Galilean transformations A basic goal is to be able to compare measurements made by observers in relative motion. If there is an observer O in frame S who has measured the time and space coordinates of an event, assigning this event three Cartesian coordinates and the time as measured on his lattice of synchronized clocks (see
Fig. 1-1). A second observer O′ in a different frame S′ measures the same event in her coordinate system and her lattice of synchronized clocks . With inertial frames, neither observer is under acceleration, and a simple set of equations allows us to relate coordinates to . Given that the two coordinate systems are in standard configuration, meaning that they are aligned with parallel coordinates and that when , the coordinate transformation is as follows: : x' = x - v t : y' = y : z' = z : t' = t . Fig. 3-1 illustrates that in Newton's theory, time is universal, not the velocity of light. • : u = {v+u'\over 1+(vu'/c^2)} . The relativistic formula for addition of velocities presented above exhibits several important features: • If and
v are both very small compared with the speed of light, then the product /
c2 becomes vanishingly small, and the overall result becomes indistinguishable from the Galilean formula (Newton's formula) for the addition of velocities:
u = +
v. The Galilean formula is a special case of the relativistic formula applicable to low velocities. • If is set equal to
c, then the formula yields
u =
c regardless of the starting value of
v. The velocity of light is the same for all observers regardless their motions relative to the emitting source. : \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}} If
v is greater than or equal to
c, the expression for \gamma becomes physically meaningless, implying that
c is the maximum possible speed in nature. For any
v greater than zero, the Lorentz factor will be greater than one, although the shape of the curve is such that for low speeds, the Lorentz factor is extremely close to one. In Fig. 3-3b, segments
OA and
OK represent equal spacetime intervals. Length contraction is represented by the ratio
OB/
OK. The invariant hyperbola has the equation , where
k =
OK, and the edges of the blue band representing the world lines of the endpoints of a rod in motion have slope 1/
β =
c/
v. Event A has coordinates (
x,
w) = (
γk,
γβk). Since the tangent line through A and B has the equation
w = (
x −
OB)/
β, we have
γβk = (
γk −
OB)/
β and : OB/OK = \gamma (1 - \beta ^ 2) = \frac{1}{\gamma}
Lorentz transformations The Galilean transformations and their consequent commonsense law of addition of velocities work well in our ordinary low-speed world of planes, cars and balls. Beginning in the mid-1800s, however, sensitive scientific instrumentation began finding anomalies that did not fit well with the ordinary addition of velocities. Lorentz transformations are used to transform the coordinates of an event from one frame to another in special relativity. The Lorentz factor appears in the Lorentz transformations: : \begin{align} t' &= \gamma \left( t - \frac{v x}{c^2} \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end{align} The inverse Lorentz transformations are: : \begin{align} t &= \gamma \left( t' + \frac{v x'}{c^2} \right) \\ x &= \gamma \left( x' + v t' \right)\\ y &= y' \\ z &= z' \end{align} When
v ≪
c and
x is small enough, the
v2/
c2 and
vx/
c2 terms approach zero, and the Lorentz transformations approximate to the Galilean transformations. t' = \gamma ( t - v x/c^2), x' = \gamma( x - v t) etc., most often really mean \Delta t' = \gamma (\Delta t - v \Delta x/c^2), \Delta x' = \gamma(\Delta x - v \Delta t) etc. Although for brevity the Lorentz transformation equations are written without deltas,
x means Δ
x, etc. We are, in general, always concerned with the space and time
differences between events. Calling one set of transformations the normal Lorentz transformations and the other the inverse transformations is misleading, since there is no intrinsic difference between the frames. Different authors call one or the other set of transformations the "inverse" set. The forwards and inverse transformations are trivially related to each other, since the
S frame can only be moving forwards or reverse with respect to . So inverting the equations simply entails switching the primed and unprimed variables and replacing
v with −
v. The derivation given here and illustrated in Fig. 3-5 is based on one presented by Bais • From the drawing,
w =
a +
b and • From previous results using similar triangles, we know that . • Because of time dilation, • Substituting equation (4) into yields . • Length contraction and similar triangles give us and • Substituting the expressions for
s,
a,
r and
b into the equations in Step 2 immediately yield \begin{align} w &= \gamma w' + \beta \gamma x' \\ x &= \gamma x' + \beta \gamma w' \end{align} The above equations are alternate expressions for the t and x equations of the inverse Lorentz transformation, as can be seen by substituting
ct for
w, for , and
v/
c for
β. From the inverse transformation, the equations of the forwards transformation can be derived by solving for and .
Linearity of the Lorentz transformations The Lorentz transformations have a mathematical property called linearity, since and are obtained as linear combinations of
x and
t, with no higher powers involved. The linearity of the transformation reflects a fundamental property of spacetime that was tacitly assumed in the derivation, namely, that the properties of inertial frames of reference are independent of location and time. In the absence of gravity, spacetime looks the same everywhere. In scenario (a), the point of closest approach is frame-independent and represents the moment where there is no change in distance versus time (i.e. dr/dt = 0 where
r is the distance between receiver and source) and hence no longitudinal Doppler shift. The source observes the receiver as being illuminated by light of frequency , but also observes the receiver as having a time-dilated clock. In frame S, the receiver is therefore illuminated by
blueshifted light of frequency : f = f' \gamma = f' / \sqrt { 1 - \beta ^2 } In scenario (b) the illustration shows the receiver being illuminated by light from when the source was closest to the receiver, even though the source has moved on. Because the source's clocks are time dilated as measured in frame S, and since dr/dt was equal to zero at this point, the light from the source, emitted from this closest point, is
redshifted with frequency : f = f' / \gamma = f' \sqrt { 1 - \beta ^2 } Scenarios (c) and (d) can be analyzed by simple time dilation arguments. In (c), the receiver observes light from the source as being blueshifted by a factor of \gamma, and in (d), the light is redshifted. The only seeming complication is that the orbiting objects are in accelerated motion. However, if an inertial observer looks at an accelerating clock, only the clock's instantaneous speed is important when computing time dilation. (The converse, however, is not true.)
