TYPES OF ENTANGLEMENT (आरादुपकारक & संनिपत्योपकारक) – 7- Shri Basudeba Mishra
WHAT IS NOT A DIMENSION
Some say: we can specify the time and place of an event in the universe by using three Cartesian coordinates for space and another number for time. This makes space-time four-dimensional. It shows that we can specify time using a number. An object remain invariant under mutual transformation of the dimensions: like rotating length to breadth or height, even though the measured value of the new axes change. Time does not fulfill these criteria. Further, we can change our directions in space, but not in time. We can measure both sides of our position in space and remember the result of measurement. But we cannot remember future. Hence time is not a dimension, though it is intricately linked to space due to the following reason.
Earlier, we have defined number as a universal quality of all substances by which we differentiate between similars. Zero is that which is not present at here-now, but is present elsewhere. Elsewhere we have proved mathematically that division of a number by zero is not infinity, but it leaves the number unchanged. Infinity is like one – without similars, with one exception. While the dimensions of one are discrete – hence clearly perceived, the dimensions of infinity are analog and not clearly perceived. Space, time, coordinates and Consciousness are the only infinities. We use their digital segments like buckets of water from ocean. Infinities do not interact as interaction involves change of position, which is possible only in discrete objects. Infinities can coexist. Thus, space and time coexist to appear as spacetime.
Some hold that the dimension of a physical quantity is defined as the power to which the fundamental quantities are raised to express the physical quantity. Suppose there is a geometric shape with some associated quantity and we scale up the lengths of all sides of the shape by 2. If the associated quantity scales 2d, then d is the dimension. For example, take a plane polygon on a graph. If we double its side-lengths, we multiply it by 22 – change in area. For a polyhedron, doubling the sides gives a factor of 23 – change in volume. But these changes have other known geometrical properties also. When we take higher values like 4 or n, can these values be derived like length, area or volume for dimensions 1, 2, and 3 respectively? There is no higher dimension with similarly increasing geometrical properties. Why should we presume higher dimensions?
Can luminous intensity be a dimension? No, because dimension is a fixed quality that depicts invariant extent in a given direction, but intensity is neither invariant nor has a direction. It is uniform within its spread area. Is the mass or the amount of substance a dimension? No, because mass is defined as a dimensionless quantity representing the amount of matter in a particle. Can an effectively ‘dimensionless dimension of one’ be defined such that it is derived as a ratio of dimensions of the same type: as in deriving angle? No, because the statement is self-contradictory.
Can the measurement change the phenomenon, body, or substance under study in such a way that the quantity actually measured differs from the measurand: like the potential difference between the terminals of a battery may decrease when using a voltmeter with a significant internal conductance to perform the measurement? No; it is a difference of intensity – not dimension. For the same reason, thermal temperature is not a dimension. The open-circuit potential difference can be calculated from the internal resistances of the battery and the voltmeter. Further, this definition differs from that in VIM, 2nd Edition, Item 2.6, and some other vocabularies, that define the measurand as the quantity subject to measurement. The description of a measurand requires specification of the state of the phenomenon, body, or substance under study. In chemistry, the measurand can be a biological activity.
Do the number of dimensions we see is limited by our senses that define our perceptions? Are sight, sound, taste, smell, and touch the only senses an organism can have? Yes; they replicate the fundamental forces of Nature. Eyes use only electromagnetic radiation (उपयाम). Sound travels between bodies separated only by a medium – like gravitational interaction (उद्याम). Smell replicates strong interaction (अन्तर्याम). Taste replicates beta decay component of weak interaction (वहिर्याम). Touch replicates the rest of weak interaction – like alpha decay (यातयाम).
Some say birds have another sense – they can perceive and navigate by the Earth’s magnetic fields. This is not a different sense, but one aspect of touch (स्पर्श). Others say: certain animals, like the mantis shrimp, see different colors than we do. These are capacity to see different wavelengths (रूप) and not a different sense. Could there be dimensions that no organism, terrestrial or otherwise, could perceive (अतीन्द्रिय)? Whether it is an issue of size (अणुपरिमाण) or our limited senses (सङ्कुचितशक्ति), could extra-dimensions be reason for science to turn to mathematics as a means of advanced exploration? No. Speculation is not science.
Some say: dimension of a physical quantity is the index of each of the fundamental quantity (Length, mass, time,) which express that quantity. The dimension of mass, length and time are represented as [M], [L] and [T] respectively. For example, the dimension of speed can be derived as: Speed= distance/time = length/time = L/T = L.T-1.
In the above expression, there is no mention of mass, current or temperature because they do not play any role in defining this quantity. Or the dimension of mass, current, luminous intensity, temperature in this expression is zero. This is the brute force approach. A system consists of several necessary parameters. By arbitrarily reducing these parameters to zero, the system no longer remains as it is. Thus, it is a wrong description.
