Covariance of the laws of motion
The second and third laws serve as fundamental definitions of force and mass in relation to a specific reference frame This section will explore the implications of these definitions by investigating whether they yield varying results when applied to different inertial frames.
Mass can be defined in the context of particle collisions observed from an inertial frame In this scenario, let m1 and m2 represent the masses of the colliding particles, with v1 and v2 indicating their velocities before the collision, and v1' and v2' representing their velocities after the collision The relationships between these variables can be expressed through relevant equations that describe the dynamics of the collision.
If the vectors "1 - u1" and "u2 - v2" are parallel, then the vectors "v1 - ii1" and "ii2 - ~2" are also parallel This implies that the third law of motion, when experimentally verified in one inertial frame, holds true in all inertial frames Let m1 and m2 represent the particle masses measured in frame S; according to equation (1.5), this relationship can be established.
But, if the first particle is the unit standard, then m1 = m 1 = I and hence
(2.4) i.e the mass of a particle has the same value in all inertial frames We can express this by saying that mass is an invariant relative to transformations between inertial frames
By differentiating equation (1.1) with respect to time, and considering that u is constant, we derive ii = a (2.5), where a represents the accelerations of a particle relative to different inertial frames, S and S' According to the second law (1.2), with ni = m, it follows that f = f (2.6), indicating that the force acting on a particle remains independent of the inertial frame from which it is measured.
Equations (1.2) and (1.4) maintain the same form in both reference frames, S and S, indicating that mass, acceleration, and force are invariant across these frames, while velocity changes according to equation (1.1) When equations retain their structure during transformation between reference frames, they are considered covariant with respect to that transformation Notably, Newton's laws of motion exhibit covariance when transitioning between inertial frames.
Special principle of relativity
The special principle of relativity posits that all physical laws are consistent across different inertial frames, meaning that observers moving uniformly relative to one another will agree on these laws This principle eliminates the concept of privileged observers, emphasizing that no individual can claim a unique relationship with the universe that others do not share Historically, this idea would have been dismissed when humanity viewed itself as the center of creation, but the shift in perspective initiated by Copernicus has led to widespread acceptance of this principle Today, it is considered a fundamental aspect of theoretical physics, with any strong evidence against it needing to be substantial to challenge its validity Consequently, this principle ensures that observers on distant planets can apply the same physical laws to explain their local phenomena, regardless of their unique celestial circumstances.
Maxwell's laws of electrodynamics, which govern non-mechanical phenomena, are more complex than Newton's laws of motion These fundamental laws can be expressed through specific equations, including curl E = -c8jtr, curl H = j + i'D, div D = p, and div B = 0.
The equations governing electric and magnetic fields, represented by E, H, D, B, j, and ρ, have been experimentally validated across all inertial frames, most notably by the Michelson-Morley experiment This landmark experiment confirmed that the speed of light is consistently measured at c (3 x 10^8 m/s) relative to a stationary apparatus on Earth, regardless of the Earth's orbital velocity This finding supports Maxwell's equations and their alignment with the special principle of relativity, despite initial resistance rooted in the belief that electromagnetic phenomena required a medium called the aether Critics argued that an 'aether wind' would distort electromagnetic wave propagation, necessitating corrections to Maxwell's equations However, the special principle is now widely accepted, as experimental results consistently align with its predictions, underscoring its universal applicability and the need to understand the historical reluctance to embrace the compatibility of electromagnetic laws with this principle.
In the context of two inertial frames, S and S, an observer in frame S measures the velocity of a light pulse as c, while another observer in frame S measures it as c According to equation (1.1), the velocities differ, leading to the conclusion that either Maxwell's equations must be modified or the special principle of relativity abandoned for electromagnetic phenomena Previous attempts to modify Maxwell's equations, such as those by Ritz, failed to yield experimentally verifiable results Given the consistent validity of the special principle, the only viable option was to reject equation (1.1) in favor of a new formulation that aligns with the experimental evidence that the speed of light remains constant across all inertial frames This necessary revision challenges our fundamental understanding of space and time, a change that has historically faced significant resistance.
