Dimensions – how many?
Most people would agree that speed has two dimensions, that is distance and time, typically measured in metres [m] and seconds [s]. (I am showing dimensions in square brackets to distinguish them from other symbols in a calculation, so I will show all dimensions from here on in [ ]). Let us now calculate the speed of light in free space with the aid of Maxwell’s wave equation,
We know the result we want will have the dimensions [m s-1] but we need to confirm this, because no calculation is deemed to be correct unless it is accompanied by a dimensional analysis that confirms that the dimensions of the result are correct.
We need to show with a dimensional analysis that the result has actually the dimensions [m s-1]; and here is how we do it:
[m] = metres, the unit length
[kg] = kilogram, the unit mass
[A] = Ampere, the unit current
[s] = second, the unit time
Solving the dimensional analysis is straight forward, if we multiply the dimensions in the square brackets, the [kg] and the [A] cancel out and we are left with [s2 m-2], taking the inverse of the square root we get [m s-1].
There is nothing we can do about it, we must represent the permeability and permittivity of free space with 6 and 10 dimensions respectively in order to get the correct result, and we need to call them “Dimensions” because so far we don’t have any other word for them.
The Oxford Dictionary does agree with this definition of the word “Dimension”, it says:
Physics, an expression for a derived physical quantity in terms of fundamental quantities such as mass, length, or time, raised to the appropriate power.
I am fully aware that ten dimensions are not very palatable to many people, firstly, because they know that there are only three “Dimensions” and secondly a “Dimension” is something that you can measure with a ruler, that is the length of a straight line between two points.
This is also consistent with the definition in the Oxford Dictionary,
Dimension, a measurable extent of a particular kind, such as length, breadth, depth, or height.
So, for the benefit of the people that prefer this definition of the word “Dimension” I will, from here on, use only this definition and use only the unit length, a metre [m] as a “Dimension” and then we will see how many “Dimensions” we will end up with.
Artists need two and three dimensions to express themselves , but as we will see further on, if we want to design the structures, or the transportation means and communication means that modern humans so desperately depend on, we need to make use of more than three linear dimensions, that is four, five and six dimensions, and I mean linear dimensions, dimensions that we can measure in metres [m], with a ruler.
In a Quiz Show on TV recently, the question was asked “What is the fourth dimension of space?”, nobody knew the answer and the host eventually elaborated that the fourth dimension of space is “Time”.
But, that does not suit us at all, because we have just established that “Time” which is measured in seconds the unit [s] is not a “Dimension” in our definition of the word. So in other words we need to argue that space has only three dimensions, but we know that this is not right either, because everyone else says it has four dimensions; so what should we do?
Ever since Einstein has published the General Theory of Relativity space became four dimensional. Most people still struggle to understand it and have trouble imagining four space dimensions. But it is not that difficult at all and the fact that the speed of light in free space is a constant, causing the whole controversy in the first place, does come quite handy now to explain the four dimensions of space. The fact that the speed of light is constant makes “Time” and “Distance” interchangeable.
Consider the equation s = c t
s is the distance in metres [m]
c is the speed of light in [m s-1]
t is time in seconds [s]
clearly if c is constant we can substitute “Time” with “Distance” and express the definition of space with four linear dimensions that we can measure in metres [m], all four of them, even time we can measure in [m] as a distance of a straight line between two points.
We do measure the distance to the stars in light-years, which is measured in metres even though a year is a time dimension, so we introduce nothing new.
It is not too difficult to depict four space dimensions on a two dimensional sheet of paper, all we need to do is use a methodical approach that differs slightly compared to our conventional three dimensional imagination. The notion that humans cannot imagine more than three dimensions is quite presumptuous and total nonsense, a very narrow minded view of some narrow minded traditionalists.
To some of us, who make a living out of using more than three dimensions in our daily lives, the method used to depict four or five dimensions is straight forward. Ever since Euler and Bernoulli published the “Beam Equation” in 1750 we are using four linear dimension to calculate beam deflections, for example.
What Euler and Bernoulli found in 1750 is that the cross-sectional area of a beam can no longer be defined with two dimensions. A two dimensional cross-section does allow us to calculate tensile and compressive stresses and even some shear stresses and deflections, but as soon as we need to calculate bending stresses and deflections we realise that the cross-sectional area does no longer have a uniformly homogeneous characteristic. A partial area of the cross-section (Figure 1) that is closer to the centre of gravity of the bending beam has an entirely different characteristic to the same partial area of the cross-section that is further away from the centre of the bending member.
