Chapter 3 Relativistic Kinematics
Chapter 3 3.1 Relativistic Transformation 3.2 Relativistic Transformation II 3.3 Four Vector, Space Time 3.4 Kinematics: Basics 3.5 Interactive Exercises
3.3 Four Vector, Space Time
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Four vectors: Introduction Energy densities
Solids 10-5
100
1K
300K
Quarks Particles Nuclei (protons...) Atoms eV
105
1010
1015 Fig 1
Stellar Interior
Densities White Solids Dwarfs
10-5
100
105
Black Holes
Neutron Stars
1010
1015 Fig 2
Water
Nucleus
Special Relativity To reach high energy densities and small distances, use probes u ≈ c Results of measurements will depend on Lorentz Frame Any sensible theory must be Co-variant
⇨
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Lorentz invariant 4
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Based on properties/ quantities that do not depend on frame e.g: rest mass proper life time Formulate in terms of 4-vector: First let's remember simple special relativity then make it a bit more elegant.
⇨
Electrons, protons, nuclei, have electric charges. Much of what we talk about will involve electromagnetic interactions of particles And it will be relativistic. x
(x, y, z, t) (x', y', z', t')
S
S'
z Fig 3
y
v
x' = x y' = y z' = γ (z – vt) t' = γ(t - β/c ·z)
Note mixing of z, and t
γ is the Lorentz boost factor β is velocity in units where C=1 γ = 1/ (1 - β2)½ β = v/c p=mγv So, v2 = p2/ m2γ2 = p2/ m2 (1- β2) = p2/ m2 (1- v2/c2) v2m2c2 = c4p2 – p2v2c2 Dayalbagh Educational Institute
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v2(m2c4 + p2 c2) = c2 · c2 p2
V2/c2 E2 = c2 p2
β = cp/E E2 We define the position-time four-vector xμ, μ = 0,1,2,3, as follows: x0 = ct, x1 = x, x2 = y, x3 = z
(1)
In terms of xμ, the Lorentz transformations take on a more symmetrical appearance: x0' = y (x0 – βx1) x1' = y(x1 – βx0) x2' = x2 x3' = x3
(2)
Where (3) More compactly: (4) The coefficients Λμυ may be regarded as the elements of a matrix Λ: (5) (i.e. rest are zero)
and all the
Einstein’s ‘summation convention’, Which says that repeated Greek indices (one as subscript, one as superscript) are to be summed from 0 to 3. Dayalbagh Educational Institute
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Thus Equation 5 becomes, finally, (6) A advantage of this tidy notation is that the same form describes Lorentz transformations that are not along the x direction. 1) The individual coordinates of an event change, in accordance with Equation 6, when we go from S to S', there is a particular combination of them that remains the same I ≡ (x0)2 - (x1)2 - (x2)2 - (x3)2 = (x0')2 – (x1')2 – (x2')2 -(x3')2
(7)
2) A quantity, which has the same value in any inertial system, is called an invariant. (In the same sense, the quantity r2 = x2 + y2 + z2 is invariant under rotations). Define the co-variant four-vector xμ (index down) as follows:
xμ ≡ gμυ xυ
(8)
(i.e. x0 = x0 , x1 = -x1, x2 = -x2, x3 = -x3 ) and the ‘original’ four-vector x μ (index up) a contra-variant four-vector. The invariant I can then be written as I = xμ xμ
(9)
(or, equivalently, xμ xμ). The invariant I can be written as a double sum: (10) Where the matrix gμυ has components displayed as a matrix g. Dayalbagh Educational Institute
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(11)
g
μυ
is the same as gμυ
In general we define a four-vector, aμ, as a fourcomponent object that transforms in the same way xμ does when we go from one inertial system to another, (12) with the same coefficients
.
