The Twin Paradox

Penrose (2004: 420):

The most dramatic of these contains the essence of the so-called ‘clock paradox’ (or ‘twin paradox’) of special relativity. Some readers may be familiar with this ‘paradox’; it refers to a space traveller who takes a rocket ship to a distant planet, travelling at close to the speed of light, and then returns to find that time on the Earth had moved forward many centuries, while the traveller might be only a few years older. … the space traveller’s clock indeed registers a shorter total elapsed time than those on Earth.

Blogger Comments:

To be clear, this 'paradox' is an expression of the finding, predicted by the Theory of Special Relativity, that the ticking of a clock takes longer the faster the clock is moving; that is, that time intervals expand as the speed of a body increases. The longer the time intervals, the fewer time intervals (units) measured.

The Time ‘Experienced’ By Photons Viewed Through Systemic Functional Linguistics

Penrose (2004: 414):

We should, however, recall … that, unlike the case for a massive particle, ∫ds is zero for a world line of a photon (so non-coincident points on the world-line can be ‘zero distance’ apart). This would also be true for any other particle that travels with the speed of light. The time ‘experienced’ by such a particle would always be zero, no matter how far it travels!

Blogger Comments:

As previously explained, the time that a photon "experiences" is the time measured by a clock that is travelling at the speed of light. If the time measured is zero, then the clock does not tick. In other words, processes do not unfold for a body travelling at the speed of light. In terms of the Special Theory of Relativity, viewed through Systemic Functional Linguistic Theory, this means that time intervals expand to infinity for a body moving at the speed of light.

Causality Structure Violations Viewed Through Systemic Functional Linguistics

Penrose (2004: 408-9):

An extreme situation arises when we have what is referred to as causality violation in which ‘closed timelike curves’ can occur, and it becomes possible for a signal to be sent from some event into the past of that same event! Such situations are normally ruled out as ‘unphysical’, and my own position would certainly be to rule them out, for a classically acceptable spacetime. Yet some physicists take a considerably more relaxed view of the matter being prepared to admit the possibility of the time travel that such closed timelike curves would allow. On the other hand, less extreme — though certainly somewhat exotic — causality structures can arise in some interesting spacetimes of great relevance to modern astrophysics, namely those which represent black holes.

Blogger Comments:

As previously explained, from the perspective of Systemic Functional Linguistic Theory, a causality structure of spacetime is actually a possibility structure of trajectories in spacetime. From this perspective, a 'causality violation' is a 'possibility violation' (an impossibility). On this basis alone, any trajectory into the past is an impossibility.

The Null-Cone Structure Of Spacetime Viewed Through Systemic Functional Linguistics

Penrose (2004: 407):

In my discussion above, I have chosen to emphasise the null-cone structure of spacetime, even more than its metric. In certain respects, the null cones are indeed more fundamental than the metric. In particular, they determine the causality properties of the spacetime. As we have just seen, material particles are to have their world lines constrained to lie within the cones, and light rays have world lines along the cones. No physical particle is permitted to have a spacelike world line, i.e. one outside its associated light cones. If we think of actual signals as being transmitted by material particles or photons, then we find that no such signal can pass outside the constraints imposed by the null cones. If we consider some point p in 𝕄, then we find that the region that lies on or within its future light cone consists of all the events that can, in principle, receive a signal from p. Likewise, the points of 𝕄 lying on or within p’s past light cone are precisely those events that can, in principle send a signal to the point p; … The null cones indeed define the causality structure of 𝕄: no material body or signal is permitted to travel faster than light; it is necessarily constrained to be within (or on) the light cones.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, null cones are not causality structures of spacetime. Instead, null cones represent all the possible trajectories (world lines) to or from a location in spacetime. That is, null cones are structures of processes (motions), not spacetime, and it is a matter of modalisation (possibility), not cause (causality).

Again, this confusion of process with circumstance lies at the heart of the misconstrual of the curvature of trajectories (geodesics) as the curvature of space-time, in the General Theory of Relativity.

Minkowskian Spacetime Viewed Through Systemic Functional Linguistics

Penrose (2004: 404, 406-7):

To complete Minkowski’s viewpoint with regard to the geometry underlying special relativity, and thereby define Minkowskian spacetime 𝕄, we must fix the scaling of g, so that it provides a measure of ‘length’ along world lines. This applies to curves in 𝕄 that we refer to as timelike which means that their tangents always lie within the null cones (Fig. 17.15a) and, according to the theory, are possible world lines for ordinary massive particles. This ‘length’ is actually a time and it measures the actual time 𝜏 that an (ideal) clock would register, between two points A and B on the curve … Photons have world lines that are called null (or lightlike), having tangents that are on the null cones (Fig. 17.15b). Accordingly the ‘time’ that a photon experiences (if a photon could actually have experiences) has to be zero!

