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Category Archives: Relativity

Relativity posts

Symmetric relativity

Although there are many experimental methods available to measure the speed of light, the underlying principle behind all methods [is] the simple kinematic relationship between constant velocity, distance and time given below:

c = D / t                     (1)

In all forms of the experiment, the objective is to measure the time required for the light to travel a given distance. (Ref.)

From the perspective of the experimenter, light is an object whose speed is to be determined. Even though the distance traversed is fixed, it is placed in the numerator because this speed is to be compared with the speeds of other objects. For the same reason the quantity to be measured, time, is placed in the denominator.

But if we take the perspective of the experiment, of what is measured, then the fixed distance is the independent variable, which is placed in the denominator. The dependent variable is the time, which is placed in the numerator, so the pace of light is measured:

= t / D                     (2)

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Mean speed and pace

Speed of a motion is the time rate of length change, that is, the length interval with respect to a timeline interval without regard to direction. Pace of a motion is the space rate of time change, that is, the time interval with respect to a baseline interval without regard to direction.

The symbol for speed is v = Δst and for pace is u = Δts. Instantaneous speed is ds/dt. Puncstancial pace is dt/ds.

There are two kinds of mean speed or pace: the time mean and the space mean. The time mean is the arithmetic mean if the denominators are a common time interval. The space mean is the arithmetic mean if the denominators are a common space interval. The time mean is the harmonic mean if the denominators are a common space interval. If the denominators are a common time interval, the space mean is the harmonic mean.

The time mean speed (TMS) is the arithmetic mean of speeds with a common time interval. The time mean pace (TMP) is the harmonic mean of paces with a common time interval. For example, the travel distance for vehicles on a highway during a time period is measured. The time mean speed or pace may then be calculated.

The space mean pace (SMP) is the arithmetic mean of paces with a common space interval. The space mean speed (SMS) is the harmonic mean of speeds with a common space interval. For example, the travel time for vehicles over a length of highway is measured. The space mean speed or pace may then be calculated.

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Three relativity transformations

Two transformations of inertial reference frames are well-known: the Galilean and the Lorentz transformations. There is a third transformation as well, which will be called the dual Galilean transformation. Below is a derivation of all three transformations, closely following the paper Getting the Lorentz transformations without requiring an invariant speed by Andrea Pelissetto and Massimo Testa (American Journal of Physics 83 (2015), p.338-340). Their approach is based on the work of von Ignatowsky in the early 20th century.

We wish to characterize the transformations that relate two different inertial frames. Let us consider two inertial observers K and K′. Let r = (x, x2, x3) and w = (t t2, t3) be space and time coordinates for K and = (x´, x2´, x3)´ and = (t´, t2´, t3´) be the corresponding quantities for K′.

In order to simplify the argument, we will restrict our considerations to the subgroup of transformations involving x and t only, setting x2´= x2, x3´ = x3, t2´ = t2, and t3´ = t3. This is equivalent to choosing coordinates so that K and K′ are in relative motion along the x and t directions in K and the x′ and t´ directions in K´.

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Simultaneity without clocks

Watches didn’t always exist. Neither did clocks that were transportable or manufactured in large quantities. I mention this because one way to determine the simultaneity of events is to have synchronized clocks transported to multiple locations – even an endless number of locations in theory.

How can an observer determine the simultaneous events from their frame of reference? Answer: simultaneous events are observed simultaneously by an observer. But how can this be reconciled with other observers who may observe the same events as non-simultaneous?

That is the point of relativity: applying transformations to coordinates from different frames of reference so that the equations of physics are the same in all reference frames. But relativity requires a convention of simultaneity (or a demonstration of what events are simultaneous events). Since I have defined time in terms of stopwatches rather than clocks, how can simultaneity be determined?

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Relativity alone

In a paper titled Nothing but Relativity (Eur. J. Phys. 24 (2003) 315-319) Palash B. Pal derived a formula for transformations between observers that is based on the relativity postulate but not a speed of light postulate. In a paper titled Nothing but Relativity, Redux (Eur. J. Phys. 28 (2007) 1145-1150) Joel W. Gannett presented an alternate derivation with fewer implicit assumptions. Here we’ll use Pal’s approach to derive the time-space version.