Lorentz transformations Minkowski space- time 6 5 The special Lorentz transformation
The argument of this section will be founded on the following three postulates: Postulate / A particle free to move under no forces has constant velocity in any inertial frame
Postulate 2 The speed of light relative to any inertial frame is c in all directions
Postulate 3 asserts that the geometry of space is Euclidean in any inertial frame, represented by the reference frame S with rectangular Cartesian axes Oxyz In this context, the coordinates of a point are measured using a stationary measuring scale within frame S, highlighting the importance of this condition since the length of an object can vary with its motion Additionally, it is assumed that standard atomic clocks, which are stationary relative to frame S, are evenly distributed throughout space and synchronized with a master clock at the origin A reliable synchronization method involves notifying observers at all clock locations about a light source at a predetermined position.
When a source begins to emit light at speed c = c0, an observer at point P will set their clock to read c0 + OP/c upon first receiving the light This assumes that light travels at speed c relative to the stationary observer, as confirmed by experimental evidence The location and timing of an event can then be defined using four coordinates (x, y, z, t), where t represents the time displayed on the clock at the event's location.
We shall often refer to the four numbers (x, y, z, c) as an erent
In the rectangular Cartesian frame S, where clocks are synchronized with a master clock at the origin, any event can be represented by four coordinates (x, y, z, T) The spatial coordinates are measured using scales at rest in frame S, while the time coordinate is indicated by the adjacent clock also at rest in S This section aims to derive the equations that connect different coordinate representations of the same event, specifically (x, y, z, T) and (x', y', z', T') These transformation equations serve as a crucial tool for translating statements about events from one coordinate system to another, facilitating communication between the two frames.
Early physical theories overlooked the impact of uniform motion on the measurements of length and time relative to a reference frame While it was acknowledged that velocity measurements vary with the reference frame, lengths and times were considered absolute However, relativity theory reveals that very few quantities are truly absolute, indicating that most measurements depend on the frame in which the measuring instruments are situated.
To adhere to Postulate I, we assume that the coordinates (x, y, z, f) are linear functions of (x, y, z, 1) Consequently, the inverse relationship maintains the same form A particle moving uniformly in space S with a velocity vector (v_x, v_y, v_z) will possess spatial coordinates (x, y, z).
By substituting linear expressions for the coordinates (x, y, z, T) instead of (x, y, z, t), it is determined that the quantities (x, y, z) are linear in terms of c, indicating that the particle's motion is uniform relative to frame S Furthermore, it can be demonstrated that only a linear transformation can fulfill the requirements of Postulate I.
Now suppose that at the instant c = c 0 a light source situated at the point P0
(x 0 , y 0 , z 0 ) in S radiates a pulse of short duration At any later instant c, the wavefront will occupy the sphere whose centre is P0 and radius c(t-c0 ) This has equation
At time T = 70, the coordinates of the light source are (x0, y0, z0), from which a short pulse is emitted According to Postulate 2, at any subsequent time T, the wavefront will form a sphere with a radius of c(T - 70) centered at (x0, y0, z0) This relationship can be expressed mathematically through the equation of the sphere.
Equations (4.2), (4.3) describe the same set of events in languages appropriate to S,
To ensure consistency between the coordinates (x, y, z, t) and their corresponding barred counterparts (.X, y, z, T), it is essential to select the equations in such a way that substituting the barred variables in equation (4.3) yields the correct linear expressions in the unbarred variables as outlined in equation (4.2).
We will utilize a mathematical device developed by Minkowski, replacing the time coordinate \( ct \) of any event observed in frame S with an imaginary coordinate \( x_4 = kt \) (where \( i = \sqrt{-1} \)) The spatial coordinates of the event, represented as \( (x, y, z) \), will be substituted with \( (x_1, x_2, x_3) \).
(4.4) and any event is then determined by four coordinates x 1 (i = I, 2, 3, 4) A similar transformation to coordinates x 1 will be carried out inS Equations (4.2), (4.3) can then be written
The X; are to be linear functions of the x 1 and such as to transform equation (4.6) into equation (4.5) and hence such that
The parameter k is determined solely by the relative velocity between frames S and S It is logical to assume a reciprocal relationship exists between these two frames, implying that applying the inverse transformation from S to S maintains this dependency.
The transformation and its inverse must keep any function of the coordinates X unchanged, leading to the conclusion that k² = I As the relative motion between S and S approaches zero, it becomes evident that k approaches +I Therefore, since k cannot equal -I, we conclude that k is consistently equal to unity.
