An Alternate Presentation of a Generalization of the Lorentz Transformation to Allow Acceleration of a Coordinate System

The traditional version of the Lorentz transformation is used to relate the coordinates of an event (a point in space-time) between different inertial coordinate systems in the absence of any gravitational effects. It is limited to those cases in which both reference frames are inertial, so they have a constant velocity relative to each other. By utilizing postulates regarding the equivalence between coordinates measured in an accelerating reference frame and coordinates measured in a reference frame instantaneously at rest with the accelerating frame, together with the traditional version of the Lorentz transformation, it is possible to derive a generalized version of the transformation applicable to the case in which one of the coordinate systems is accelerating. This is not new, and this report does not produce any new final conclusions. What is different in this report is the style of analysis and the format in which the relevant equations are presented. The advantage is a simpler way to derive conclusions, albeit the conclusions are not new. The conventional approach derives basis vectors (a tetrad) for an accelerating system from assumptions regarding orthonormality, being nonrotating, and some assumption about relative orientations with an inertial system. Then event coordinates in the accelerating system are defined in terms of expansions in basis vectors. The approach used here starts by deriving coordinate transformations directly, without any need for basis vectors, by utilizing a to-be-determined inertial reference frame. This reference frame is one of the entities to be solved but equations for it are derived and very simple. Having solved that part, the coordinate transformation is established. From the coordinate transformation, basis vectors are shown to be tangent vectors to coordinate curves and the equations for them are very simple. By defining suitable quantities and introducing suitable notation, the generalized version of the Lorentz transformation can be written in a way that is almost as simple as the traditional (constant velocity) version when calculating the coordinates of an event in an inertial system when given the coordinates in an accelerating system. Unfortunately, calculations of the inverse transformation, i.e., calculating the coordinates in the accelerating system when given coordinates in an inertial system, are more cumbersome. Worse yet, while a suitably selected history and future ensures the existence of an inverse transformation, there can exist space-time points for which it is not unique. However, the metric tensor can be derived in the accelerating system for the general case and is included in this report. This is used to calculate time dilations and Doppler effects that are outside the scope of inertial coordinate systems and have gravitational interpretations.

The traditional version of the Lorentz transformation is used to relate the coordinates of an event (a point in space-time) between different inertial coordinate systems in the absence of any gravitational effects. It is limited to those cases in which both reference frames are inertial, so they have a constant velocity relative to each other. By utilizing a postulate (stated below in (1.1)) regarding the equivalence between coordinates measured in an accelerating reference frame and coordinates measured in a reference frame instantaneously at rest with the accelerating frame, together with the traditional version of the Lorentz transformation, it is possible to derive a generalized version of the transformation applicable to the case in which one of the coordinate systems is accelerating. This is not new, and this report does not produce any new final conclusions. What is different in this report is the style of analysis and the format in which the relevant equations are presented. The advantage is a simpler way to derive conclusions, albeit the conclusions are not new. The conventional approach, e.g. in [1, pp. 170-172] derives basis vectors (a tetrad) for an accelerating system from assumptions regarding orthonormality, being nonrotating, and some assumption about relative orientations with an inertial system. Then event coordinates in the accelerating system are defined in terms of expansions in basis vectors. The approach used here starts by deriving coordinate transformations directly, without any need for basis vectors, by utilizing a to-be-determined inertial reference frame. This reference frame is one of the entities to be solved but equations for it are derived and very simple. Having solved that part, the coordinate transformation is established. From the coordinate transformation, basis vectors are shown to be tangent vectors to coordinate curves and the equations for them are very simple. This report is self-contained in that derivations are provided for all conclusions. By defining suitable quantities and introducing suitable notation, the generalized version of the Lorentz transformation can be written in a way that is almost as simple as the traditional (constant velocity) version when calculating the coordinates of an event in an inertial system when given the coordinates in an accelerating system. Unfortunately, calculations of the inverse transformation, i.e., calculating the coordinates in the accelerating system when given coordinates in an inertial system, are more cumbersome. Worse yet, while a suitably selected history and future ensures the existence of an inverse transformation, there can exist space-time points for which it is not unique. However, the metric tensor can be derived in the accelerating system for the general case and is included in this report. This is used to calculate time dilations and Doppler effects that are outside the scope of inertial coordinate systems and have gravitational interpretations.

This report frequently uses the phrase “as seen by an observer”. In the context of this report, the word “seen” does not refer to a literal visual image. A visual image of an event (an “event” is a point in space-time) is received by an observer after a time delay, the time required for a light signal sent by the event to reach the observer. In the context of this report, the phrase “as seen by an observer” does not refer to a visual image but, instead, means that the observer assigned space-time coordinates to the event1. For example, the phrase “the observer sees a clock to be running slow” means that if the observer assigns a time coordinate to one tick of that clock, and another time coordinate to the next tick of that clock, the time between consecutive ticks is longer than the time between consecutive ticks of the observer’s own clock.

1If the only information available to the observer with which to assign space-time coordinates to an event is a visual image, it is assumed that the observer accounted for the travel time of the light signal when assigning these coordinates so that the assigned time coordinate does not include this travel time.

This report considers two observers. One, called the home observer, uses a clock at rest at the origin of his coordinate system, which is an inertial coordinate system (recognized to be inertial by freely moving particles having constant velocities)2. The other observer, called the traveler (with travel defined to be relative to home), uses a clock at the origin of his coordinate system, called the traveler’s system. The traveler’s system is not rotating but may have a translational acceleration relative to the home system. A space-time point denoting the start of the traveler’s journey has an arbitrary initial spatial translation relative to the home system and an arbitrary initial velocity in the home system, but the clocks are synchronized so that the traveler’s time coordinate and home time coordinate of the start of the journey are both zero. The goal is to relate, for an arbitrary event, the traveler’s coordinates of that event to the home coordinates of the same event.

2Gravity is not considered here so we say freely moving instead of free falling.

The traveler’s clock traces out a worldline as seen by the home observer. Information that is assumed to be known includes the three-dimensional (spatial) vector function, denoted X(t), which is a parameterization of the traveler’s worldline expressing the spatial coordinates x, in the home system, of a point on the traveler’s worldline, in terms of the coordinate time t in the home system3. From this parametrization, the velocity of the traveler’s clock as seen by the home observer is the function V(t) calculated from V(t) = dX(t)/dt.

3The notation used here uses bold block font for three-dimensional spatial vectors. Bold cursive font will be used for 4-vectors discussed later.

