麻辣社区营山论坛干部:爱因斯坦的相对论下载

相对论分狭义相对论和广义相对论。
本科时学过的，现在就引用当时的话来概括一下内容。
狭义相对论：所有物理规律对于一切惯性参照系都具有相同的表现形式。
广义相对论：所有物理规律对于一切参照系（包括非惯性参照系）都具有相同的表现形式。
狭义相对论的两条公设：
a)，光速不变原理：光速在一切惯性参照系中保持不变；
b)，相对性原理：对于一切惯性系，运用该参照系的空间和时间所表达的物理规律，它们的形式都是相同的；
狭义相对论的所有定理、推论以及一切理论都是在这两个简化得不能再简化的公设的基础上推导出来的。
广义相对论扩大了狭义相对论的考察范围，把参照系从惯性参照系发展到非惯性参照系。其中，用到了一个很核心的公设，那就是等效原理。等效原理的大意是这样的：由引力场造成的参照系与加速度的运动造成的参照系是等效的。从这个原理出发可直接推导出引力场附近的空间是“弯曲”的，它从根本上解决了一个难题，即光经过引力场时表现出了“弯曲”的现象。然而光线总是走直线的，不会弯曲的，真正弯曲的是空间，因为空间弯曲了，所以光线经过引力场时表现出弯曲，实际上，光仍然走直线，不会弯曲的。这也解决了水星今日点的进动剩余难题。

如果还想了解更多，需要学习更专业的物理书籍。

ON THE ELECTRODYNAMICS OF MOVING
BODIES
By A. EINSTEIN
June 30, 1905
It is known that Maxwell’s electrodynamics—as usually understood at the
present time—when applied to moving bodies, leads to asymmetries which do
not appear to be inherent in the phenomena. Take, for example, the reciprocal
electrodynamic action of a magnet and a conductor. The observable phenomenon
here depends only on the relative motion of the conductor and the
magnet, whereas the customary view draws a sharp distinction between the two
cases in which either the one or the other of these bodies is in motion. For if the
magnet is in motion and the conductor at rest, there arises in the neighbourhood
of the magnet an electric field with a certain definite energy, producing
a current at the places where parts of the conductor are situated. But if the
magnet is stationary and the conductor in motion, no electric field arises in the
neighbourhood of the magnet. In the conductor, however, we find an electromotive
force, to which in itself there is no corresponding energy, but which gives
rise—assuming equality of relative motion in the two cases discussed—to electric
currents of the same path and intensity as those produced by the electric
forces in the former case.
Examples of this sort, together with the unsuccessful attempts to discover
any motion of the earth relatively to the “light medium,” suggest that the
phenomena of electrodynamics as well as of mechanics possess no properties
corresponding to the idea of absolute rest. They suggest rather that, as has
already been shown to the first order of small quantities, the same laws of
electrodynamics and optics will be valid for all frames of reference for which the
equations of mechanics hold good.1 We will raise this conjecture (the purport
of which will hereafter be called the “Principle of Relativity”) to the status
of a postulate, and also introduce another postulate, which is only apparently
irreconcilable with the former, namely, that light is always propagated in empty
space with a definite velocity c which is independent of the state of motion of the
emitting body. These two postulates suce for the attainment of a simple and
consistent theory of the electrodynamics of moving bodies based on Maxwell’s
theory for stationary bodies. The introduction of a “luminiferous ether” will
prove to be superfluous inasmuch as the view here to be developed will not
require an “absolutely stationary space” provided with special properties, nor
1The preceding memoir by Lorentz was not at this time known to the author.
1
assign a velocity-vector to a point of the empty space in which electromagnetic
processes take place.
The theory to be developed is based—like all electrodynamics—on the kinematics
of the rigid body, since the assertions of any such theory have to do
with the relationships between rigid bodies (systems of co-ordinates), clocks,
and electromagnetic processes. Insucient consideration of this circumstance
lies at the root of the diculties which the electrodynamics of moving bodies
at present encounters.
I. KINEMATICAL PART
§ 1. Definition of Simultaneity
Let us take a system of co-ordinates in which the equations of Newtonian
mechanics hold good.2 In order to render our presentation more precise and
to distinguish this system of co-ordinates verbally from others which will be
introduced hereafter, we call it the “stationary system.”