Energy and momentum Extending momentum to four dimensions In classical mechanics, the state of motion of a particle is characterized by its mass and its velocity.
Linear momentum, the product of a particle's mass and velocity, is a
vector quantity, possessing the same direction as the velocity: . It is a
conserved quantity, meaning that if a
closed system is not affected by external forces, its total linear momentum cannot change. In relativistic mechanics, the momentum vector is extended to four dimensions. Added to the momentum vector is a time component that allows the spacetime momentum vector to transform like the spacetime position vector . In exploring the properties of the spacetime momentum, we start, in Fig. 3-8a, by examining what a particle looks like at rest. In the rest frame, the spatial component of the momentum is zero, i.e. , but the time component equals
mc. We can obtain the transformed components of this vector in the moving frame by using the Lorentz transformations, or we can read it directly from the figure because we know that {{tmath|1=(m c)^{\prime}=\gamma m c}} and {{tmath|1=p^{\prime}=-\beta \gamma m c}}, since the red axes are rescaled by gamma. Fig. 3-8b illustrates the situation as it appears in the moving frame. It is apparent that the space and time components of the four-momentum go to infinity as the velocity of the moving frame approaches
c. old-fashioned color television sets, etc.), has nevertheless not proven to be a
fruitful concept in physics in the sense that it is not a concept that has served as a basis for other theoretical development. Relativistic mass, for instance, plays no role in general relativity. For this reason, as well as for pedagogical concerns, most physicists currently prefer a different terminology when referring to the relationship between mass and energy. "Relativistic mass" is a deprecated term. The term "mass" by itself refers to the rest mass or
invariant mass, and is equal to the invariant length of the relativistic momentum vector. Expressed as a formula, : E^2 - p^2c^2 = m_\text{rest}^2 c^4 This formula applies to all particles, massless as well as massive. For photons where
mrest equals zero, it yields, . The fact that physical processes do not care
where in space they take place (
space translation symmetry) yields
conservation of momentum, the fact that such processes do not care
when they take place (
time translation symmetry) yields
conservation of energy, and so on. In this section, we examine the Newtonian views of conservation of mass, momentum and energy from a relativistic perspective.
Total momentum To understand how the Newtonian view of conservation of momentum needs to be modified in a relativistic context, we examine the problem of two colliding bodies limited to a single dimension. In Newtonian mechanics, two extreme cases of this problem may be distinguished yielding mathematics of minimum complexity: • The two bodies rebound from each other in a completely elastic collision. • The two bodies stick together and continue moving as a single particle. This second case is the case of completely inelastic collision. For both cases (1) and (2), momentum, mass, and total energy are conserved. However, kinetic energy is not conserved in cases of inelastic collision. A certain fraction of the initial kinetic energy is converted to heat. In case (2), two masses with momentums {{tmath|1=\boldsymbol{p}_{\boldsymbol{1} }=m_{1} \boldsymbol{v}_{\boldsymbol{1} } }} and {{tmath|1=\boldsymbol{p}_{\boldsymbol{2} }=m_{2} \boldsymbol{v}_{\boldsymbol{2} } }} collide to produce a single particle of conserved mass {{tmath|1=m=m_{1}+m_{2} }} traveling at the
center of mass velocity of the original system, \boldsymbol{v_{c m}}=\left(m_{1} \boldsymbol{v_1}+m_{2} \boldsymbol{v_2}\right) /\left(m_{1}+m_{2}\right) . The total momentum {{tmath|1=\boldsymbol{p=p_{1}+p_{2} } }} is conserved. Fig. 3-10 illustrates the inelastic collision of two particles from a relativistic perspective. The time components {{tmath|E_{1} / c}} and {{tmath|E_{2} / c}} add up to total
E/c of the resultant vector, meaning that energy is conserved. Likewise, the space components {{tmath|1=\boldsymbol{p_{1} } }} and {{tmath|1=\boldsymbol{p_{2} } }} add up to form
p of the resultant vector. The four-momentum is, as expected, a conserved quantity. However, the invariant mass of the fused particle, given by the point where the invariant hyperbola of the total momentum intersects the energy axis, is not equal to the sum of the invariant masses of the individual particles that collided. Indeed, it is larger than the sum of the individual masses: {{tmath|1=m>m_{1}+m_{2} }}.
Energy and momentum conservation In a Newtonian analysis of interacting particles, transformation between frames is simple because all that is necessary is to apply the Galilean transformation to all velocities. Since , the momentum . If the total momentum of an interacting system of particles is observed to be conserved in one frame, it will likewise be observed to be conserved in any other frame. so Fig. 3-12b presents instead a graphing calculator solution. The energy of the neutrino is 29.79 MeV, and the energy of the muon is . Most of the energy is carried off by the near-zero-mass neutrino. }} == Introduction to curved spacetime ==