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the null cones of Minkowskian spacetime represent processes (all possible trajectories) in spacetime, not spacetime. 

To be clear, the time that a photon "experiences" is the time measured by a clock that is travelling at the speed of light. If the time measured is zero, then the clock does not tick. In other words, processes do not unfold for a body travelling at the speed of light. In terms of the Special Theory of Relativity, viewed through Systemic Functional Linguistic Theory, this means that time intervals expand to infinity for a body moving at the speed of light.

The World Lines Of Spacetime Viewed Through Systemic Functional Linguistic

Penrose (2004: 394):

How do we incorporate Einstein’s notion of an ‘inertial’ motion into the structure of spacetime? As a step in the direction of the full Einstein theory, it will be helpful to consider a reformulation of Newton’s gravitational theory according to Einstein’s perspective. … Roughly speaking, in Cartan’s scheme, it is the inertial motions in this Einsteinian, rather than the Newtonian sense, that provide the ‘straight’ world lines of spacetime.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the notion that spacetime is structured by world lines confuses circumstance (spacetime) with process (world line). From this perspective, world lines are trajectories in spacetime, not structures of spacetime.

The Curvature Of A World Line Viewed Through Systemic Functional Linguistics

Penrose (2004: 389):

We can also consider world lines that are not geodesics. In ordinary spatial terms, these represent particle motions that accelerate. The actual magnitude of this acceleration is measured, in spacetime terms, as a curvature of the world-line. According to Newton’s second law, this acceleration is equal to the total force on the particle, divided by its mass. (This is Newton’s f = ma, in the form a = f ÷ m, where a is the particle’s acceleration, m is its mass, and f is the total force acting upon it.) Thus, the curvature of a world line, for a particle of given mass, provides a direct measure of the total force acting on that particle.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the curvature of a world line is not the curvature of spacetime, but the curvature of a process (motion) in spacetime. This confusion of process with circumstance accounts for the misconstrual of the curvature of trajectories (geodesics) as the curvature of spacetime, in the General Theory of Relativity.

Newton’s First Law In Spacetime Terms Viewed Through Systemic Functional Linguistics

Penrose (2004: 388):

Newton’s first law … states that the motion of a particle, upon which no forces act, must be uniform and in a straight line. This is called an inertial motion. In spacetime terms, the motion (i.e. ‘history’) of any particle, whether in inertial motion or not, is represented by a curve, called the world line of the particle.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the equation of motion with history confuses process with time. This confusion lies behind the misconstrual of the curvature of trajectories (geodesics) as the curvature of spacetime, in the General Theory of Relativity.

The Spacetime Of Galilean Relativity Viewed Through Systemic Functional Linguistics

Penrose (2004: 386-7):

What Galileo teaches us is that the dynamical laws are precisely the same when referred to any uniformly moving frame. … There is nothing to distinguish the physics of the state of rest from that of uniform motion. In terms of what has been said above, what this tells us is that there is no dynamical meaning to saying that a particular point in space is, or is not, the same point as some chosen point in space at a later time. In other words, our cinema-screen analogy is inappropriate! There is no background space — a ‘screen’ — which remains fixed as time evolves. 

We cannot meaningfully say that a particular point p in space (say, the point of the exclamation mark on the keyboard of my laptop) is, or is not, the same point in space as it was a minute ago. To address this issue more forcefully, consider the rotation of the Earth. According to this motion, a point fixed to the Earth’s surface (at the latitude of Oxford, say) will have moved by some 10 miles in the minute under consideration. Accordingly, the point p that I had just selected will now be situated somewhere in the vicinity of the neighbouring town of Witney, or beyond. But wait! I have not taken the Earth’s motion about the sun into consideration. If I do that, then I find that p will now be about one hundred times farther off, but in the opposite direction (because it is a little after mid-day, and the Earth’s surface, here, now moves oppositely to its motion about the Sun), and the Earth will have moved away from p to such an extent that p is now beyond the reach of the Earth’s atmosphere! But should I not have taken into account the sun’s motion about the centre of our Milky Way galaxy? Or what about the ‘proper motion’ of the galaxy itself within the local group? Or the motion of the local group about the centre of the Virgo cluster of which it is a tiny part, or of the Virgo cluster in relation to the vast Coma supercluster, or perhaps the Coma cluster towards ‘the Great Attractor’? 