The x 1 will now be interpreted as rectangular Cartesian coordinates in a four- dimensional Euclidean space which we shall refer to as ~4 This space is termed
Minkowski space-time allows us to understand the 'distance' between two points with coordinates x and x' through the square of their difference, as indicated in equation (4.5) The coordinates x can be interpreted in relation to a different set of rectangular Cartesian axes in four-dimensional space This interpretation aids in meeting the requirements outlined in equation (4.7) with k set to 1, establishing a connection between the coordinates x and x' through specific relational equations.
In this article, we explore a linear relationship characterized by constants aiJ and b, where i ranges from 1 to 4 The coordinates of the origin of the first set of rectangular axes, denoted as h;, are defined in relation to a second set of axes Chapter 2 will demonstrate that the constants a;1 meet specific identities outlined in equations (8.14) and (8.15) Additionally, algebraic principles confirm that the relationship between the variables x; and X; adheres to the assumed linear form, provided it satisfies the criteria established in equation (4.7).
The general Lorentz transformation allows for the conversion of event coordinates from one reference frame (S) to another (S) using equations (4.4) and (4.9) This transformation establishes a relationship between spatial and temporal measurements in both frames, provided that certain conditions are met, such as ensuring that an event with real coordinates in frame S also has real coordinates in frame S.
We will explore the unique Lorentz transformation derived from the assumption that the x1 axes in Iff 4 are obtained by rotating the X1 axes through an angle α in the x1-x4 plane This rotation leaves the origin and the x2 and x3 axes unchanged As illustrated in Fig I, the transformations can be expressed as x1 = x1 cos(α) + x4 sin(α) and x4 = -x1 sin(α) + x4 cos(α).
Employing equations (4.4), these transformation equations may be written x = xcoscx + ict sin:x id= -xsin:x+ictcoscx ~ = y} z = z (5.2)
To interpret the equations (5.2), consider a plane which is stationary relative to the f frame and has equation ax+by+cz+{T = o for all f Its equation relative to the S frame will be
(5.4) at any fixed instant c In particular, if ii = b = d = 0, this is the coordinate plane
Oxy and its equation relative to Sis z = 0, i.e it is the plane Oxy Again, if b = c
Fitzgerald contraction Time dilation
In the next two sections, we shall explore some of the more elementary physical consequences of the transformation equations (5.8)
Consider first a rigid rod stationary in Sand lying along the x-axis Let x
== x 1 , x = x 2 at the two ends of the bar so that its length as measured in Sis given by
In frame S, a bar moves at speed u, and to accurately measure its length, it's essential to observe the positions of its two ends simultaneously at a specific time t By marking the x-axis at x = x1 and x = x2, corresponding to the ends of the bar at time t, we create a pair of events with space-time coordinates (x1, t) and (x2, t) in frame S According to the principles of relativity, this pair of events will have different coordinates (x1, T1) and (x2, T2) in the moving frame S The equations governing these transformations necessitate careful consideration of the relative motion and time dilation effects.
But x 2 - x 1 = I is the length of the bar as measured m S and it follows by subtraction of equations (6.2) that
The length of a bar accordingly suffers contraction when it is moved longitudin- ally relative to an inertial frame This is the Fitzgerald contraction
The contraction of a rod is not a physical reaction akin to that of a metal rod cooling; rather, it arises from the altered relationship between the rod and the measuring instruments When measuring with stationary scales relative to the rod, the length differs from measurements taken with scales moving with respect to the rod, which require simultaneous observation and the use of clocks Classical physics assumed these measurement methods would yield identical results, based on the belief that a rigid bar has an intrinsic length unaffected by measurement procedures However, it is now recognized that length, like all physical quantities, is defined by the measurement process itself and lacks meaning outside this context Consequently, changing the measurement procedure can lead to different results, illustrating that the length of the rod can be adjusted by merely switching to a different frame of reference, with no physical implications resulting from this change in description.
In examining the two events marked by chalk on the x-axis, we can apply equations (5.8) to the space-time coordinates of these events across two different frames, leading to the derivation of significant equations that illustrate their relationship.
The equations demonstrate that while events may appear simultaneous in one frame of reference (S), they are not simultaneous in another (S') This indicates that simultaneity is a relative concept rather than an absolute one, challenging previous assumptions about its meaning.