If a material point is at rest relatively to this system of co-ordinates, its
position can be defined relatively thereto by the employment of rigid standards
of measurement and the methods of Euclidean geometry, and can be expressed
in Cartesian co-ordinates.
If we wish to describe the motion of a material point, we give the values of
its co-ordinates as functions of the time. Now we must bear carefully in mind
that a mathematical description of this kind has no physical meaning unless
we are quite clear as to what we understand by “time.” We have to take into
account that all our judgments in which time plays a part are always judgments
of simultaneous events. If, for instance, I say, “That train arrives here at 7
o’clock,” I mean something like this: “The pointing of the small hand of my
watch to 7 and the arrival of the train are simultaneous events.”3
It might appear possible to overcome all the diculties attending the definition
of “time” by substituting “the position of the small hand of my watch” for
“time.” And in fact such a definition is satisfactory when we are concerned with
defining a time exclusively for the place where the watch is located; but it is no
longer satisfactory when we have to connect in time series of events occurring
at dierent places, or—what comes to the same thing—to evaluate the times of
events occurring at places remote from the watch.
We might, of course, content ourselves with time values determined by an
observer stationed together with the watch at the origin of the co-ordinates,
and co-ordinating the corresponding positions of the hands with light signals,
given out by every event to be timed, and reaching him through empty space.
But this co-ordination has the disadvantage that it is not independent of the
standpoint of the observer with the watch or clock, as we know from experience.
2i.e. to the first approximation.
3We shall not here discuss the inexactitude which lurks in the concept of simultaneity of
two events at approximately the same place, which can only be removed by an abstraction.
2
We arrive at a much more practical determination along the following line of
thought.
If at the point A of space there is a clock, an observer at A can determine the
time values of events in the immediate proximity of A by finding the positions
of the hands which are simultaneous with these events. If there is at the point B
of space another clock in all respects resembling the one at A, it is possible for
an observer at B to determine the time values of events in the immediate neighbourhood
of B. But it is not possible without further assumption to compare,
in respect of time, an event at A with an event at B. We have so far defined
only an “A time” and a “B time.” We have not defined a common “time” for
A and B, for the latter cannot be defined at all unless we establish by definition
that the “time” required by light to travel from A to B equals the “time” it
requires to travel from B to A. Let a ray of light start at the “A time” tA from
A towards B, let it at the “B time” tB be reflected at B in the direction of A,
and arrive again at A at the “A time” t0A.
In accordance with definition the two clocks synchronize if
tB − tA = t0A − tB.
We assume that this definition of synchronism is free from contradictions,
and possible for any number of points; and that the following relations are
universally valid:—
1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes
with the clock at B.
2. If the clock at A synchronizes with the clock at B and also with the clock
at C, the clocks at B and C also synchronize with each other.
Thus with the help of certain imaginary physical experiments we have settled
what is to be understood by synchronous stationary clocks located at different
places, and have evidently obtained a definition of “simultaneous,” or
“synchronous,” and of “time.” The “time” of an event is that which is given
simultaneously with the event by a stationary clock located at the place of
the event, this clock being synchronous, and indeed synchronous for all time
determinations, with a specified stationary clock.
In agreement with experience we further assume the quantity
2AB
t0A − tA
= c,
to be a universal constant—the velocity of light in empty space.
It is essential to have time defined by means of stationary clocks in the
stationary system, and the time now defined being appropriate to the stationary
system we call it “the time of the stationary system.”
§ 2. On the Relativity of Lengths and Times
The following reflexions are based on the principle of relativity and on the
principle of the constancy of the velocity of light. These two principles we define
as follows:—
3
1. The laws by which the states of physical systems undergo change are not
aected, whether these changes of state be referred to the one or the other of
two systems of co-ordinates in uniform translatory motion.
2. Any ray of light moves in the “stationary” system of co-ordinates with
the determined velocity c, whether the ray be emitted by a stationary or by a
moving body. Hence
velocity =
light path
time interval
where time interval is to be taken in the sense of the definition in § 1.
Let there be given a stationary rigid rod; and let its length be l as measured
by a measuring-rod which is also stationary. We now imagine the axis of the
rod lying along the axis of x of the stationary system of co-ordinates, and that
a uniform motion of parallel translation with velocity v along the axis of x in
the direction of increasing x is then imparted to the rod. We now inquire as to
the length of the moving rod, and imagine its length to be ascertained by the
following two operations:—
(a) The observer moves together with the given measuring-rod and the rod
to be measured, and measures the length of the rod directly by superposing the
measuring-rod, in just the same way as if all three were at rest.