According to the principle of homogeneity of dimensional equations, the dimensions of fundamental quantities on LHS of an equation must be equal to the dimensions of the fundamental quantities on the RHS of that equation. The famous equation e = mc2 fails this test. Let us consider three quantities A, B and C such that C = A + B. According to this principle, the dimensions of C are equal to the dimensions of A and B. For example: we can write the dimensional first equation of kinematics: v = u + at as: [M0 L T-1] = [M0 L T-1] + [M0 L T-1] X [M0 L0 T] = [M0 L T-1].
Apart from the fact that mass and time are not dimensions as shown above (also being variables or emergent properties), the equation does not give information about the dimensional constant common to all parameters like mass, length and time. If a quantity depends on more than three factors having dimension, the formula cannot be derived. From the above equation, we cannot derive the formulae containing trigonometric function, exponential functions, logarithmic function, etc. The exact form of relation cannot be developed when there are more than one part in any relation. It gives no information whether a physical quantity is scalar or vector.
Others say: high-dimensional abstract spaces (independent of the physical space we live in) like parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics exist. This implies that position coordinates are not the only dimensions. For example, if a system consisting of homogenous ideal gas particles following the postulates of Kinetic Theory of Gases contained in an ideal confinement, the Pressure P; Volume V; Temperature T; and amount of gas i.e. no. of moles n, are the only required dimensions to state all the properties of that system. These are mere words. What is the proof in support of this argument? Has these spaces been discovered?
Some say: dimension is basically a number needed to specify something. For example the surface of a sheet of paper is two-dimensional because we can specify a point on the sheet of paper using the Cartesian coordinate system. But a graph is not the same as the real object it represents. The paper itself is three dimensional with varying thickness. We use one of its surfaces for plotting the graph. The real object that the graph represents has three dimensions. The graph gives only partial information. Further, what we “see” is the radiation emitted by a body – not the body proper. What we touch is the body proper and not the radiation emitted by it. Thus, both give incomplete information, which needs to be mixed to get a complete picture. For this reason, we have two eyes.
Dimension is not a sequence of addresses existing at different address locations along the street at different years. A fixed physical address and time does uniquely identify a specific house, but that is an arbitrary nomenclature – not a universal rule to qualify as dimension.
THE 10 DIMENSIONS
Dimension is an existential description. Change in dimension changes the existential description of the body irrespective of time and space. It never remains the same thereafter. Since everything is in a state of motion with reference to everything else at different rates of displacement, these displacements could not be put into any universal equation. Any motion of a body can be described only with reference to another body. Poincare and other have shown that even three body equations cannot be solved precisely. Our everyday experience shows that the motion of a body with reference to other bodies can measure different distances over the same time interval and same distance over different time intervals. Hence any standard equation for motion including time variables for all bodies or a class of bodies is totally absurd.
Dimension is generally understood as the number of independent coordinates needed to specify any point in a given space. For describing the size of an object, we use three numbers: length, breadth and elevation. For describing any position on Earth, we use three numbers: longitude, latitude and elevation, which also express the same information for a spherical structure. Photon and other radiation that travel at uniform velocity, are massless or without a fixed background structure – hence, strictly, are not “bodies”.
The three or six dimensions (including their negative directions from the origin) are not absolute terms, but are related to the order of placement of the object in the coordinate system of the field in which the object is placed. Since
- dimension of an object (वयुन) is related to the spread of the object, i.e., the relationship between its “confined structural inner space” and its “outer space” through which it is related to others (प्रचय संयोग),
- the outer space (वयोनाध) is infinite,
- the outer space does not affect inner space without breaking the dimension (वय),
the three or six dimensions remain invariant under mutual transformation of the axes (पर्यायवाची). If we rotate the object so that x-axis changes to the y-axis or z-axis, there is no effect on the structure (spread – विस्तार) of the object, i.e. the relative positions between different points on the body and their relationship to the space external to it remain invariant.
Based on the positive and negative directions (spreading out from or contracting towards) the origin, these describe six unique functions of position, i.e. (x,0,0), (-x,0,0), (0,y,0), (0,-y,0), (0,0,z), (0,0,-z), that remain invariant under mutual transformation. Besides these, there are four more unique positions, namely (x, y), (-x, y), (-x, -y) and (x, -y) where x = y for any value of x and y, which also remain invariant under mutual transformation. These are the ten dimensions and not the so-called “mathematical structures”. Since time does not fit in this description, it is not a dimension.
Our ancients named these 10 dimensions as: 1) Maahendree (माहेन्द्री), 2) Vaishwaanaree (वैश्वानरी), 3) Yaamyaa (याम्या), 4) Nairhtee (नैऋती), 5) Vaarunee (वारुणी), 6) Vaayavee (वायवी), 7) Kouveree (कौवेरी), 8) Aishaani (ऐशानी), 9) Braahmee (ब्राह्मी) and 10) Naagee (नागी). The nomenclature indicates their confining character (संस्त्यान).
(to be continued