The registration of events at coordinates (0, 0, 0, 1) and (0, 0, 0, T2) in frame S indicates two distinct occurrences By applying the inverse transformation from equation (5.8), the times t1 and t2 for these events, as measured in frame S, can be determined.
This equation shows that the clock moving with 0 will appear from S to have its rate reduced by a factor J (I - u 2 jc 2 ) This is the rime dilation effect
Any physical process can serve as a clock, indicating that all processes will appear to evolve more slowly when viewed from a moving frame of reference This phenomenon is exemplified by the observed decay rates of radioactive particles in cosmic rays, which move at high velocities relative to Earth, showing a reduction in decay rate precisely as predicted by the relevant equations.
When a human passenger is launched from Earth in a rocket traveling near the speed of light, both metabolic and physiological processes within the passenger's body experience a retardation relative to Earth Although the passenger remains unaware of this effect, upon returning to Earth, they will find that their perception of the flight duration is shorter than the time recorded on Earth This situation raises the "dock paradox," where the passenger might argue they were at rest while the Earth moved However, this paradox is resolved by recognizing that the rocket's frame is not inertial due to its acceleration relative to an inertial frame, meaning the principles of special relativity cannot be applied by the passenger in their frame.
The methods of general relativity are universally applicable, ensuring that calculations made by a passenger using these techniques will align with the results observed by someone on Earth.
The clock paradox presents an intriguing scenario where the clock at rest (0) appears to run slower compared to the moving clock in frame S By considering frame S as the reference frame, a similar analysis reveals that the moving clock operates differently, challenging our understanding of time perception in varying frames of reference.
In the context of special relativity, the assertion that "0 runs slow compared to 0" requires clarification, as it presents a contradiction when only inertial frames are considered A direct comparison of clocks at different spatial points is not feasible, leading to the need for a more nuanced interpretation Specifically, "0 is observed to run slow when compared to the successive synchronized clocks of frame S that it aligns with during its motion." This understanding eliminates any contradiction, as the expanded statements provide a coherent explanation of the observed time discrepancies.
Spacelike and timelike intervals Light cone 14
We have proved in section 4 that if xi, Xm are the coordinates in Minkowski space-time of two events, then
The quantity L (xi - Xiol 2 (7.1) i = I remains constant for all observers using inertial frames and rectangular axes in space-time By transforming back to the standard space and time coordinates in an inertial frame, we can derive the corresponding results.
(x- Xo)z + (y- Yolz + (z- Zo)z- cz (I- lo)z (7.2) is invariant for all inertial observers
Thus, if (x, y, z, 1), (x0 , y 0 , z 0 , 10 ) are the coordinates of two events relative to any inertial frameS and we define the proper time intenãa/ r between the events by the equation
In the context of two events, the value of r remains invariant, meaning that while observers in different inertial frames may assign different coordinates to these events, they will consistently agree on the value of r.
Denoting the time interval between the events by 1t and the distance between them by M, both relative to the same frame S and positive, it follows from equation (7.3) that
(7.4) Suppose that a new inertial frame Sis now defined, moving in the direction of the line joining the events with speed M I dl This will only be possible if M jdt < c
Relative to this frame the events will occur at the same point and hence M = 0
The proper time interval between two events is defined as the ordinary time interval measured in a frame where the events occur at the same spatial point In this scenario, it is evident that r² is greater than zero, indicating that the proper time interval between the events is considered time-like.
Suppose, if possible, that a frame Scan be chosen relative to which the events are simultaneous In this frame ~~ = 0 and
Thus r 2 < 0, and, in any frame, ~dj~t >c r is then purely imaginary and the interval between the events is said to be space/ike
In the context of spacetime intervals, a timelike interval occurs when the time difference between two events is less than the speed of light, allowing a material body to exist at both events Conversely, a spacelike interval arises when the time difference exceeds the speed of light, making it impossible for a material body to be present at both locations The special case of a light pulse occurs when the interval is exactly equal to the speed of light, resulting in a proper time interval of zero between the transmission and reception of the signal.