Consider the area “w dy” shown in Figure 1. It’s contribution to the rigidity of the beam is negligibly small when “y = 0”, whereas if “y” is at its maximum the same area, with exactly the same physical dimensions, makes a significantly greater contribution to the bending rigidity of the beam, so we need to define this difference and we can do that only with another dimension.
The characteristic of the beam cross-section that we need to calculate stresses and deflections is called the “sectional moment of inertia” and it is calculated as the sum of all areas “w dy” multiplied by “y² ”. The smaller we make the dimension “dy” the more accurate the result will be, but at the same time it does increase the number of calculations we need to carry out because we have to add up more “w dy y² ”. This is where the infinitesimal calculus comes handy, because now we can make “dy” infinitely small and we can add up an infinite number of “w dy y² ” giving us a perfectly accurate result.
We use the letter “d” (for differential) to define something that is infinitely small, so if we make “y” infinitely small we refer to it as “dy”. But it does not matter how we calculate it, the dimensional analysis of the calculation will always be the same and does show that the dimension of the result is [m4], because [m2] [m2] = [m4].
If the sectional moment of inertia would have been developed one hundred years later, Maxwell would have probably described it as a vector field of the rigidity potential of a surface. Since it is a surface field it does require only two dimensions to define the origin of a vector and since all vectors are tangential to the axis XX, the direction and magnitude of the vector does also require only two dimensions to define it, giving a total of four dimensions [m4]. More about Maxwell’s fields in “Six Dimensions?” below.
Integrating around x-x between -h/2 and +h/2 gives
I would like to come back to the four dimensions of space again, because there is some similarity to the sectional moment of inertia. What we know is that space does also have a different characteristic depending on its proximity to the centre of gravity of a mass. If we look at a differential volume of space “dV” as shown in Figure 2, we know that the characteristic of this differential volume does change as a function of the distance “r ” from the centre of the mass “m”. The gravity in a space volume is greater in the proximity of the mass than if it is further away from the mass. That is why light is bending more when it passes closer to the mass than when it passes further away from the mass. Since the differential space “dV” has already three dimensions we need another dimension to define the difference between the space volumes and we can measure it by measuring the distance “r” and end up with the four dimensional space.
Differential Volume of Space “dV” in the proximity of a Mass “m”
The four space dimensions are typically depicted in a “space-time” diagram as shown in Figure 3. Where the x-axis represents the space dimension and the y-axis represents the time dimension in [m]. (as mentioned above, since the speed of light “c” is constant the time “t ” can be represented by “ct ” which of course has the dimension [m]. The four hyperbolas show in Figure 3 are representing the so called Minkowsky space-time, meaning that any point along a hyperbola, no matter where it is located at the hyperbolic line, like point “A” has the same space-time distance from “O”. The position of point “A” along the hyperbola is dependant only on the speed of the observer. If the observer travels at close to the speed of light, the time dimension will be equal to the space dimension, that means point A would be touching the line at 45°. Expressing the distance OA mathematically OA2= ct2 – x2, which is the function of the hyperbola.
The area “B” is located before point “O” in the space-time direction, meaning we would travel backwards in time to get there, which is demonstrably possible. The 45° lines of the Minkowsky space-time do, however, ensure that we cannot travel backwards in time to cause “Causality”. Causality is an expression used to describe for example a situation where we would travel backwards in time to prevent our own birth, the Minkowsky space-time does make that impossible.
Minkowsky space-time diagram
Brian Cox & Jeff Forshaw, “Why Does E=mc2?” Perseus Books Group.
Richard Feynman uses the four dimensional space-time diagram, referred to appropriately as the Feynman Diagram, very successfully in making the most complex events in Quantum Mechanics understandable.
The Feynman Diagram
Richard Feynman “Quantum Electro Dynamics”, Princeton Science Library
Similar to the diagram used in Figure 2, to describe the difference in the space characteristic in the proximity of a mass as a function of the distance “r”, we now move the Differential Volume “dV” inside the spherical mass to calculate the rotational moment of inertia of the spherical mass as shown in Figure 5.