All conclusions in this report are derived from two postulates. One, taken from [1, p. 164] with minor paraphrasing, is the statement:

$\left[\begin{array}{l}\text{Carefully constructed clocks (e}\text{.g}\text{., atomic clocks), when accelerated, will tick}\\ \text{at the same rate as unaccelerated clocks moving momentarily along with them,}\\ \text{and sufficiently rigid accelerated measuring rods will measure the same lengths}\\ \text{as unaccelrated rods moving momentarily along with them}\text{. }\end{array}\right]$ (1.1)

The second postulate that all results in this report are derived from is the Lorentz transformation applicable to the case in which the traveler has a constant velocity v as seen by the home observer. This is presented in various formats in [2] through [3-8] but the format preferred for this work is taken from [9] and given as follows. Consider an event with home coordinates denoted $(x_1,t_1)$ and traveler coordinates denoted $({\overline{x}}_1,{\overline{t}}_1)$ , and a second event with home coordinates denoted $(x_2,t_2)$ and traveler coordinates denoted $({\overline{x}}_2,{\overline{t}}_2)$ . The standard (constant velocity) Lorentz transformation in [9] relates the displacement between these events expressed in each of the two systems according to4

4The open symbol ○ denotes the Euclidean dot product between three-dimensional (spatial) vectors. The unit vector n is a unit vector in the context of this dot product. This dot product is distinguished from a four-dimensional dot product, denoted by a solid dot, defined later. ___________________________________________________________________________________________________

$t_2-t_1=\gamma \left({\overline{t}}_2-{\overline{t}}_1+\frac{v\circ \left[{\overline{x}}_2-{\overline{x}}_1\right]}{c^2}\right)$ (1.2a)

$x_2-x_1={\overline{x}}_2-{\overline{x}}_1+\left(\gamma -1\right)\left[\left({\overline{x}}_2-{\overline{x}}_1\right)\circ n\right]n+\gamma v\left({\overline{t}}_2-{\overline{t}}_1\right)$ (1.2b)

where the parameter γ and the unit vector n are defined by

$\gamma \equiv \frac{1}{\sqrt{1-\frac{v^2}{c^2}}},n\equiv \frac{v}{v}$ (1.3)

Let P denote an arbitrary space-time point on the traveler’s worldline. The coordinates of P in the home system are denoted $(x_P,t_P)$ , and the coordinates in the traveler’s system are denoted $({\overline{x}}_P,{\overline{t}}_P)$ . The traveler’s clock is at the traveler’s origin and the worldline considered here is the worldline of the traveler’s clock so

${\overline{x}}_P=0$ (2.1)

Also, even if the traveler’s clock is accelerating, the hypothesis (1.1) states that it is still true that increments of proper time for displacements along the worldline of the traveler’s clock equal increments of time measured by the traveler’s clock. With proper time and the traveler’s clock both set to zero when the traveler starts his journey, the proper time along the worldline up to the point P equals ${\overline{t}}_P$ . The proper time is an invariant that can be calculated in the home system, giving

${\overline{t}}_P=\tau \left(t_P\right)$ (2.2)

where τ is the proper time calculated in the home system according to

$\tau \left(t\right)={\displaystyle {\int }_0^t\frac{1}{\gamma (\xi )}d\xi }$ (2.3)

with γ(t) defined by

$\gamma (t)\equiv \frac{1}{\sqrt{1-\frac{V^2(t)}{c^2}}}$ (2.4)

Now consider an inertial system that is “local” with the traveler’s system at the arbitrary point P on the traveler’s worldline. This statement is defined to mean that there exists a time at which the origins of the two systems are at the same location and with each axis of the two systems aligned, the two systems are at rest relative to each other at this time, the space-time coordinates of the origins at this time are the space-time coordinates of P, and the clock in the inertial system is set to ${\overline{t}}_P$ at this time. We will call this new inertial system the P-system. We use the postulate, stated in (1.1), that the traveler’s clock will momentarily (while at P) tick at the same rate as the P-system’s clock, and the traveler’s measuring rods will momentarily (while at P) measure the same lengths as the P-system’s measuring rods. To utilize this local inertial system in the analysis we use the following approach. Instead of selecting an arbitrary event and attempting to calculate traveler coordinates of it, we work in the opposite direction by selecting traveler coordinates and then identify what the event is that has those coordinates (this identification can be done by finding the home coordinates). Because the space-time point P is arbitrary, it is sufficiently general to confine our attention to those events that the P-system declares to be simultaneous with the event P. The event to be constructed will be called E, with traveler coordinates denoted $({\overline{x}}_E,{\overline{t}}_E)$ and home coordinates denoted $(x_E,t_E)$ . With the P-system declaring E to be simultaneous with P, we conclude from the hypothesis (1.1) that the traveler also declares E to be simultaneous with P,5 implying that

5While the P-system declares the events E and P to be simultaneous, it is not necessarily true that the home observer declares them to be simultaneous. ___________________________________________________________________________________________________

${\overline{t}}_E={\overline{t}}_P$ (2.5)

A constraint that the event i must satisfy to be treated in this analysis is that there must exist a point P on the traveler’s worldline such that the inertial system local with the traveler’s system at P sees E to be simultaneous with P. Given that E is such an “allowed” event, so that E is simultaneous with P as seen by the P-system, the coordinates of E in the P-system are the same as the coordinates in the traveler’s system. Therefore, the coordinates in the P-system are also $({\overline{x}}_E,{\overline{t}}_E)$ . Also, the coordinates of P in the P-system are $({\overline{x}}_P,{\overline{t}}_P)$ = $(0,{\overline{t}}_P)$ . The P-system has velocity V(tP) as seen in the home system, so the transformation from barred coordinates to unbarred coordinates is obtained from (1.2) by replacing subscripts (with 2 replaced by E and 1 replaced by P), replacing v with V(tP), and replacing γ with γ(tP). The results, after simplification by using (2.1) and (2.5) are

$t_E-t_P=\gamma \left(t_P\right)\frac{V\left(t_P\right)\circ {\overline{x}}_E}{c^2}$ (2.6a)

$x_E-x_P={\overline{x}}_E+\left[\gamma \left(t_P\right)-1\right]\left({\overline{x}}_E\circ n\right)n$ . (2.6b)

Information assumed to be available includes the functions X(t) and V(t) explained in Section 1. From these we construct the functions γ(t) and τ(t) via (2.4) and (2.3), so all these functions, which refer to the traveler’s worldline as seen by the home observer, are regarded as given. The goal here is to calculate the home coordinates of an arbitrary event E when the given information, in addition to the above functions, consists of the traveler’s coordinates $({\overline{x}}_E,{\overline{t}}_E)$ . The following steps are used:

The first step calculates t_P. This is done by combining (2.5) with (2.2) so t_P is calculated from the given ${\overline{t}}_E$ via

${\overline{t}}_E=\tau \left(t_P\right)$ . (3.1)

Note that γ(t) is a strictly increasing function so there exists a unique t_P satisfying (3.1) providing only that ${\overline{t}}_E$ \ is between zero and the largest display of the traveler’s clock (finite if its worldline has a stopping point). With t_P now solved, we calculate x_P via

$x_P=X\left(t_P\right)$ . (3.2)

With this done, all terms appearing in (2.6) have been evaluated except for the terms, x_E and t_E, to be solved. They can now be solved by writing (2.6) as

$t_E=t_P+\gamma \left(t_P\right)\frac{V\left(t_P\right)\circ {\overline{x}}_E}{c^2}$ (3.3a)

$x_E=x_P+{\overline{x}}_E+\left[\gamma \left(t_P\right)-1\right]\left[{\overline{x}}_E\circ V\left(t_P\right)\right]\frac{V\left(t_P\right)}{V^2\left(t_P\right)}$ . (3.3b)

The final results are (3.1) through (3.3). The paragraphs below discuss some implications.

