Accelerated Motion in Special Relativityby Dr. phil. Friedrich Futschik retiredThe NETHERLANDSe-mail: fflo@tele2.nl

In 2001, professor Dr. Vladimir L. Kalashnikov published his "Clock Paradox and Accelerated Motion in Special Relativity" as a pedagogical analysis of accelerated motion and gave the invariant description of the so-called "clock paradox", in the framework of the theory of special relativity. Following A. A. Logunov's statements, he cautioned against an incorrect and too broad interpretation of the equivalence principle for the accelerated motion.The author of this publication has thoroughly edited the original MAPLE 6 version of professor Kalashnikov (see www.geocities.com/optomaplev ) by using the Classic Worksheet of MAPLE 9.01 and its Tensor Package.

Usually, the content of the theory of special relativity is reduced to the description of uniform motion and the corresponding Lorentz Transformations. The pure geometrical interpretation of the theory is attributed to the theory of general relativity with its assumptions concerning general coordinate transformations. That this is not a correct opinion will be shown. Both theories are theories of a space-time world and admit arbitrary permissible coordinate transformations. In a sense, "general relativity" is not a good name for the theory, the main meaning of which is the interpretation of gravitation as a curvature of space-time. The difference between the two theories concerns the space-time structure: the flat, pseudo-Euclidian (topologically simple) manifold of special relativity and the curved, Riemannian (topologically nontrivial) manifold of the general relativity.The gravitation is the curvature and, as a result of its tensor nature, we cannot "turn it off" in some finite region by means of a coordinate transformation.Therefor, the equivalence principle has only a local and heuristic meaning. An observer can always distinguish gravitation from acceleration. It should be emphasized that the acceleration is not allied to the curvature of space-time and, as a result, it can be described in the framework of the special relativity.Referencies:MAPLE 9.01, Learning Guide and Help MenuJ.L. Synge, Relativity: The General Theory, Amsterdam (1960)L.D. Landau, E,M. Lifshitz, The Classical Theory of Fields, Pergamon Press, Oxford (1962)A. Duschek, A. Hochrainer, Grundz\374ge der Tensorrechnung in analytischer Darstellung, III. Teil, Wien, Springer (1955) A.A. Logunov, Lectures on Relativity and Gravitation : A Modern Analysis of Problems , Moscow, Nauka (1987, in Russian)

restart; with(tensor): with(difforms):We take two coordinate systems, one system "in rest" , the inertial system, with space coordinates (NiMlIlhH,NiQlIllHJSJaRw==) and the time coordinate T ; the other system moving uniformly accelerated along the NiMlIlhH-axis of the inertial system, with coordinates (t,x,y,z). The directions of all corresponding axes are equal, v = velocity and a = constant acceleration, both relative to an observer in the resting system; NiMvJSdsYW1iZGFHKiYiIiJGJi0lJXNxcnRHNiMsJkYmRiYqJiUidkciIiMqJCUiY0dGLSEiIkYwRjA= , c the constant velocity of light .A light signal, starting at NiMvJSJURyUidEc== NiMiIiE= in the origin and travelling in the resting system during time NiMlIlRH to a point (X,Y,Z) , i.e. a distance for which holds :sqrt(X^2+Y^2+Z^2) = c*T;which meansc^2*T^2-(X^2+Y^2+Z^2) = 0;

NiMvKiQsKCokSSJYRzYiIiIjIiIiKiRJIllHRihGKUYqKiRJIlpHRihGKUYqI0YqRikqJkkiY0dGKEYqSSJUR0YoRio=NiMvLCoqJkkiY0c2IiIiI0kiVEdGJ0YoIiIiKiRJIlhHRidGKCEiIiokSSJZR0YnRihGLSokSSJaR0YnRihGLSIiIQ==

Then, according to the experimental fact that the velocity of light is independant of relative movements of both, the source of light and the observer, the same equations must also hold for the coordinates in the moving system: c^2*t^2-(x^2+y^2+z^2)=0;

NiMvLCoqJkkiY0c2IiIiI0kidEdGJ0YoIiIiKiRJInhHRidGKCEiIiokSSJ5R0YnRihGLSokSSJ6R0YnRihGLSIiIQ==

This means that the space-time distance S , withS^2 = c^2*T^2-(X^2+Y^2+Z^2);ord(S)^2 = c^2*d(T)^2-(d(X)^2+d(Y)^2+d(Z)^2);