Like the differential area of the sectional moment of inertia, Figure 1, the characteristic of the differential volume “dV” in the rotational moment of inertia changes also with “r²”. Thus multiplying “dV” [m³] by “r²” [m²] does give us a result with the dimension [m5]. If we integrate “dV” spherically between the centre of the mass and R we do get the familiar equation of the rotational moment of inertia as shown below in [m5]. That is five dimensions, all measured in metres [m] as a straight line between two points, as we did with the four space dimensions.
Differential Volume inside a Mass “m”
Integrating between centre of m and R gives
Our earth is such a spherical mass, (approximately spherical) and here is a bit of history of how we learned that we need five and six linear dimensions to define the characteristic of our earth or any other object in the universe.
In 1735 a French expedition led by Pierre Bouguer and Charles Marie de La Condamine set out on a journey to South America to measure amongst other things the weight of the earth. The method used to do that had already been suggested by Isaac Newton, but nobody before was bold enough to actually attempt such an expensive and risky experiment.
The idea was to measure the weight of a salient mass like a single mountain and use a second mass hung like a plumb bob near the mountain and then measure the angular deflection of the plumb bob line. If the earth would not have any mass, the plumb bob line would be nearly horizontal and pointing directly towards the centre of gravity of the mountain mass. The earth however has a significant mass, so the deflection is quite small, but measurable. Knowing the mass of the mountain and the mass of the plumb bob we then can calculate the mass of the earth by measuring the angular deflection of the plumb bob line. Mount Chimborazo was seen as a suitable contender for this experiment, as it is reasonably simple in shape to calculate it’s mass and it is massive.
Bouguer measured an angle of 8 arc seconds between the plumb bob line and the vertical line pointing towards the centre of the earth and although he was not overconfident about the result of the experiment, it did conclusively prove that the earth was not hollow like some very eminent scientists of the time like Edmond Halley suggested. Isaac Newton has predicted an angular deflection of 2 arc minutes, but of course he did not expect that the earth was that much heavier in the centre as it indeed is.
Mount Chimborazo, Ecuador
In 1774 the same experiment was repeated in a more amiable environment, using Mount Schiehallion in Scotland for the experiment, led by Nevil Maskelyne, Charles Hutton and Reuben Burrow. The result of this experiment showed a deflection in the plumb bob line of 11.6 arc second (not much different from the Chimborazo deflection of 8 arc seconds) and proved conclusively that the density of the earth was about twice that of Mount Schiehallion. In other words the earth is definitely not hollow but evidently much heavier in the centre than on the surface.
Mount Schiehallion, Scotland
In 1798 Henry Cavendish used a fascinatingly simple device with four defined masses hanging on a torsional balance to calculate the density of earth to an extraordinary accuracy, very close to the current accepted value of 5515 [kg/m³], the greatest density of any planet in the solar system.
The Cavendish Experiment see:
The fact that Schiehallion has about half the density of earth was an indication that the density in the earth is not evenly distributed, the earth must have a greater density in the centre. Even today we know the density of earth with a reasonable accuracy only down to a depth of 500 [km], the rest we can only estimate through seismic earth quake waves. Figure 6 shows the best estimate we have to date of the density distribution of the earth. The straight lines and the homogeneous curves are clearly not convincing that we know all about it.
Approximate density distribution of planet earth
If we would now calculate the rotational inertia of the earth in a three dimensional world with the average equatorial radius “R” of the earth, approximately 6.378 x 106[m] and the mass of the earth, 5.97219 x 1024 [kg] we will get a rotational inertia of approximately 9.72 x 1037[kg m²].
If, however, we integrate the inertia from the centre of the earth to the maximum radius considering the varying densities of “dV r² ” we get an inertia of only 8.03 x 1037 [kg m²] due to the density concentration in the core of the earth. That means if we limit our definition of the earth to three dimensions we do get a 20% error in the moment of inertia.
In order to physically demonstrate that it is not sufficient to define a solid body with only three dimensions, I have manufacture two solid disks with a varying density distribution, and freely roll them down an inclined plane to show the difference in acceleration; a difference we cannot detect with three dimensions, because both disks have exactly the same size, all three dimensions are exactly identical, and the weight of both disks is exactly the same.