To each such (contra variant) four-vector we associate a covariant four-vector aμ, obtained by simply changing the signs of the spatial components, or, more formally aμ = gμυaυ
(13)
aμ = g μ υaυ
(14)
where g μ υ are technically the elements in the matrix g-1 (however, since our metric is its own inverse). Given any two four-vectors, aμ and bμ, the quality aμbμ = aμbμ = a0b0 - a1b1 - a2b2 - a3b3
(15)
is invariant (the same number in any inertial system) i.e. the scalar product of a and b. (16) (17) Using the notation a2 for the scalar product of aμ with itself: (18) Dayalbagh Educational Institute
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Notice, however, that a2 need not be Positive classify all four-vectors according to the sign of a2: If a2 > 0, If a2 < 0, If a2 = 0,
aμ is called timelike aμ is called spacelike aμ is called lightlike
(19)
In this hierarchy, a vector is a tensor of rank one, and a scalar (invariant) is a tensor of rank zero. We construct covariant and ‘mixed’ tensors by lowering indices (at the cost of a minus sign for each spatial index), for example (20) and so on. Notice that the product of two tensors is itself a tensor: (aμ bυ) is a tensor of second rank; (aμtυλσ) is a tensor of fourth rank; and so on. Finally, we can obtain from any tensor of rank n + 2a ‘contracted’ tensor of rank n, by summing like upper and lower indices. Thus sμμ is a scalar; tυμυ is a vector; aμ tμυλ is a secondrank tensor.
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Space-Time Interval The Lorentz transformations reveal relation between space and time.
an
intimate
For this reason, it is useful to think of a fourdimensional space-time with coordinates x, y, z, and t. Since it is dimensional events for events take
difficult to imagine, or to draw, a fourspace-time, we limit our discussion to which the space intervals between two place in the “t, x-plane”, that is, Δy = Δz = 0.
In Fig. 6(a), the spatial dimension in units of 1/ c is plotted along the abscissa and the time dimension is plotted along the ordinate. The three lines in the figure represent the history of three experimental observations. C B Line A represents a A t particle at rest in reference frame S since, as t increases, the displacement coordinate D remains the same. Line B represents particle traveling with a constant speed ux.
Fig 6
x/c
The speed of an object in terms of c in this reference frame is the ratio of an interval along the ordinate Δt.
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(It is important note the difference between the “two dimensional world” of Fig. 6 and displacement versus time. When we plot displacement along the ordinate and time along the abscissa, the speed is the ratio of an interval along the ordinate to an interval along the abscissa.) Since line C is at an angle of 45°, the slope of line C is or Line C represents the “world line” of a light pulse sent out from the origin. Lines that make an angle greater than 45° with the x-axis represent speeds lower than c. The special theory of relativity predicts that no material particle can travel with a speed greater than c, so the world line D in the figure is not possible. The world lines of A and B appear different to observers in reference frame S' moving to the right relative to S. B' t' The particle at rest A' C appears to observers in S' to move with velocity v to the left (Fig. 6(b)). The world line B' has a greater slope than B because observers in S' measure the particle’s speed to be u', which is less than u. Dayalbagh Educational Institute
Fig 6
x/c
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The world line C' representing the light pulse, however, also has a slope of one because the speed of light is the same for all observers. While observers in reference frames S and S' measure different space intervals Δx and Δx', respectively, and different time intervals, Δt and Δt', respectively, there is a space-time interval ΔI2 that is common to both, where
The invariance of the space-time interval, that is,
can be demonstrated by substituting the values of Δt' and Δx' from Eq. (4) and (3) of Section 5.1. The space-time interval ΔI2 is interesting because, unlike space intervals and time intervals, which are not invariant, all observers in any inertial frame agree on the magnitude of ΔI2. In addition, the use of the space-time interval allows us to make predictions about casualty. For the purpose of this discussion, we define casualty in such a way, that event a can cause event b if a precedes b, or b can be the cause of effect a precedes a. We examine three cases: 1. ΔI2 > 0, 2. ΔI2 = 0,
and
3. ΔI2 < 0 . Dayalbagh Educational Institute
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Although it is troublesome to imagine that any quantity that is squared is less than one, we ignore this difficulty and explore the usefulness of the space-time interval. Let event a take place at t = 0 and x = 0, and event b at t and x. The space interval between a and b is Δx = x – 0 = x and the time interval Δt = t – 0 = t. If ΔI2 = Δt2 – Δx2/c2 = Δt'2 – Δx'2/c2 is positive, then the magnitude of the distance between the two events Δx is less than c Δt. The two events occur in a space interval that can be traversed by light or by a signal that travels slower than light. Event a then can be the cause of event b. Because of the invariance of the space-time interval, when ΔI2 > 0, the distance between the two events Δx' in any other inertial system is less than c Δt' and the events may be causally connected. For any observer, event b is in the future of a and a is in the past of b. Whenever ΔI2 is positive, no observer will see the events reversed in time.