Consider two inertial timeframes S and , where the second one moves with legerity u, along the t-axis, with respect to the first one. There are two other time axes. The coordinates and radial distance in the S-timeframe will be denoted by t and x, and in the timeframe will be denoted with a prime. The time-space transformation equations have the form:

= T(t, x, u) and = X(t, x, u),

and out task is to determine these functions. A few properties of these functions can readily be observed. First, the principle of relativity tells us that if we invert the legerity in these equations, we must obtain the same functional forms:

t = T(t´, x´, –u) and x = X(t´, x´, –u).

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6D invariant interval

Since one may associate either the arclength (travel length) or the arctime (travel time) with direction, one might think that the full coordinates for every event are of the form (s, t, ê), with arclength s, arctime t, and unit vector ê. Since the direction is a function of either the arclength or the arctime, the coordinates would be either (s, t, ê(s)) or (s, t, ê(t)).

However, since s = ∫ || r′(τ) || , where the integral is from 0 to t, and t = ∫ || w′(σ) || , where the integral is from 0 to s (see here), this reduces to either (t, r) or (s, w).

But science seeks unification and so must combine these forms into one. In that case, both s and t are redundant, and the full coordinates for every event are of the form [r, w]. That is, there are three dimensions of space and three dimensions of time. The arclength and arctime are implicit, and may be made explicit through integration.

The standard exposition of special relativity looks at one dimension of space and one dimension of time. This is convenient and makes Δs = Δx and Δt = Δw1. But in general Δs and Δt will either be measured directly or found through integration.

What is the distance-like invariant interval then between two events? The interval in length units (proper length) is (dσ)² = (cdw)² – (dr)²,  where c is the speed of light. The interval in time units (proper time) is (dτ)² = (dw)² – (dr/c)².

This appears different from special relativity because it substitutes the vector dw for the scalar dt. However, the scalar (dt)² = (dw1)² + (dw2)² + (dw3)² so there is no discrepancy.

In order to demonstrate that this interval is invariant for two observers traveling at different rates, one must either convert dw to dt or convert dr to ds, which reduces the six dimensions to four.

The intervals above may be generalized for general relativity with the relation L = cP √(–gμν dxμ dxν), where P is the path, gμν is the metric tensor, and there are six coordinates xμ and xν.

Ten meanings of time

Carlo Rovelli’s “Analysis of the Distinct Meanings of the Notion of “Time” in Different Physical Theories” (Il Nuovo Cimento B, Jan 1995, Vol 110, No 1, pp 81–93) describes ten distinct versions of the concept of time, which he arranges hierarchically. Here are excerpts from his article:

We find ten distinct versions of the concept of time, all used in the natural sciences, characterized by different properties, or attributes, ascribed to time. We propose a general terminology to express these differences. p.81

… our aim is to emphasize the general fact that a single, pure and sacred notion of “Time” does not exist in physics. p.82

The real line is a traditional metaphor for the idea of time. Time is frequently represented by a variable t in R. The structure of R corresponds to an ensemble of attributes that we naturally associate to the notion of time. These are the following:
a) The existence of a topology on the set of the time instants, namely the existence of a notion of two time instants being close to each other, and the characteristic “one dimensionality” of time;
b) The existence of a metric. Namely the possibility of stating that two distinct time intervals are equal in magnitude. We denote this possibility as metricity of time.
c) The existence of an ordering relation between time instants. Namely, the possibility of distinguishing the past direction from the future direction;
d) The existence of a preferred time instant, namely the present, the “now”. p.83

In the natural language, when we use the concept of time we generally assume that time is one-dimensional, metrical, external, spatially global, temporally global, unique, directed, that it implies a present, and that it allows memory and expectations. The concept of time used in Newtonian physics is one-dimensional, metrical, external, spatially global, temporally global, unique, but it is not directed and it does not have a present. In thermodynamics, time has the additional property of being directed. Proper time along world line in general relativity is one-dimensional, metrical, temporally global but it is not external, not spatially global, not unique; on the other side, the time determined by a matter clock is one-dimensional, metrical, but not temporally global, an so on. p.87

… the notion of present, of the “now” is completely absent from the description of the world in physical terms. This notion of time can be described by the structure of an affine line A. p.88

… our list does not include the possibility of considering a non-metric but directional notion of time. p.89

Table I. [without the fourth column]

