern laws of physics can be very conveniently modeled by 4D geometry, using time at the 4th dimension. It is a convenience or a useful model. Einstein once said something like the following. Modern physics (at least relativity theory) finds it useful to deal with Events. An event takes place at coordinates (x, y, z, t), where (x,y,z) specifies a postion in space & t specifies a time. Using the above notation, one can use the mathematics of geomery to describe the laws of physics.
We can both agree that there are no alternative theories to GR, which work better to model Nature? If yes, then we will call GR THE theory of nature. If no, then you should suggest an alternative and tell me why it is better than GR.
There is a lot of mathematics relating to n-dimensional geometry for which time is not used as a dimension. Some of the more trivial results/theorems include the following. The length of the diagonal of an n-dimension unit hyper-cube is SquareRoot(n). For 4D, this is 2. It is startling to realize that in 441 dimensions, the diagonal is 21 units long. An inscribed hyper-sphere would touch all the sides but only one unit of the diagonal would be inside that sphere. Formulae for volumes & surface areas of Hyper-spheres, hyper-cones, & other geometric objects are known. BenTheMan: I hope you typed the following in haste. While General Relativity is the best theory of gravity we have, the use of time as a fourth dimension is only part of a very useful model. It would be more correct to say that the Relativity uses the mathematics of 4D differential geometry to model the laws of physics. In this model time is assigned as a fourth dimension. An interesting aspect of the model is that there is no motion. A moving particle is modeled as a 4D curve called a World Line. Other mathematical models could be used with the time variable not being treated as a fourth coordinate. Such models are more difficult to describe & understand. BTW: Classical physics could be modeled using a 4D Space-Time continuum & this model might be better than the ones actually used. Such a model would use a space-time continuum with different geometric properties than those in the model used for relativity.
Can you formulate general relativity in a different number of dimensions, and still model our universe? Hmm, I don't know. I'd agree with you if GR was based on an Euclidean manifold, but it is based on a Lorentzian manifold. In Euclidean space, the signature of the metric is (++++), whereas in a Lorentzian manifold, the metric is (+---), making time somehow different. There is a sense in which GR treats time differently than space.
A dimension is really just a degree of freedom, a numerical value that can be varied independently of everything else. You can even treat temperature as a dimension if you wanted. From a mathematical standpoint, time behaves in many ways like a dimension of space, so we call it "the 4th dimension".
Isn't a time dimension a prerequisite for having any of the others? If you didn't have time, what would be the difference between a point and a line?
No. A point is zero dimensional, a line is one dimensional. You can have both points and lines in an Euclidean space, where there is no time direction.
BenTheMan: General Relativity is fundamentally a 4D model. String Theory is based on more dimensions, but there seems to be no observational support making it a better model than GR. I know little about String Theory, but I have read that is it consistent with our current knowledge. I do not think there are any proposed experiments which could make it more suitable than GR. Nobody claims that time in the GR model is or should be treated as the same as one of the 3D space dimensions. Some use i (Square root of minus 1) in the GR model to make all the signs the same, but this is a gimmick. Note that when using spherical coordinates for 3D geometry, one must treat the radius coordinate differently than the Latitude & Longitude coordinates. This is conceptually equivalent to treating time differently in the GR model of reality. For those not familiar with the concept of World Lines & Events, consider the following. Physics deals with events which occur at 4D coordinates : (x, y, z, t), which are analogous to points in a 4D space. GR uses the 4D location of an event, while classical physics uses a 3D point in space & a 1D instant of time as the location of an event. In situations dealing with more than one event. Classical physics deals with spatial distances between events & time intervals between events. GR deals with what are called intervals between events. In the absence of both gravity & acceleration, Interval[sup]2[/sup] = DeltaX[sup]2[/sup] + DeltaY[sup]2[/sup] + DeltaZ[sup]2[/sup] - DeltaT[sup]2[/sup], where DeltaX, DeltaY, DeltaZ & DeltaT are differences between corresponding coordinates. This very much like the extended Pythagorean Theorem which gives the distance between 3D Points: Distance[sup]2[/sup] = DeltaX[sup]2[/sup] + DeltaY[sup]2[/sup] + DeltaZ[sup]2[/sup] In the presence of gravity or accelerated motion, the GR formula for an Interval between events is more complicated. However, it is analogous to formulae for distances between points in a curved space. The mathematical discipline called Differential Geometry is used for analyzing curved spaces. This discipline was developed as an intellectually interesting field of mathematics. When GR was developed, Minkowski (& others) recognized that this mathematics was a useful tool. The World Line concept is easy to understand if you ignore one spatial dimension. Consider using World Lines to represent orbits/paths in a solar system like ours where almost all objects move in an approximation to a plane. Sol is always on the vertical axis. Coordinates (x, y, t) are used with t (time) taking the place of the Z-coordinate. A circular orbit looks like a helix on the surface of a cylinder. A decaying orbit falling toward Sol looks like a helix on the surface of a cone (Apex on the vertical axis & base in the XY-Plane). An object falling directly toward Sol looks like a straight line from a point in the XY-Plane to the vertical axis. Note that the above models a moving object as a static curve or a straight line. This model contains all the information inherent in equations like: X = Rcos(Angle) & y = Rsin(Angle). The above concept using World Lines allows GR to answer a philosophical question that was the subject of much debate for hundreds of years. What does it mean to say that this object is the same object that it was yesterday or ten years ago? For a living organism, there are differences from day to day & a majority of the atoms are replaced over a period of years. For an inanimate object, there are differences in position and usually minor differences in composition (when do you say the inanimate object is not the same?). The GR answer to the question is as follows. While not a definitive answer to the question, it is better than the answers that various philosophers gave & fought over.