(b) By means of stationary clocks set up in the stationary system and synchronizing
in accordance with § 1, the observer ascertains at what points of the
stationary system the two ends of the rod to be measured are located at a definite
time. The distance between these two points, measured by the measuring-rod
already employed, which in this case is at rest, is also a length which may be
designated “the length of the rod.”
In accordance with the principle of relativity the length to be discovered by
the operation (a)—we will call it “the length of the rod in the moving system”—
must be equal to the length l of the stationary rod.
The length to be discovered by the operation (b) we will call “the length
of the (moving) rod in the stationary system.” This we shall determine on the
basis of our two principles, and we shall find that it diers from l.
Current kinematics tacitly assumes that the lengths determined by these two
operations are precisely equal, or in other words, that a moving rigid body at
the epoch t may in geometrical respects be perfectly represented by the same
body at rest in a definite position.
We imagine further that at the two ends A and B of the rod, clocks are
placed which synchronize with the clocks of the stationary system, that is to say
that their indications correspond at any instant to the “time of the stationary
system” at the places where they happen to be. These clocks are therefore
“synchronous in the stationary system.”
We imagine further that with each clock there is a moving observer, and
that these observers apply to both clocks the criterion established in § 1 for the
synchronization of two clocks. Let a ray of light depart from A at the time4 tA,
4“Time” here denotes “time of the stationary system” and also “position of hands of the
moving clock situated at the place under discussion.”
4
let it be reflected at B at the time tB, and reach A again at the time t0A. Taking
into consideration the principle of the constancy of the velocity of light we find
that
tB − tA = rAB
c − v
and t0A − tB = rAB
c + v
where rAB denotes the length of the moving rod—measured in the stationary
system. Observers moving with the moving rod would thus find that the two
clocks were not synchronous, while observers in the stationary system would
declare the clocks to be synchronous.
So we see that we cannot attach any absolute signification to the concept of
simultaneity, but that two events which, viewed from a system of co-ordinates,
are simultaneous, can no longer be looked upon as simultaneous events when
envisaged from a system which is in motion relatively to that system.
§ 3. Theory of the Transformation of Co-ordinates and
Times from a Stationary System to another System in
Uniform Motion of Translation Relatively to the Former
Let us in “stationary” space take two systems of co-ordinates, i.e. two systems,
each of three rigid material lines, perpendicular to one another, and issuing
from a point. Let the axes of X of the two systems coincide, and their axes of
Y and Z respectively be parallel. Let each system be provided with a rigid
measuring-rod and a number of clocks, and let the two measuring-rods, and
likewise all the clocks of the two systems, be in all respects alike.
Now to the origin of one of the two systems (k) let a constant velocity v
be imparted in the direction of the increasing x of the other stationary system
(K), and let this velocity be communicated to the axes of the co-ordinates, the
relevant measuring-rod, and the clocks. To any time of the stationary system K
there then will correspond a definite position of the axes of the moving system,
and from reasons of symmetry we are entitled to assume that the motion of k
may be such that the axes of the moving system are at the time t (this “t” always
denotes a time of the stationary system) parallel to the axes of the stationary
system.
We now imagine space to be measured from the stationary system K by
means of the stationary measuring-rod, and also from the moving system k
by means of the measuring-rod moving with it; and that we thus obtain the
co-ordinates x, y, z, and , ,  respectively. Further, let the time t of the
stationary system be determined for all points thereof at which there are clocks
by means of light signals in the manner indicated in § 1; similarly let the time
 of the moving system be determined for all points of the moving system at
which there are clocks at rest relatively to that system by applying the method,
given in § 1, of light signals between the points at which the latter clocks are
located.
To any system of values x, y, z, t, which completely defines the place and
time of an event in the stationary system, there belongs a system of values ,
5
, ,  , determining that event relatively to the system k, and our task is now
to find the system of equations connecting these quantities.
In the first place it is clear that the equations must be linear on account of
the properties of homogeneity which we attribute to space and time.
If we place x0 = x − vt, it is clear that a point at rest in the system k must
have a system of values x0, y, z, independent of time. We first define  as a
function of x0, y, z, and t. To do this we have to express in equations that  is
nothing else than the summary of the data of clocks at rest in system k, which
have been synchronized according to the rule given in § 1.