Clearly we should take Galileo seriously. There is no meaning to be attached to the notion that any particular point in space a minute from now is to be judged as the same point in space as the one that I have chosen.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the general confusion here is of 'location' with 'thing'. First, a second-order location (space realised in a projected image) is confused with a first-order thing onto which the image is projected (a cinema screen). Then, a location ('point in space') is confused with a thing ('the point of an exclamation mark'), which naturally leads to the confused notion that a location changes its location ('is not the same point in space') as processes unfold.

If 'location' and 'thing' are not confused, then it is things that change their locations — relative to other things — as processes unfold.

Aristotelian Simultaneity Viewed Through Systemic Functional Linguistics

Penrose (2004: 384-5):

In Aristotelian physics — and, indeed, in the later dynamical scheme(s) of Galileo and Newton — there is an absolute notion of temporal simultaneity. Thus, it has absolute meaning to say, according to such dynamical schemes, that the time here, at this very moment, as I sit typing this in my office at home in Oxford, is ‘the same time’ as some event taking place on the Andromeda galaxy (say the explosion of some supernova star). 

To return to our analogy of the cinema screen, we can ask whether two projected images, occurring at two widely separated places on the screen, are taking place simultaneously or not. The answer here is clear. The events are to be taken as simultaneous if and only if they occur in the same projected frame. Thus, not only do we have a clear notion of whether or not two (temporally separated) events occur at the same spatial location on the screen, but we also have a clear notion of whether or not two (spatially separated) events occur at the same time. 

Moreover, if the spatial locations of the two events are different, we have a clear notion of the distance between them, whether or not they occur at the same time (i.e. the distance measured along the screen); also, if the times of the two events are different, we have a clear notion of the time interval between them, whether or not they occur at the same place.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the notion of 'simultaneity' is concerned with the ordering of processes in time. Importantly, in the case of the visual representation of processes projected onto a cinema screen, the temporal ordering of such processes is relative to — and construed by — the unfolding of the mental processes of viewers.

Euclidean Distance Viewed Through Systemic Functional Linguistics

Penrose (2004: 384):

In Euclidean geometry, whether 1-dimensional or 3-dimensional, there is a notion of distance. In the 3-dimensional spatial case, this is to be ordinary Euclidean distance (measured in metres, or feet, say); in the 1-dimensional case, this distance is the ordinary time interval (measured, say, in seconds).

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, distance is the extent, the interval, between locations in space or time. Because time is the dimension of the unfolding of processes, time is measured by the interval between locations in the unfolding of a (relatively) constant cyclical process, such as the ticking of a clock, the rotation of the Earth on its axis, or the revolution of the Earth around the Sun. It is important to distinguish the dimension, time, from the process which provides the unit (interval) of measurement, the ticking of a clock.

The Infinite Viewed Through Systemic Functional Linguistics

Penrose (2004: 357):

It appears to be a universal feature of the mathematics normally believed to underlie the workings of our physical universe that it has a fundamental dependence on the infinite. In the times of the ancient Greeks, even before they found themselves to be forced into considerations of the real-number system, they had already become accustomed, in effect, to the use of rational numbers. Not only is the system of rationals infinite in that it has the potential to allow quantities to be indefinitely large (a property shared with the natural numbers themselves), but it also allows for an unending degree of refinement on an indefinitely small scale. There are some who are troubled with both of these aspects of the infinite. They might prefer a universe that is, on the one hand, finite in extent and, on the other, only finitely divisible, so that a fundamental discreteness might begin to emerge at the tiniest levels.

Although such a standpoint must be regarded as distinctly unconventional, it is not inherently inconsistent. Indeed, there has been a school of thought that the apparently basic physical role for the real-number system ℝ is some kind of approximation to a ‘true’ physical number system which has only a finite number of elements.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, limitlessness in the physical universe and limitlessness in mathematical systems pertain to distinct orders of experience: first-order meaning (phenomena) and second-order meaning (metaphenomena), respectively. Mathematics is meaning construed of the meaning of the physical universe, but, as different systems, limitlessness in mathematics does not logically entail limitlessness in the physical universe.