It is interesting that (3.3a) implies that t_E = t_P if either V(t_P) = 0 or ${\overline{x}}_E$ = 0. This could have been anticipated even before deriving (3.3). Recall that the events P and E are always simultaneous as seen by the P-system, because this is a condition that was imposed on E. If V(t_P) = 0, the P-system is stationary relative to home, so P and E are also simultaneous as seen by the home system. Therefore, we could have anticipated that tE = tP when V(t_P) = 0 even before deriving (3.3). Now suppose ${\overline{x}}_E$ = 0. Recall again that the event E is simultaneous with P as seen by the P-system, and therefore also as seen by the traveler, so they have the same time coordinates in the traveler’s system. Also, P is on the traveler’s worldline, so its spatial coordinates are zero in the traveler’s system. Therefore ${\overline{x}}_E$ , = 0 if and only if P and E have the same space-time coordinates in the traveler’s system, in which case they are the same space-time points and therefore have the same space-time coordinates in all systems. Therefore, we could have anticipated that t_E = t_P and x_E = x_P when ${\overline{x}}_E$ = 0 even before deriving (3.3).

A topic that can be discussed here is time dilation. However, there are two versions of time dilation because the traveler and home observer can disagree on simultaneity6. Specifically, they can disagree on which readings on their two clocks are simultaneous. One way to clearly state which version of time dilation is being considered starts with the realization that a clock display is not only a time coordinate for some observer, but it can also be regarded as an event (think of a 5:00 o’clock whistle which is clearly an event). The version of time dilation that is being considered is clearly stated when stating which clock is the one whose displays are the events for which coordinates are to be determined. Here we discuss the version of time dilation in which displays on the traveler’s clock are not only traveler time coordinates of the displays but also events. The goal is to calculate home time coordinates of these events, so we will calculate how fast the traveler’s clock is ticking as seen by the home observer. This calculation is very simple. The traveler’s worldline is taken to be the worldline of the traveler’s clock (the clock defines the traveler’s origin) so traveler clock displays are events for which ${\overline{x}}_E$ = 0. From the previous paragraph we conclude that t_E = t_P so (3.1) becomes

6There is a third version of time dilation in Section 14. __________________________________________________________________________________________________

${\overline{t}}_E=\tau \left(t_E\right)\text{(when the events are traveling clock displays denoted }{\overline{t}}_E\text{)}$ (3.4a)

which can be inverted via (2.3) and (2.4) to solve for t_E. Also, differentiating (3.4a) while using (2.3) gives

$\frac{d{t}_E}{d{\overline{t}}_E}=\gamma \left(t_E\right)\text{(when the events are traveling clock displays denoted }{\overline{t}}_E\text{)}$ . (3.4b)

The instantaneous result (3.4b) is the same simple result that is obtained without acceleration. This is a time dilation in that dt_E, which is the time seen by the home observer that is needed to change the display of the traveler’s clock, is greater than $d{\overline{t}}_E$ , which is the change of the display of the traveler’s clock and therefore also equal to the time seen by the traveler needed to change the display of the traveler’s clock. A larger time between displays corresponds to a slower clock, so the home observer sees the traveler’s clock to be running slow. Unfortunately, the version of time dilation in which displays on the home clock are events for which traveler time coordinates are to be calculated, i.e., that calculates how fast the home clock is ticking as seen by the traveler, is not so simple but an example is given in Section 12.

Note that (3.1) through (3.3) are as simple to write as (1.2) while having enough generality to apply to an accelerating traveler. However, (3.1) through (3.3) have a different format than (1.2). As a check for consistency, we show that the two formats are equivalent for the case in which the traveler has a constant velocity so that (1.2) also applies. Specifically, we will show that (3.1) through (3.3) produce (1.2) when the traveler’s velocity is constant.

Given a constant traveler velocity there is no loss of generality by orienting the coordinate axis so that the velocity is in the x-direction with x-component denoted v. In this case, γ is a constant given by (1.3), so the definition (2.3) of τ together with (3.1) gives

$t_P=\gamma {\overline{t}}_E\text{(constant velocity)}$ . (4.1)

Also, the vector equation (3.2) has components

$x_P=v{t}_P,y_P=0,z_P=0\text{(constant velocity)}$

where we treat the case in which the traveler starts the journey from the home origin. The above can be written via (4.1) as

$x_P=\gamma v{\overline{t}}_E,y_P=0,z_P=0\text{(constant velocity)}$ . (4.2)

Next, the scaler equation (3.3a) becomes

$t_E=t_P+\gamma \frac{v{\overline{x}}_E}{c^2}\text{(constant velocity)}$ (4.3)

while the vector equation (3.3b) has the components

$x_E=x_P+\gamma {\overline{x}}_E,y_E={\overline{y}}_E,z_E={\overline{z}}_E,\text{(constant velocity)}$ . (4.4)

The last step substitutes (4.1) into (4.3) and uses (4.2) together with (4.4) to get

$t_E=\gamma \left({\overline{t}}_E+\frac{v{\overline{x}}_E}{c^2}\right)\text{(constant velocity)}$ (4.5a)

$x_E=\gamma \left({\overline{x}}_E+v{\overline{t}}_E\right),y_E={\overline{y}}_E,z_E={\overline{z}}_E,\text{(constant velocity)}$ (4.5b)

which is the familiar format for expressing the Lorentz transformation applicable to inertial reference frames.

A simple application of the transformation given by (3.1) through (3.3) is the case in which the traveler departs from home with a constant speed v but changes direction as needed to return home. The goal is to compare clock readings (home versus traveler) when the traveler returns home. This comparison was already given by the time-dilation equation (3.4a) but the goal here is to provide another derivation of the same result. We first continue with the derivation that starts with (3.4a). When v is constant this equation reduces to

$t_E=\gamma {\overline{t}}_E\text{(constant }v\text{, }E\text{ is display of }{\overline{t}}_E\text{ on traveler's clock)}$ . (5.1)

In (5.1), the display ${\overline{t}}_E$ on the traveler’s clock is the event (also the traveler’s time coordinate of the event) while t_E is the home time coordinate of the event.