From the origin of system k let a ray be emitted at the time 0 along the
X-axis to x0, and at the time 1 be reflected thence to the origin of the coordinates,
arriving there at the time 2; we then must have 1
2 (0 + 2) = 1, or,
by inserting the arguments of the function  and applying the principle of the
constancy of the velocity of light in the stationary system:—
1
2  (0, 0, 0, t) +  0, 0, 0, t + x0
c − v
+ x0
c + v=  x0, 0, 0, t + x0
c − v.
Hence, if x0 be chosen infinitesimally small,
1
2  1
c − v
+
1
c + v@
@t
= @
@x0
+
1
c − v
@
@t
,
or
@
@x0
+ v
c2 − v2
@
@t
= 0.
It is to be noted that instead of the origin of the co-ordinates we might have
chosen any other point for the point of origin of the ray, and the equation just
obtained is therefore valid for all values of x0, y, z.
An analogous consideration—applied to the axes of Y and Z—it being borne
in mind that light is always propagated along these axes, when viewed from the
stationary system, with the velocity pc2 − v2 gives us
@
@y
= 0,
@
@z
= 0.
Since  is a linear function, it follows from these equations that
 = at −
v
c2 − v2 x0
where a is a function (v) at present unknown, and where for brevity it is
assumed that at the origin of k,  = 0, when t = 0.
With the help of this result we easily determine the quantities , ,  by
expressing in equations that light (as required by the principle of the constancy
of the velocity of light, in combination with the principle of relativity) is also
6
propagated with velocity c when measured in the moving system. For a ray of
light emitted at the time  = 0 in the direction of the increasing 
 = c or  = act −
v
c2 − v2 x0.
But the ray moves relatively to the initial point of k, when measured in the
stationary system, with the velocity c − v, so that
x0
c − v
= t.
If we insert this value of t in the equation for , we obtain
 = a
c2
c2 − v2 x0.
In an analogous manner we find, by considering rays moving along the two other
axes, that
 = c = act −
v
c2 − v2 x0
when
y
pc2 − v2
= t, x0 = 0.
Thus
 = a
c
pc2 − v2
y and  = a
c
pc2 − v2
z.
Substituting for x0 its value, we obtain
 = (v)(t − vx/c2),
 = (v)(x − vt),
 = (v)y,
 = (v)z,
where
 =
1
p1 − v2/c2
,
and  is an as yet unknown function of v. If no assumption whatever be made
as to the initial position of the moving system and as to the zero point of  , an
additive constant is to be placed on the right side of each of these equations.
7
We now have to prove that any ray of light, measured in the moving system,
is propagated with the velocity c, if, as we have assumed, this is the case in the
stationary system; for we have not as yet furnished the proof that the principle
of the constancy of the velocity of light is compatible with the principle of
relativity.
At the time t =  = 0, when the origin of the co-ordinates is common to the
two systems, let a spherical wave be emitted therefrom, and be propagated with
the velocity c in system K. If (x, y, z) be a point just attained by this wave,
then
x2 + y2 + z2 = c2t2.
Transforming this equation with the aid of our equations of transformation
we obtain after a simple calculation
2 + 2 + 2 = c2 2.
The wave under consideration is therefore no less a spherical wave with
velocity of propagation c when viewed in the moving system. This shows that
our two fundamental principles are compatible.5
In the equations of transformation which have been developed there enters
an unknown function  of v, which we will now determine.
For this purpose we introduce a third system of co-ordinates K0, which relatively
to the system k is in a state of parallel translatory motion parallel to
the axis of ,† such that the origin of co-ordinates of system K0 moves with
velocity −v on the axis of . At the time t = 0 let all three origins coincide, and
when t = x = y = z = 0 let the time t0 of the system K0 be zero. We call the
co-ordinates, measured in the system K0, x0, y0, z0, and by a twofold application
of our equations of transformation we obtain
t0 = (−v)(−v)( + v/c2) = (v)(−v)t,
x0 = (−v)(−v)( + v ) = (v)(−v)x,
y0 = (−v) = (v)(−v)y,
z0 = (−v) = (v)(−v)z.
Since the relations between x0, y0, z0 and x, y, z do not contain the time t,
the systems K and K0 are at rest with respect to one another, and it is clear that
the transformation from K to K0 must be the identical transformation. Thus
(v)(−v) = 1.
5The equations of the Lorentz transformation may be more simply deduced directly

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