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If ΔI2 = 0, Δx = c Δt and Δx' = c Δt'. The two events are separated by a distance that light can travel in the time interval, and the two events can be causally related. Finally, we look at the case for which ΔI is negative. For this case, the magnitude of Δx is greater than c Δt and the events are separated by a space interval that is greater than the distance light can travel during the time interval. Since no signal can travel with a speed greater than the speed of light, there is no cause-and-effect relation between a and b. An interesting ordering of events occurs when ΔI2 is negative. Using the time transformation,
we see that Δt' has the same sign as Δt if (υ/c)(Δx/c) is less than Δt, and the opposite sign if (υ/c)(Δx/c) is greater than Δt. In the latter case, if a occurs before b in reference frame S, b may occur before a in other reference frames. There is still no causal relationship, however, between a and b. Since no signal can travel with a speed greater than the speed of light, a cannot cause b and b cannot cause a.
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The results of this discussion are summarized in Fig. 7 Events are located in terms of their coordinates (t, x/c). Event a is located at (0, 0). For ΔI2 > 0, the events are located in space-time intervals between the two dashed lines at an angle of 45° with the , x-axes. t
b1
b3 b4 x/c
a
b2
Fig 7: Events a and b, can be casually related; events a and b4 cannot be casually related.
Event b1 lies in the absolute future of event a and event b2 in the absolute past of a. Events in this region can be causally related. Event a can cause b1 and event b2 can cause a. Since there must be a reference frame in which both a and b1 happen in the same place (Δx' = 0), the space-time interval then equals Δt'.
For this reason the space-time interval when ΔI2 > 0, is called timelike. When ΔI2 = 0, a and b3 are connected by a light signal and can be cause and effect. Such a space-time interval is called lightlike. Dayalbagh Educational Institute
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For ΔI2 < 0, a and b4 can not he cause and effect. ln some reference frames, event b4 happens after a, but in others b4 happens before a. There exists some reference frame for which a and b4 happen simultaneously (Δt' = 0) and the space interval between them is Δx' = c (-ΔI2) 1/2. When ΔI2 is negative, the space-time interval is called spacelike.
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Example 1 The space-time coordinates (t, x/c) for event a are (0, 0) and for event b (2, 1) in reference frame S. Find the space-time interval for the events in (a)reference frame S and (b)reference frame S', which moves with velocity υ = 0.6c to the right relative to S. a) In S, Δx/c = 1 and Δt = 2. Therefore,
ΔI2 = Δt2 - Δx2/c2 = 4-1 = 3. b) Using the Lorentz transformations, we find
And
While Δx' < 1, Δt' cannot be less than one for a positive value of ΔI2 since Δt' is greater than Δx'/c, and therefore greater than υ Δx/c2 :
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Example 2 Chandra Shekar and Raman, passengers on a rocket ship S', which is moving relative to a space station with velocity Ď&#x2026;, are considering sending each other presents. Observers in a space station S witness the following events with coordinates (t, x/c). Event a. Chandra Shekar hands a messenger a gift for Raman (-45 x 10-8, -75 x 10-8), Fig. 8(a). Event b. Raman hands his messenger a gift for Chandra Shekar -8 -8 (90 x 10 , 150 x l0Chandra ), Fig. 8(b). Raman Shekar
Fig 8 (a and b)
Event c. Raman receives his present (330 x 10-8, 150 x 10-8), Fig. 8(c). Event d. Chandra Shekar receives his present (465 x 10-8, 375 x 10-8), Fig. 8(d). Dayalbagh Educational Institute
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Raman
Chandra Shekar
Fig. 8 (c and d)
Q1.Did Chandra Shekar send Raman a present because he knew Raman was sending him one? What is the spatial separation for which the events a and b, happen simultaneously? A1.For events a and b, Δt = 135 x 10-8s and
225 x 10-8 s ;
Therefore,
The space-time interval is space like so there is no cause and effect. In some reference frame, simultaneous, Δt' = 0 and
the
events
were
Δx' = c ΔI = 3 x 108 m s-1 x 180 x 10-8 s = 540 m.