In four-dimensional space, represented by the coordinates (x, y, z, c), we utilize Minkowski space-time, which differs from the Euclidean space-time continuum discussed previously This representation allows for real-valued coordinates, making it more effective for diagrammatic illustrations Consider a particle positioned at the origin (0, 0, 0, 0) that begins to move along the x-axis with a constant speed u; its y- and z-coordinates remain zero Consequently, the particle's motion is restricted to the xt-plane, where it is depicted as a straight line, with point Q representing its position along the x-axis.
In the context of particle motion, the world-line of a particle is represented by QP, where if L PQ1 = 0, then tan() = u, with the condition that |u| < c Consequently, the particle's world-line must reside within the sector AQB, defined by L AQB = 2:x and tan IX = c A particle that arrives at point O at time t = 0 after moving along the x-axis must exist within the sector A'QB' Events in these sectors are separated from event Q by a timelike interval, indicating the possibility of the particle being present at both In contrast, events in sectors AQB' and A'QB are separated from Q by spacelike intervals, making simultaneous occurrence impossible The lines A'A and B'B represent the world-lines of light signals passing through O at t = 0, propagating in both positive and negative x-axis directions Notably, any event in AQB occurs in the future relative to event Q in frame S, and cannot be made simultaneous with Q, as this would suggest a spacelike interval Thus, the sector AQB exclusively contains future events.
In the context of event Q, all occurrences within the sector A' Q B' are classified as being in the absolute past Conversely, events located in the sectors AQB' and A'QB are separated from Q by spacelike intervals and can be synchronized with Q through an appropriate selection of inertial frames Depending on the chosen frame, these events may transpire either before or after Q Collectively, these two sectors delineate a region of space-time referred to as the conditional present.
Since no physical signal can have a speed greater than c, the world-line of any such signal emanating from Q must lie in the sector AQB It follows that the event
Q can only be the physical cause of events that lie in its absolute future and can only be the effect of events from its absolute past It cannot be causally linked to events occurring in its conditional present.
In classical physics, the signal velocity has no upper limit, and events along the x-axis, such as AA' and BB', coincide perfectly This creates a clear distinction between past and future, with a precisely defined present where all observers agree that the time coordinate is t = 0.
In the four-dimensional space Qxyzt, the three regions of absolute past, absolute future and conditional present are separated from one another by the hyper-cone
A light pulse emitted from an event Q creates a light cone that defines its world-line This light cone serves as a framework for categorizing the space-time continuum into three distinct regions relative to the event Q, highlighting that any event can serve as the apex of its own light cone.
In an inertial frame 0xy, a particle of mass m is influenced by a force f while moving within the plane The rotating frame 0x'y' is defined such that the angle Lx'Ox equals 1/1 and is characterized by an angular velocity w The particle's position is described using polar coordinates (r, θ) relative to this rotating frame The polar components of the force are denoted as (f_r, f_θ), while the particle's acceleration components are represented as (a_r, a_θ) The particle's speed in the rotating frame is v, and the angle between its direction of motion and the radius vector is φ The equations of motion can be expressed as m*a_r = f_r + 2*m*w*v*sin(φ) + m*r*w² and m*a_θ = f_θ - 2*m*w*v*cos(φ) - m*r.
In a rotating frame, the motion of a particle complies with the second law of motion when considering additional forces acting on it These include the centrifugal force, represented as mw²r, which acts radially outward, and the Coriolis force, denoted as 2mwv, which acts perpendicular to the direction of motion Additionally, a transverse force, mniJ, is also present, although it becomes negligible if the rotation is uniform.
In the context of special relativity, when a stationary bar along the Ox axis is observed in a frame S, the length measured from simultaneous observations at the ends of the bar will appear longer than its proper length in frame S This phenomenon occurs due to the effects of time dilation and the relativity of simultaneity, resulting in a length contraction factor that increases the observed length of the bar.
In the context of special relativity, consider a bar that takes a time T to pass a stationary point on the x-axis, as measured by a clock at that point If the length of the bar in the stationary frame (S-frame) is defined as uT, where u represents the speed of the bar, we can derive the Fitzgerald contraction This phenomenon illustrates how the length of an object moving at a significant fraction of the speed of light appears contracted in the direction of motion when observed from a stationary frame.
Swill's measuring rod appears to be shortened by a factor of (1 - u²/c²)^(1/2) when observed from S Consequently, when S measures the length of the bar fixed in She, the expected result will reflect this contraction.