Five Dimensions http://edwardwechner6.blogspot.com.au
We cannot change the fact that mass has three linear dimensions, there is no such thing as a mass without a volume, even the most dense object in the universe, the black hole has three dimensions, which we can readily calculate with the Schwarzschild equation. Accordingly, if we would condense the earth to the density of a black hole, the earth would have a diameter of 0.0176 [m]. And we cannot change the fact that if we multiply the three dimensional volume of a mass, [m³] by the radius squared, [m²] of an individual density layer we do end up inevitably with five dimensions, [m5], all linear dimensions that we can measure with a ruler.
In the Chimborazo and Schiehallion experiments we have found that gravity does not always point to the centre of the earth. If the plumb bob line is not vertical, then there must be a horizontal component, tangential to the surface of the earth, because the earth does not have a smooth surface. In Figure 2, we did depict a mass of a smooth spherical surface and found that four dimensions are sufficient to describe the characteristic of space. In general this is an acceptable model if we look at a macro analysis of the universe, where the unevenness of a planets surface has a negligible small effect.
However we cannot escape the fact that in a smaller scale we must consider the effects of unevenness of a surface that does cause disturbances in the gravitational field or any other field. Gauss has detected this behaviour in an electromagnetic field and Maxwell has consequently developed a mathematical description of electromagnetic fields, independent of how many current sources or magnetic sources there are present. Largely thanks to Maxwell’s work we now have equations, that are capable of describing any three dimensional field. It was Maxwell that coined the term “field” as I have mentioned already above in the discussion of the sectional moment of inertia.
Figure 7 does show such a three dimensional vector field. This is a way to define electromagnetic fields or gravitational fields for example. Each vector has a position in a three dimensional space, it also has an individual length and it has a specific direction. This means we need six dimensions to define each individual vector as shown in Figure 8. Every single dimension can be measured with a ruler in [m] and since there are six dimensions required to define a single vector we end up with the dimension [m6]. It does not matter if we define it with six Cartesian coordinates as shown in Figure 8 or if we define its origin and the direction with angles and its length relative to the origin, we will still end up with six dimensions.
This is where Maxwell’s equations come handy, Maxwell was able to capture the behaviour of three dimensional vector fields with equations that allow us to mathematically quantify all six dimensions of any individual vector throughout the field.
Maxwell’s Equations http://edwardwechner8.blogspot.com.au
The six dimensions required to define a vector in a 3 dimensional field.
I would be amiss not to mention that the majority of people on earth don’t really care about any of this multidimensional-ism, their world consists of one dimension only and that is their religion, typically either the Bible or the Quran.
Both books conveniently lend themselves to interpretations that suits any life style you want to choose. Murder and violence is encouraged or discouraged, whichever way you like to interpret these so called “holy” scripts.
In the Quran you can definitely find an interpretation that you must shoot a 14 year old girl in the head because she believes that she has the right for an education.
In the Bible you can find an interpretation that you must invade Iraq for the sole purpose of murdering and maiming one million innocent people.
This senseless violence will not end until countries like Pakistan and America do allow their children to choose their own education. As soon as all children on earth have the freedom to choose there own education, these insanities will all fade away into oblivion within a few generations.
Earth will be round and beautiful, Neanderthals will be an intricate part of our heritage, even though they lived 50,000 years before God made Adam and before Allah made the flat earth.
Homo Erectus http://edwardwechner7.blogspot.com.au
We have reasons to be optimistic about our future given the extraordinary progress we make in science. Common sense will eventually prevail and people will become more interested in dimensions — that is more than one dimension!
Physics and Religion http://edwardwechner4.blogspot.com.au
The Bible, god has created Adam 6016 years ago (in a pink night gown).
The Quran, god made the earth as flat as a carpet and pegged it down with some mountains.
Just as I referred to the philosophical similarity between Pakistan and America, I came across some interesting supporting statistics, Figure 9, courtesy of GlobeScan/PIPA & BBC News. Both Pakistan and America seem to be the only two countries on earth that would favour Mitt Romney as president of America, rather than Barrack Obama.
France and Spain are on the other extreme of the scale with the greatest percentage support for Obama. Could it be that their own excesses in religious based violence during the “Inquisition” in the middle ages is still on their conscience and that they do what intelligent humans do, learn from their own mistakes, while others still make them?
I believe the answer is yes, and I am encouraged to see that my own country, Australia, is not far behind France.
25.10.2012 Edward Wechner
7. November 2012, after the election of President Obama
I take everything back, reason has prevailed in America over pure evil, Pakistan is now alone.