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Q2.Can Raman send a present that Chandra Shekar receives in any inertial frame? A2.For events a and d, and therefore,
The space-time interval is time like so there can be cause and effect. Q3.Can Raman send and receive a present from Chandra Shekar in any inertial frame? In a reference frame in which a and c happen at the same place, what is the time interval between the two events? A3.For events a and c, Therefore,
and
.
The space-time interval is time like, so both events can happen to Raman. In some reference frame, events a and c happen in the same place and Î&#x201D;t = 300 x 10-8 s.
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Q4.Does Raman receive his gift first in every reference frame? In a reference frame in which the gifts are received simultaneously, what is the spatial separation between Raman and Chandra Shekar? A4.For events c and d, and therefore,
The space-time interval is space like so there are some frames in which Chandra Shekar receives his gift first. For the reference frame Sâ&#x20AC;&#x2122; in which the gifts are received simultaneously (Î&#x201D;t = 0), or
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x' = 540 m.
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Example 3 Use the Lorentz transformations to find (a) the coordinates of the events from the point of view of observers in the rocket ship S’ traveling with velocity v = 3/5c with respect to the space station described in Example 2, and (b) show that the space-time intervals between events a and d are equal; that is, ΔI' 2 = ΔI2. (Note that (1 – v2/c2)1/2 = 0.8.) a) Coordinates for event a:
x' = -180 m ;
Coordinates for event b:
Coordinates for event c:
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Coordinates for event d:
Chandra Shekar
Raman
Fig 9
Figure 9(a) shows the events a and b as viewed by observers on the rocket ship. Raman is at x' = -180 m and Chandra Shekar is at x' = 360 m. Both hand their gifts to the messengers simultaneously when t' = 0. Both receive their presents from the messengers simultaneously (Fig. 9(b)). For events a and d, Î&#x201D;x' = 540 m and Î&#x201D;t' = 300 x 10-8 s.
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Raman
Chandra Shekar
Fig 9
ΔI'2 = Δt'2 – Δx' 2 = (300 x 10-8 s)2 - (180 x 10-8 s)2 = (240 x 10-8 s)2 = ΔI2, as in Example 2(Q2).
Example 4 Is pμ, space-like, time-like or light like for a real particle of mass m ? How about a virtual particle
Solution 4 Now, ,
= Since
, hence
If m=0, then Dayalbagh Educational Institute
is time like.
So it is light like. 24
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The 4-momentum anything
of a virtual particle could be
Example 5
Given 2 four vectors aμ = (1,4,2,1) and bμ = (7,0,2,3) find aμ = (1,-4,-2,-1) , bμ = (7,0,-2,-3)
Solution 5 →2
a = 42 + 22 + 12 = 21 →
b2 = 02 + 22 + 32=13 →
→
a . b = (4.0+2.2+1.3) = 7 →
a2 = 12 – a2 = 1-21 = -20 →
b = 49 – b2 = 49 -13 = 36 2
2
→
→
a.b = 1.7 – a . b = 7 – 7 =0 So aμ is space like and bμ is time like
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