Next, we use a different derivation of the same result (5.1) just to demonstrate consistency. Instead of the event E consisting of the traveler’s clock displaying the reading ${\overline{t}}_E$ , which is the interpretation of E in (3.4) and (5.1), here we take the event E to be the traveler’s return home. In this interpretation, ${\overline{t}}_E$ is the traveler’s time coordinate of his return home while t_E is the home time coordinate of the traveler’s return home. The first step in the analysis notes that a constant γ in (2.3) together with (3.1) gives

$t_P=\gamma {\overline{t}}_E\text{(constant }v\text{)}$ . (5.2)

But the event E is a point on the traveler’s worldline, so ${\overline{x}}_E$ = 0 and (3.3a) gives

$t_E=t_P\text{(}E\text{ on traveler's world line)}$ . (5.3)

Combining (5.3) with (5.2) produces agreement with (5.1).

When traveler coordinates of an event are given, the home coordinates are obtained from a simple procedure in Section 3. We now consider the case in which the home coordinates are given, and the goal is to calculate the traveler’s coordinates. Using the transformation in either direction, one of the unknowns to be solved is t_P. When the traveler coordinates are given, t_P is solved via (3.1). When the home coordinates x_E and t_E are the givens, we need to construct an equation that contains the unknown t_P with all other terms known. This is done by taking the dot product of (3.3b) with V(t_p) to get

$V\left(t_P\right)\circ x_E=V\left(t_P\right)\circ x_P+V\left(t_P\right)\circ {\overline{x}}_E+\left[\gamma \left(t_P\right)-1\right]\left[{\overline{x}}_E\circ V\left(t_P\right)\right]=V\left(t_P\right)\circ x_P+\gamma \left(t_P\right)\left[{\overline{x}}_E\circ V\left(t_P\right)\right]$

which gives

$\gamma \left(t_P\right)V\left(t_P\right)\circ {\overline{x}}_E=V\left(t_P\right)\circ \left(x_E-x_P\right)$

Substituting this into (3.3a) while using (3.2) gives

$t_E=t_P+\frac{1}{c^2}V\left(t_P\right)\circ \left(x_E-X(t_P)\right)$ . (6.1)

With the parameters t_E and x_E given, and the functions V and X given, (6.1) is the equation governing t_P. An unfortunate property of (6.1) is that there are example traveler worldlines in which (6.1) does not have a unique solution for t_P for some inputs t_E and x_E. In some cases, no solution exists. For some other cases, solutions exist but are not unique. This is discussed in more detail in Section 8. For the remainder of this discussion, we assume that event E is one in which (6.1) has a unique solution for t_P and this solution has been found (by some numerical root finding routine if necessary).

Given that t_P satisfying (6.1) can be found and has been found, we calculate ${\overline{t}}_E$ via (3.1). The last step of the inverse transformation solves for ${\overline{x}}_E$ as follows. To shorten the notation, we define, for an arbitrary vector W, a parallel part $W_{\parallel }$ and a perpendicular part $W_{\perp }$ , with respect to the direction of V(t_P), according to

$W_{\parallel }\equiv \left(W\circ V\left(t_P\right)\right)\frac{V\left(t_P\right)}{V^2\left(t_P\right)},W_{\perp }\equiv W-W_{\parallel }$ . (6.2)

The advantage of this notation is a less cumbersome way to write (3.3b), which is

$x_E-x_P={\overline{x}}_E+\left[\gamma \left(t_P\right)-1\right]{\overline{x}}_{E,\parallel }={\overline{x}}_{E,\perp }+\gamma \left(t_P\right){\overline{x}}_{E,\parallel }$ . (6.3)

The parallel part of the left side equals the parallel part of the far right side, and the perpendicular part of the left side equals the perpendicular part of the far right side so

${\left(x_E-x_P\right)}_{\parallel }=\gamma \left(t_P\right){\overline{x}}_{E,\parallel },\text{or}{\overline{x}}_{E,\parallel }=\frac{1}{\gamma \left(t_P\right)}{\left(x_E-x_P\right)}_{\parallel }$ (6.4a)

and

${\overline{x}}_{E,\perp }={\left(x_E-x_P\right)}_{\perp }$ . (6.4b)

Now express ${\overline{x}}_E$ as the sum of the parallel part plus perpendicular part while using (6.4) to get

${\overline{x}}_E={\overline{x}}_{E,\perp }+{\overline{x}}_{E,\parallel }={\left(x_E-x_P\right)}_{\perp }+\frac{1}{\gamma \left(t_P\right)}{\left(x_E-x_P\right)}_{\parallel }$ .

Now use the second equation in (6.2) to write this as

${\overline{x}}_E=x_E-x_P-{\left(x_E-x_P\right)}_{\parallel }+\frac{1}{\gamma \left(t_P\right)}{\left(x_E-x_P\right)}_{\parallel }=x_E-x_P+\left[\frac{1}{\gamma \left(t_P\right)}-1\right]{\left(x_E-x_P\right)}_{\parallel }$ .

Finally, we use the first equation in (6.2) to write this as

${\overline{x}}_E=x_E-x_P+\left[\frac{1}{\gamma \left(t_P\right)}-1\right]\left[\left(x_E-x_P\right)\circ V\left(t_P\right)\right]\frac{V\left(t_P\right)}{V^2\left(t_P\right)}$ .

Before summarizing the above results, we digress by noting two interesting properties of (6.1). One property is that t_E = t_P if V(t_P) = 0. This could have been anticipated even before deriving (6.1) as already explained in Section 3. The second interesting property deserves more discussion. That property is the implication that if x_E = X(t_P) then t_E = t_P. This could have been anticipated even before deriving (6.1) as follows. Suppose x_E = X(t_P), i.e., x_E = x_P. Then the events E and P have the same spatial coordinates in the home system. They also have the same time coordinates in the P-system (by choice of point P) so it is not surprising that E and P are the same space-time point, implying t_E = t_P. This conclusion is made rigorous by (6.1), showing that it is true that the condition x_E = X(t_P) implies t_E = t_P. Therefore, the condition x_E = X(t_P) implies the simultaneous conditions x_E = x_p and t_E = t_P, which implies that E and P are the same space-time points, which implies that they have the same coordinates in all systems. A variety of implications follow by combining this implication with others already stated (e.g., that the event E is on the traveler’s worldline if and only if ${\overline{x}}_E$ = 0, and the conclusion from Section 3 that t_E = t_P and x_E = x_P when ${\overline{x}}_E$ = 0). Putting the implications all together, we obtain the following summary of implications:

$\begin{array}{l}\left[x_E=X\left(t_P\right)\right]\text{ if and only if }\left[E\text{ is the same spacetime point as }P\right]\\ \text{if and only if }\left[E\text{ is a point on the traveler's worldline}\right]\text{ if and only if}\\ \left[{\overline{x}}_E=0\right].\\ \\ \text{If any of the above are satisfied then }{t}_E=t_P.\\ \text{If }V\left(t_P\right)=0\text{ then }{t}_E=t_P.\end{array}\}$ (6.5)

We now summarize the results derived in this section. It is assumed that we are given x_E and t_E, and the goal is to calculate ${\overline{x}}_E$ and ${\overline{t}}_E$ . The first step calculates t_P from

(6.6)

where it is understood that we are treating the case in which there exists such a t_P (see Section 8 for more discussion). Having done this, the quantities X(t_P), V(t_P), γ(t_P), and τ(t_P) all become knowns quantities, via the given worldline of the traveler in the home system together with (2.3) and (2.4), in addition to x_E and t_E being known quantities. The next set of steps calculate x_P, ${\overline{t}}_E$ , and ${\overline{x}}_E$ from

$x_P=X\left(t_P\right)$ (6.7)

${\overline{t}}_E=\tau \left(t_P\right)$ (6.8a)

${\overline{x}}_E=x_E-x_P+\left[\frac{1}{\gamma \left(t_P\right)}-1\right]\left[\left(x_E-x_P\right)\circ V\left(t_P\right)\right]\frac{V\left(t_P\right)}{V^2\left(t_P\right)}$ . (6.8b)

The goal of this section is to verify, merely as a check for mistakes, that (6.6) through (6.8) reduce to the familiar Lorentz transformation when the traveler has a constant velocity.

Given a constant traveler velocity there is no loss of generality by orienting the coordinate axis so that the velocity is in the x-direction with x-component denoted v. For this case, only x-components of terms in (6.6) have contributions and using X(t_P) = vt_P with (6.6) gives

$t_E=t_P+\frac{1}{c^2}v\left(x_E-v{t}_P\right)$

with solution

$t_P=\frac{1}{1-\frac{v^2}{c^2}}\left(t_E-\frac{v{x}_E}{c^2}\right)$

or

$t_P={\gamma }^2\left(t_E-\frac{v{x}_E}{c^2}\right)\text{(constant velocity)}$ . (7.1)

Also, the vector equation (6.7) has the components

$x_P=v{t}_P,y_P=0,z_P=0\text{(constant velocity)}$

which can be written via (7.1) as

$x_P=v{\gamma }^2\left(t_E-\frac{v{x}_E}{c^2}\right),y_P=0,z_P=0\text{(constant velocity)}$ . (7.2)

Next, the scaler equation (6.8a), together with a constant γ in (2.3) gives

${\overline{t}}_E=\frac{1}{\gamma }{t}_P$

and using (7.1) gives

${\overline{t}}_E=\gamma \left(t_E-\frac{v{x}_E}{c^2}\right)\text{(constant velocity)}$ . (7.3a)

Finaly, the vector equation (6.8b) has the components

${\overline{x}}_E=\frac{1}{\gamma }\left(x_E-x_P\right),{\overline{y}}_E=y_E-y_P,{\overline{z}}_E=z_E-x_P,\text{(constant velocity)}$ .

Substituting (7.2) into this and simplifying terms via the identity

$\frac{1}{{\gamma }^2}+\frac{v^2}{c^2}=1$

gives

${\overline{x}}_E=\gamma \left(x_E-v{t}_E\right),{\overline{y}}_E=y_E,{\overline{z}}_E=z_E,\text{(constant velocity)}$ . (7.3b)

The transformation given by (7.3) is the familiar form for the case of a constant velocity.

Whether or not the inverse transformation in Section 6 exists and is unique is completely determined by the very first step of the calculation, which is to solve (6.6) for t_P. As pointed out in Section 2, a constraint that the event E must satisfy to be treated in this analysis is that there must exist a point P on the traveler’s worldline such that the inertial system local with the traveler’s system at P sees E to be simultaneous with P. There exists a solution for t_P to (6.6) if and only if E is such an “allowed” event. However, it is also possible that E is such an allowed event, so there exists a point P satisfying the above condition, but such a point P is not unique. In this case there are multiple solutions for t_P to (6.6). Lack of existence is possible if the traveler’s acceleration continues forever, and an example is given later in Section 12. However, it is shown in this section that if the acceleration always has a finite magnitude and has a finite time duration, solutions must exist but are not always unique.

The goal of this section is to show that if the acceleration always has a finite magnitude and has a finite time duration, there necessarily exists a solution for tP to (6.6) (albeit not necessarily unique). The first step towards this goal shortens the notation in (6.6) by defining the function F(t_P, x_E), a function of t_P and the vector x_E, by

$F\left(t_P,x_E\right)\equiv t_P+\frac{1}{c^2}V\left(t_P\right)\circ \left[x_E-X(t_P)\right]$ (8.1)

so that (6.6) becomes

$t_E=F\left(t_P,x_E\right)$ . (8.2)

The traveler’s journey begins at t = 0 but nothing was said about his prior history. It is convenient to stipulate that, before starting the journey, the traveler was at rest in the home system, so V(t_P) = 0 if t_P < 0. Also, the acceleration is taken to have a finite time duration so there exists some cutoff time in the home system, denoted t_C, such that the acceleration is zero when t > t_C. We therefore have

$V\left(t_P\right)=\{\begin{array}{l}V\left(t_C\right)\text{if}{t}_P>t_C\\ 0\text{if}{t}_P<0\end{array}$ . (8.3a)

The cutoff for the velocity implies that X(t_P) satisfies

$X(t_P)=X(t_C)+V\left(t_C\right)\left[t_P-t_C\right]\text{if}{t}_P>t_C$ . (8.3b)

Substituting (8.3) into (8.1) and regrouping terms for the case in which t_P > t_C gives

$F\left(t_P,x_E\right)=\{\begin{array}{l}\frac{1}{{\gamma }^2\left(t_C\right)}\left(t_P-t_C\right)+F\left(t_C,x_E\right)\text{if}{t}_P>t_C\\ \\ t_P+\frac{1}{c^2}V\left(t_P\right)\circ \left[x_E-X(t_P)\right]\text{if}0\le t_P\le t_C\\ \\ t_P\text{if}{t}_P<0\end{array}$ (8.4)

where γ is defined by (2.4). It is evident from inspection of (8.4) that

$F\left(t_P,x_E\right)\to -\infty \text{as}{t}_P\to -\infty ,\text{and}F\left(t_P,x_E\right)\to +\infty \text{as}{t}_P\to +\infty$ . (8.5)

Given that the magnitude of acceleration is always finite, so the velocity is a continuous function of time, F>(t_P, x_E) is a continuous function of each argument. Continuity together with the mapping property (8.5) implies that (8.2) has a solution for t_P for any value of t_E.