Curled Up Internal Spatial Dimensions Viewed Through Systemic Functional Linguistics

Penrose (2004: 325-6):

The spaces that we need for the gauge theories of particle interactions (other than gravity), are different from these (and so they are something new), and it is best to think of them as referring to a kind of ‘spatial’ dimension that is additional to those of ordinary space and time. These extra ‘spatial’ dimensions are frequently referred to as internal dimensions, so that moving along in such an ‘internal direction’ does not actually carry us away from the spacetime point at which we are situated. …

Indeed, in many (or most?) of the current attempts at finding a deeper framework for fundamental physics (e.g. supergravity or string theory), the very notion of ‘spacetime’ is extended to higher dimensionality. The ‘internal dimensions’ then come about through the agency of these extra spatial dimensions, where these extra spatial dimensions are put on an essentially equal footing with those of ordinary space and time. The resulting ‘spacetime’ thus acquires more dimensions than the standard four. Ideas of this nature go back to about 1919, when Theodor Kaluza and Oskar Klein provided an extension of Einstein’s general relativity in which the number of spacetime dimensions is increased from 4 to 5. The extra dimension, enables Maxwell’s superb theory of electromagnetism to be incorporated, in a certain sense, into a ‘spacetime geometrical description’. However, this ‘5th dimension’ has to be thought of as being ‘curled up into a tiny loop’ so that we are not directly aware of it as an ordinary spatial dimension.

The analogy is often presented of a hosepipe, which is to represent a Kaluza–Klein-type modification of a 1-dimensional universe. When looked at on a large scale, the hosepipe indeed looks 1-dimensional: the dimension of its length. But when examined more closely, we find that the hosepipe surface is actually 2-dimensional, with the extra dimension looping tightly around on a much smaller scale than the length of the hosepipe. This is to be taken as the direct analogy of how we would perceive only a 4-dimensional physical spacetime in a 5-dimensional Kaluza–Klein total ‘spacetime’. The Kaluza–Klein 5-space is to be the direct analogue of the hosepipe 2-surface, where the 4-spacetime that we actually perceive is the direct analogue of the basically 1-dimensional appearance of the hosepipe.

 

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the notion of the internal dimensions of spacetime being curved ('curled up in a tiny loop') confuses circumstance (spacetime) with thing (surface). This confusion invalidates theories that depend on this notion, including all versions of string theory.

Geodesics Viewed Through Systemic Functional Linguistics

Penrose (2004: 303, 295):

Geodesics are important to us for other reasons. They are the analogues of the straight lines of Euclidean geometry. In our example of the sphere S2, considered above (Figs. 14.2–14.4), the geodesics are great circles on the sphere. More generally, for a curved surface in Euclidean space, the curves of minimum length (as would be taken up by a string stretched taut along the surface) are geodesics. We shall be seeing later (§17.9) that geodesics have a fundamental significance for Einstein’s general relativity, representing the paths in spacetime that describe freely falling bodies.

Blogger Comments:

Systemic Functional Linguistic Theory makes an important distinction between things, processes and circumstances. From this perspective, a geodesic on the surface of a sphere is the curvature of a thing, and a geodesic path in spacetime is the curvature of a process. Neither of these is the curvature of space, since space is a circumstance, not a thing or process. It is from this fundamental confusion that the mistaken notion of curved spacetime in General Relativity derives.

The Spatial Symmetry Of Objects And Physical Interactions Viewed Through Systemic Functional Linguistics

Penrose (2004: 247):

Spaces that are symmetrical have a fundamental importance in modern physics. Why is this? It might be thought that completely exact symmetry is something that could arise only exceptionally, or perhaps just as some convenient approximation. Although a symmetrical object, such as a square or a sphere, has a precise existence as an idealised (‘Platonic’; see §1.3) mathematical structure, any physical realisation of such a thing would ordinarily be regarded as merely some kind of approximate representation of this Platonic ideal, therefore possessing no actual symmetry that can be regarded as exact. Yet, remarkably, according to the highly successful physical theories of the 20th century, all physical interactions (including gravity) act in accordance with an idea which, strictly speaking, depends crucially upon certain physical structures possessing a symmetry that, at a fundamental level of description, is indeed necessarily exact!

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, objects are things that participate in processes, physical interactions are processes, and space is a circumstance of things participating in processes. From this perspective, the symmetry of an object, such as a square or a sphere, is not a symmetry of space, but a symmetry of a thing in space, and the symmetry of physical interactions is not a symmetry of space, but a symmetry of processes in space. 

This failure to distinguish space from things and processes in space has very important negative consequences, because, as explained elsewhere, it results in the false notion of curved space(-time) in the interpretation of the General Theory of Relativity.

In this view also, the 'Platonic ideal' of a physical square or sphere is a reconstrual of a first-order meaning (phenomenon) as a second-order meaning (metaphenomenon).