Time concept Attributes Example
time of natural language memory brain
time with a present present biology
thermodynamical time directional thermodynamics
Newtonian time uniqueness Newton mechanics
special relativistic time being external special relativity
cosmological time space global proper time in cosmology
proper time time global world line proper time
clock time metricity clocks in general relativity
parameter time 1-dimensional coordinate time
no time none quantum gravity

… our hypothesis concerning time is that the concepts of time with more attributes are higher-level concepts that have no meaning at lower levels. p.91

If this hypothesis is correct, then we should deduce from it that most features of time are genuinely meaningless for general systems. p.91

… we suggest that the very notion of time, with any minimal characterization, is likely to disappear in a consistent theory that includes relativistic quantum-gravitational systems. p.91

… the concept of time, with all its attributes, is not a fundamental concept in nature, but rather that time is a progressively more specialized concept that makes sense only for progressively more special systems. p.92

Algebraic relativity

Relativity may be derived as an algebraic relation among differentials. Consider motion in the x spatial dimension, with a differential displacement, dx, differential velocity displacement, dv, and arc (elapsed) time t:

dx² = (dx/dt)²dt² = dv²dt² =  d(vt)².

Let there be a constant, c:

dx² = d(vt)² = d(cvt)²/c² = d(ct)² (v/c)² = d(ct)² (1 – (1 – v²/c²)).

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Four space and time dimensions

Since the development of relativity theory, space and time have been combined in a four-dimensional continuum. Because the speed of light is an absolute value in relativity theory, it acts as a conversion factor between space and time. Accordingly, the four dimensions may be understood as any combination of space and time:

4 + 0: Four dimensions of space and none of time. The invariant spacetime interval is commonly expressed in spatial terms only as s² = x² + y² + z² – ct²  with signature (+++–). The opposite signature is also used (+–––). The factor c converts the time coordinate into a distance coordinate. Note that there is an implicit 1-3 split in dimensions.

3 + 1: Three dimensions of space and one of time. This has been the common conception of space and time for centuries.

2 + 2: Two dimensions of space and two of time. This was discussed in the previous post here. It is a pictorial representation of four dimensions, two at a time.

1 + 3: One dimension of space and three of time. This has been discussed in many posts such as here. It may have been the ancient conception.

0 + 4: No dimension of space and four of time. This is the invariant spacetime interval expressed in temporal terms only as τ² = t² – x²/c² – y²/c² – z²/c². There is an implicit 1-3 split in dimensions.

The above should be understood in the context of an implicit six-dimensionality, as was discussed here. As soon as rates are considered, three dimensions of either space or time must be compressed to one so that only four dimensions remain.

A new geometry for space and time

This blog has described how as the distances between places cover three dimensions of space, so the durations between events cover three dimensions of time. One way of looking at this is as a map with the distance and duration given between places, such as this from the Interstate Drive Times and Distances:

There are two numbers for each leg or link in the map; one number in red for the distance in miles and another in blue for the time in hours and minutes. The durations are not proportional to the distance, which reflects the differing local conditions and topography.

It is common to adopt a speed that represents the typical speed for the mode of travel, which I’m calling the modal speed (or rate). This serves various purposes, one of which is for pre-trip planning to estimate the travel time to a destination. It can also serve as the conversion of distance and duration for the mode, much as the speed of light serves for relativity.

Divergence from the modal rate may be represented through a third dimension (z), similar to a topographic map. The modal rate then would be represented by a flat topography. A link with greater distance than the modal one would be like a positive z coordinate, and a link with less than the modal one would be like a negative z coordinate. The resultant link would be determined by the Pythagorean theorem for positive values of z, and by a hyperbolic version (as if z were imaginary) for negative values of z.

That is, |A|² = x² + y² + z² if z > 0 (Pythagorean), and |A|² = x² + y² – z² if z < 0 (hyperbolic). For positive values of z, the magnitude of the resultant link is greater than unity, that is, greater than the modal rate. But for negative values of z, the magnitude of the resultant link is less than unity, that is, less than the modal rate.

The result may be mapped like a contour or isoline map, except that negative values have a hyperbolic geometry. The simplest way to see this is to take Δt = 1 and Δr = > 1, which uses the Pythagorean theorem. If either the time interval changes to Δt < 1 or the space interval is reduced to Δr < 1, the hyperbolic version is used for the link.