Having established that there necessarily exists a solution for t_P to (6.6) if the acceleration always has a finite magnitude and has a finite time duration, the next question asks for the conditions in which the solution is unique under the assumed conditions regarding the acceleration. Note from (8.4) that F>(t_P, x_E) is strictly increasing in t_P when t_P < 0 and when t_P > t_C. If (a big “if”) F>(t_P, x_E) is strictly increasing in t_P when 0 < t_P < t_C, then a solution for t_P to (6.6) is unique. However, if (another big “if”) a solution for t_P to (6.6) is a value of t_P at which F>(t_P, x_E) is strictly decreasing in t_P, then this solution for t_P is not unique. Therefore, the question of uniqueness is answered by considerations of whether F>(t_P, x_E) is increasing or decreasing in t_P. It will be shown below that (given a nonzero acceleration at some point in time) there necessarily exists home coordinates (t_E, x_E) for which a solution for t_P to (6.6) is not unique. A sufficient condition, imposed on x_E, for uniqueness will also be given. The arguments below will refer to the derivative of F>(t_P, x_E) obtained from (8.1) and given by

$\frac{\partial }{\partial t_P}F\left(t_P,x_E\right)=1-\frac{1}{c^2}{V}^2\left(t_P\right)+\frac{1}{c^2}A\left(t_P\right)\circ \left(x_E-X(t_P)\right)$ . (8.6)

We will show that there exist home coordinates (t_E, x_E) for which a solution for t_P to (6.6) is not unique by constructing them. Select a t_P for which A(t_P) ≠ 0 but is otherwise arbitrary. Now select any x_E satisfying

$1-\frac{1}{c^2}{V}^2\left(t_P\right)+\frac{1}{c^2}A\left(t_P\right)\circ \left(x_E-X(t_P)\right)<0\text{(for discussion)}$ . (8.7)

Note that such an xE satisfying (8.7) necessarily exists when A(t_P) ≠ 0. Also note, from (8.6), that this choice for xE makes F>(t_P, x_E) strictly decreasing in t_P at the selected value of t_P. Now use (8.2) (equivalent to (6.6)) to define t_E. These steps have constructed a set of home coordinates (t_E, x_E) such that F>(t_P, x_E) is strictly decreasing in t_P at the solution t_P, implying that the solution is not unique. Therefore, given that there is some point in time at which the acceleration is not zero, we conclude that there exist home coordinates (t_E, x_E) for which a solution for t_P to (6.6) is not unique.

A fairly simple constraint imposed on the home coordinates (t_E, x_E) that is a sufficient condition for the solution for t_P to (6.6) to be unique is for x_E to be selected to make the right side of (8.6) positive for all t_P. This constraint makes F>(t_P, x_E) strictly increasing in t_P for all t_P, ensuring uniqueness.

The next section calculates the metric tensor in the traveling system. That analysis is less cumbersome if some identities are available, and one goal of this section is to make them available. A few other miscellaneous identities useful in later sections are also derived.

The first goal is to derive expressions for the derivatives of parallel parts and perpendicular parts of arbitrary spatial vectors as they are defined in (6.2). This is accomplished in several steps. The first step calculates the derivative of the magnitude of the velocity vector. The second step uses this result to calculate the derivative of the unit vector in the direction of the velocity vector, and the last step calculates derivative of parallel and perpendicular parts of arbitrary vectors.

Note from (6.2) that parallel and perpendicular parts of an arbitrary spatial vector W can be written as

$W_{\parallel }\equiv \left(W\circ n\left(t_P\right)\right)n\left(t_P\right),W_{\perp }\equiv W-W_{\parallel }$ . (9.1)

where the unit vector n is defined by

$n\left(t_P\right)\equiv \frac{V\left(t_P\right)}{V\left(t_P\right)}$ . (9.2)

The first step that is used to calculate derivatives of parallel and perpendicular parts of W notes that

$2V\left(t_P\right)\frac{dV\left(t_P\right)}{d{t}_P}=\frac{d{V}^2\left(t_P\right)}{d{t}_P}=\frac{dV\left(t_P\right)\circ V\left(t_P\right)}{d{t}_P}=2V\left(t_P\right)\circ \frac{dV\left(t_P\right)}{d{t}_P}=2V\left(t_P\right)\circ A\left(t_P\right)$

which gives

$\frac{dV\left(t_P\right)}{d{t}_P}=A\left(t_P\right)\circ n\left(t_P\right)$ . (9.3)

Now use

$\frac{dn\left(t_P\right)}{d{t}_P}=\frac{d}{d{t}_P}\left(\frac{V\left(t_P\right)}{V\left(t_P\right)}\right)=\frac{1}{V\left(t_P\right)}\frac{dV\left(t_P\right)}{d{t}_P}-\frac{V\left(t_P\right)}{V^2\left(t_P\right)}\frac{dV\left(t_P\right)}{d{t}_P}=\frac{1}{V\left(t_P\right)}A\left(t_P\right)-\frac{n\left(t_P\right)}{V\left(t_P\right)}\frac{dV\left(t_P\right)}{d{t}_P}$ .

Substituting (9.3) into the above gives

$\frac{dn\left(t_P\right)}{d{t}_P}=\frac{1}{V\left(t_P\right)}\left[A\left(t_P\right)-\left(A\left(t_P\right)\circ n\left(t_P\right)\right)n\left(t_P\right)\right]$ . (9.4)

From the definition (9.1) of parallel and perpendicular parts, we recognize the square bracket in (9.4) as the perpendicular part of the acceleration so

$\frac{dn\left(t_P\right)}{d{t}_P}=\frac{1}{V\left(t_P\right)}{A}_{\perp }\left(t_P\right)$ . (9.5)

The next step takes the derivative of the parallel part of an arbitrary vector W using

$\frac{d{W}_{\parallel }\left(t_P\right)}{d{t}_P}=\frac{d}{d{t}_P}\left(W\left(t_P\right)\circ n\left(t_P\right)\right)n\left(t_P\right)=\left(\frac{dW\left(t_P\right)}{d{t}_P}\circ n\left(t_P\right)\right)n\left(t_P\right)+$

$\left(W\left(t_P\right)\circ \frac{dn\left(t_P\right)}{d{t}_P}\right)n\left(t_P\right)+\left(W\left(t_P\right)\circ n\left(t_P\right)\right)\frac{dn\left(t_P\right)}{d{t}_P}$ .

The first term on the right is seen to be the parallel part of the derivative of W. Substituting (9.5) into the second and third terms on the right produces

$\frac{d{W}_{\parallel }\left(t_P\right)}{d{t}_P}={\left(\frac{dW\left(t_P\right)}{d{t}_P}\right)}_{\parallel }+\frac{1}{V\left(t_P\right)}\left[\left(W\left(t_P\right)\circ A_{\perp }\left(t_P\right)\right)n\left(t_P\right)+\left(W\left(t_P\right)\circ n\left(t_P\right)\right)A_{\perp }\left(t_P\right)\right]$ . (9.6)

The derivative of the perpendicular part is obtained from

$\frac{d{W}_{\perp }\left(t_P\right)}{d{t}_P}=\frac{dW\left(t_P\right)-W_{\parallel }\left(t_P\right)}{d{t}_P}=\frac{dW\left(t_P\right)}{d{t}_P}-\frac{d{W}_{\parallel }\left(t_P\right)}{d{t}_P}$ .

Substituting (9.6) into the above far right term while using

${\left(\frac{dW\left(t_P\right)}{d{t}_P}\right)}_{\perp }=\frac{dW\left(t_P\right)}{d{t}_P}-{\left(\frac{dW\left(t_P\right)}{d{t}_P}\right)}_{\parallel }$

gives

$\frac{d{W}_{\perp }\left(t_P\right)}{d{t}_P}={\left(\frac{dW\left(t_P\right)}{d{t}_P}\right)}_{\perp }-\frac{1}{V\left(t_P\right)}\left[\left(W\left(t_P\right)\circ A_{\perp }\left(t_P\right)\right)n\left(t_P\right)+\left(W\left(t_P\right)\circ n\left(t_P\right)\right)A_{\perp }\left(t_P\right)\right]$ . (9.7)

Another identity useful in later sections is an expression for the derivative of γ(t_P). Using (2.4) gives

$\frac{d\gamma \left(t_P\right)}{d{t}_P}=\frac{d}{d{t}_P}{\left(1-\frac{V^2(t_P)}{c^2}\right)}^{-1/2}=\frac{1}{2c^2}{\left(1-\frac{V^2(t_P)}{c^2}\right)}^{-3/2}\frac{d{V}^2(t_P)}{d{t}_P}=\frac{1}{2c^2}{\left(1-\frac{V^2(t_P)}{c^2}\right)}^{-3/2}\frac{dV(t_P)\circ V(t_P)}{d{t}_P}$

which finally gives

$\frac{d\gamma \left(t_P\right)}{d{t}_P}=\frac{1}{c^2}{\gamma }^3\left(t_P\right)V(t_P)\circ A(t_P)$ . (9.8)

Still another identity that will be useful later and is trivial to prove is

${\gamma }^2\left(t_P\right)-1=\frac{1}{c^2}{\gamma }^2\left(t_P\right)V^2\left(t_P\right)$ . (9.9)

Now consider an arbitrary trajectory, or worldline, seen in an arbitrary inertial system. This could be the worldline of the traveler’s clock in the home system but not necessarily. Along this worldline we can define increments of proper time, denoted dτ, and increments of coordinate time, denoted dt. From these increments we can define the derivative dt/dτ, which is a derivative defined on a curve. Two different expressions for this derivative are derived. Both start with the equation

${\left(d\tau \right)}^2={\left(dt\right)}^2-\frac{1}{c^2}\left(dx\circ dx\right)$ (9.10a)

and then divide by (dτ)² to get

$1={\left(\frac{dt}{d\tau }\right)}^2-\frac{1}{c^2}\frac{dx}{d\tau }\circ \frac{dx}{d\tau }$ . (9.10b)

One expression for dt/dτ is obtained by rearranging terms in (9.10) to get

$\frac{dt}{d\tau }=\sqrt{1+\frac{1}{c^2}\frac{dx}{d\tau }\circ \frac{dx}{d\tau }}$ . (9.11a)

A second expression for the same derivative is obtained by using the chain rule to write (9.10) as

$1={\left(\frac{dt}{d\tau }\right)}^2-\frac{1}{c^2}\frac{dt}{d\tau }\frac{dx}{dt}\circ \frac{dt}{d\tau }\frac{dx}{dt}=\left[1-\frac{1}{c^2}\frac{dx}{dt}\circ \frac{dx}{dt}\right]{\left(\frac{dt}{d\tau }\right)}^2$

and rearranging terms to get

$\frac{dt}{d\tau }=\frac{1}{\sqrt{1-\frac{1}{c^2}\frac{dx}{dt}\circ \frac{dx}{dt}}}=\gamma$ (9.11b)

with γ defined by the second equality in (9.11b), consistent with the definition in earlier sections. Combining (9.11a) with (9.11b) gives

$\frac{1}{\sqrt{1-\frac{1}{c^2}\frac{dx}{dt}\circ \frac{dx}{dt}}}=\sqrt{1+\frac{1}{c^2}\frac{dx}{d\tau }\circ \frac{dx}{d\tau }}$ (9.11c)

when dτ is defined by (9.10).

This section calculates the metric tensor in the traveling system. This metric, denoted $\overline{g}$ , is obtained by starting with the Lorentz metric, denoted g, applicable to the home system and then transforming it via the transformation of a double covariant tensor. This transformation is defined by

${\overline{g}}_{i,j}={\displaystyle \sum _{k,l=0}^3\frac{\partial x^k}{\partial {\overline{x}}^i}\frac{\partial x^l}{\partial {\overline{x}}^j}{g}_{k,l}}\text{for }i,j=0,1,2,3$ (10.1)

where we use the notation $x^0=c{t}_E,x^{\text{1}}=x_E,x^{\text{2}}=y_E,x^{\text{3}}=z_E$ , with corresponding notation for the traveler coordinates. The Lorentz metric is diagonal with g_0,0 = 1 and all other diagonal elements equal to ‒1, so (10.1) becomes

${\overline{g}}_{i,j}=\frac{\partial x^0}{\partial {\overline{x}}^i}\frac{\partial x^0}{\partial {\overline{x}}^j}-{\displaystyle \sum _{k=1}^3\frac{\partial x^k}{\partial {\overline{x}}^i}\frac{\partial x^k}{\partial {\overline{x}}^j}}\text{for }i,j=0,1,2,3$

Recognizing the sum on the right as a three dimensional (spatial) vector dot product gives

${\overline{g}}_{i,j}=c^2\frac{\partial t_E}{\partial {\overline{x}}^i}\frac{\partial t_E}{\partial {\overline{x}}^j}-\frac{\partial x_E}{\partial {\overline{x}}^i}\circ \frac{\partial x_E}{\partial {\overline{x}}^j}\text{for }i,j=0,1,2,3$ . (10.2)

It is evident from (3.1) that partial derivatives that treat ${\overline{t}}_E,{\overline{x}}_E,{\overline{y}}_E,\text{and }{\overline{z}}_E$ as independent variables (meaning that a partial derivative that varies one variable in the list does so with all others in the list held fixed) also treat $t_P,{\overline{x}}_E,{\overline{y}}_E,\text{and }{\overline{z}}_E$ as independent variables. Therefore, a derivative with respect can be expressed in terms of a derivative with respect to t_P by using

$\frac{\partial }{\partial {\overline{t}}_E}=\frac{d{t}_P}{d{\overline{t}}_E}\frac{\partial }{\partial t_P}={\left(\frac{d{\overline{t}}_E}{d{t}_P}\right)}^{-1}\frac{\partial }{\partial t_P}$

Using (3.1) and (2.3) to evaluate the derivative in the parenthesis on the right allows the above to be written as

$\frac{\partial }{\partial {\overline{t}}_E}=\gamma \left(t_P\right)\frac{\partial }{\partial t_P}$ (10.3)

Using (10.3) with (10.2), components of the metric tensor can be written as

${\overline{g}}_{0,0}={\gamma }^2\left(t_P\right)\left[{\left(\frac{\partial t}{\partial t_P}\right)}^2-\frac{1}{c^2}\frac{\partial x}{\partial t_P}\circ \frac{\partial x}{\partial t_P}\right],{\overline{g}}_{0,1}=\gamma \left(t_P\right)\left[c\frac{\partial t}{\partial t_P}\frac{\partial t}{\partial \overline{x}}-\frac{1}{c}\frac{\partial x}{\partial t_P}\circ \frac{\partial x}{\partial \overline{x}}\right]$ (10.4a)

${\overline{g}}_{0,2}=\gamma \left(t_P\right)\left[c\frac{\partial t}{\partial t_P}\frac{\partial t}{\partial \overline{y}}-\frac{1}{c}\frac{\partial x}{\partial t_P}\circ \frac{\partial x}{\partial \overline{y}}\right],{\overline{g}}_{0,3}=\gamma \left(t_P\right)\left[c\frac{\partial t}{\partial t_P}\frac{\partial t}{\partial \overline{z}}-\frac{1}{c}\frac{\partial x}{\partial t_P}\circ \frac{\partial x}{\partial \overline{z}}\right]$ (10.4b)

${\overline{g}}_{1,1}=c^2{\left(\frac{\partial t}{\partial \overline{x}}\right)}^2-\frac{\partial x}{\partial \overline{x}}\circ \frac{\partial x}{\partial \overline{x}},{\overline{g}}_{1,2}=c^2\frac{\partial t}{\partial \overline{x}}\frac{\partial t}{\partial \overline{y}}-\frac{\partial x}{\partial \overline{x}}\circ \frac{\partial x}{\partial \overline{y}}$ (10.4c)

${\overline{g}}_{1,3}=c^2\frac{\partial t}{\partial \overline{x}}\frac{\partial t}{\partial \overline{z}}-\frac{\partial x}{\partial \overline{x}}\circ \frac{\partial x}{\partial \overline{z}},{\overline{g}}_{2,2}=c^2{\left(\frac{\partial t}{\partial \overline{y}}\right)}^2-\frac{\partial x}{\partial \overline{y}}\circ \frac{\partial x}{\partial \overline{y}}$ (10.4d)

${\overline{g}}_{2,3}=c^2\frac{\partial t}{\partial \overline{y}}\frac{\partial t}{\partial \overline{z}}-\frac{\partial x}{\partial \overline{y}}\circ \frac{\partial x}{\partial \overline{z}},{\overline{g}}_{3,3}=c^2{\left(\frac{\partial t}{\partial \overline{z}}\right)}^2-\frac{\partial x}{\partial \overline{z}}\circ \frac{\partial x}{\partial \overline{z}}$ (10.4e)

where we shortened the notation by omitting the subscripts E to the event coordinates ${\overline{x}}_E,{\overline{y}}_E,{\overline{z}}_E$ and $t_E,x_E,y_E,\text{and }{z}_E$ . The remaining off-diagonal elements of the metric tensor are implied by the symmetry condition ${\overline{g}}_{i,j}={\overline{g}}_{j,i}$ .

The next step is to calculate the derivatives in (10.4). One equation needed for this is (3.3a), which is written below with the E subscripts omitted as

$t=t_P+\gamma \left(t_P\right)\frac{V\left(t_P\right)\circ \overline{x}}{c^2}$ . (10.5)

The other needed equation is (3.3b). It is convenient to write it in terms of parallel and perpendicular parts. Doing so while omitting the E subscripts to shorten the notation gives

$x=X\left(t_P\right)+{\overline{x}}_{\perp }+\gamma \left(t_P\right){\overline{x}}_{\parallel }$ . (10.6)

The first derivative to be calculated is obtained from (10.5), (9.8), and a direct application of the product rule which gives

$\frac{\partial t}{\partial t_P}=1+\frac{1}{c^4}{\gamma }^3\left(t_P\right)\left[V(t_P)\circ A(t_P)\right]\left[V\left(t_P\right)\circ \overline{x}\right]+\frac{1}{c^2}\gamma \left(t_P\right)A\left(t_P\right)\circ \overline{x}$

which can also be written as

$\frac{\partial t}{\partial t_P}=1+\frac{1}{c^4}{\gamma }^3\left(t_P\right)V^2(t_P)\left[n(t_P)\circ A(t_P)\right]\left[n\left(t_P\right)\circ \overline{x}\right]+\frac{1}{c^2}\gamma \left(t_P\right)A\left(t_P\right)\circ \overline{x}$ .

Now use

$A_{\parallel }(t_P)\circ \overline{x}=\left[A(t_P)\circ n(t_P)\right]n(t_P)\circ \overline{x}=\left[n(t_P)\circ A(t_P)\right]\left[n\left(t_P\right)\circ \overline{x}\right]$

to write the above as

$\frac{\partial t}{\partial t_P}=1+\frac{1}{c^4}{\gamma }^3\left(t_P\right)V^2(t_P)A_{\parallel }(t_P)\circ \overline{x}+\frac{1}{c^2}\gamma \left(t_P\right)A\left(t_P\right)\circ \overline{x}$ .

Next, use

$A\left(t_P\right)\circ \overline{x}=A_{\parallel }(t_P)\circ \overline{x}+A_{\perp }(t_P)\circ \overline{x}$

to write the above as

$\frac{\partial t}{\partial t_P}=1+\frac{1}{c^2}\gamma \left(t_P\right)\left[{\gamma }^2\left(t_P\right)\frac{V^2(t_P)}{c^2}+1\right]A_{\parallel }(t_P)\circ \overline{x}+\frac{1}{c^2}\gamma \left(t_P\right)A_{\perp }(t_P)\circ \overline{x}$ .

Finally, use the identity (9.9) to write the above as