NiMvKiRJIlNHNiIiIiMsKiomSSJjR0YmRidJIlRHRiZGJyIiIiokSSJYR0YmRichIiIqJEkiWUdGJkYnRi8qJEkiWkdGJkYnRi8=NiMvKiQtSSJkRzYiNiNJIlNHRiciIiMsKiomSSJjR0YnRiotRiY2I0kiVEdGJ0YqIiIiKiQtRiY2I0kiWEdGJ0YqISIiKiQtRiY2I0kiWUdGJ0YqRjYqJC1GJjYjSSJaR0YnRipGNg==

must be invariant.The necessary transformations of the coordinates from one system to the other have been derived independently by Einstein , who proposed to name them Lorentz-Transformations :LT1:={x[1]=lambda*(X[1]-v*T),x[2]=X[2],x[3]=X[3],t=lambda*(-v/c^2*X[1]+T)};solve({t=lambda*(-v/c^2*X[1]+T),x[1]=lambda*(X[1]-v*T)},{T,X[1]});

NiM+SSRMVDFHNiI8Ji8mSSJ4R0YlNiMiIiIqJkknbGFtYmRhR0YlRissJiZJIlhHRiVGKkYrKiZJInZHRiVGK0kiVEdGJUYrISIiRisvJkYpNiMiIiMmRjBGNy8mRik2IyIiJCZGMEY8L0kidEdGJSomRi1GKywmKihGMkYrSSJjR0YlISIjRi9GK0Y0RjNGK0Yr

NiM8JC9JIlRHNiIqKCwmKiZJInRHRiYiIiJJImNHRiYiIiNGKyomSSJ2R0YmRismSSJ4R0YmNiNGK0YrRitGK0knbGFtYmRhR0YmISIiLCYqJEYsRi1GKyokRi9GLUY0RjQvJkkiWEdGJkYyKipGLEYtLCZGMEYrKiZGL0YrRipGK0YrRitGM0Y0RjVGNA==

i.e. the coordiates in the resting system, expressed through the coordinates in the moving system:LT2:={X[1]=lambda*(x[1]+v*t),X[2]=x[2],X[3]=x[3],T=lambda*(v/c^2*x[1]+t)};

NiM+SSRMVDJHNiI8Ji8mSSJYR0YlNiMiIiMmSSJ4R0YlRiovJkYpNiMiIiQmRi1GMC8mRik2IyIiIiomSSdsYW1iZGFHRiVGNiwmJkYtRjVGNiomSSJ2R0YlRjZJInRHRiVGNkY2RjYvSSJUR0YlKiZGOEY2LCYqKEY8RjZJImNHRiUhIiNGOkY2RjZGPUY2RjY=

N.B.1 : The change from the coordinates in the resting to those in the moving system is, of course accomplished by replacing v by - v !N.B.2 : Since only X and x matter in the following considerations, henceforth mostly they will be used from or for the space vectors (X,Y,Z) and (x,y,z) , respectively.N.B.3 : In MAPLE's tensor package the indexing functions use (NiYvKiYlImNHIiIiJSJUR0YmJiUiWEc2I0YmJkYpNiMiIiMmRik2IyIiJCZGKTYjIiIl) , so that the invariant expression has to be understood as -S^2 = X[1]^2-(X[2]^2+X[3]^2+X[4]^2);

NiMvLCQqJEkiU0c2IiIiIyEiIiwqKiQmSSJYR0YnNiMiIiJGKEYvKiQmRi02I0YoRihGKSokJkYtNiMiIiRGKEYpKiQmRi02IyIiJUYoRik=

N.B.4 : We consider a and NiMtJSJ2RzYjIiIh vectors parallel to the vector X , with NiMqJCUiYUciIiM= , NiMqJC0lInZHNiMiIiEiIiM= and NiMvKiYlImFHIiIiLSUidkc2IyIiIUYmKiZGJ0YmRiVGJg== the respective scalar products.For a point (= observer), resting in the moving system holds d(x) = 0 and from the formulae aboved(X)=lambda*v*d(t); d(T)=lambda*d(t);d(s)^2=c^2*t^2;s:=c*t=c/lambda*T;

NiMvLUkiZEc2IjYjSSJYR0YmKihJJ2xhbWJkYUdGJiIiIkkidkdGJkYrLUYlNiNJInRHRiZGKw==NiMvLUkiZEc2IjYjSSJUR0YmKiZJJ2xhbWJkYUdGJiIiIi1GJTYjSSJ0R0YmRis=NiMvKiQtSSJkRzYiNiNJInNHRiciIiMqJkkiY0dGJ0YqSSJ0R0YnRio=NiM+SSJzRzYiLyomSSJjR0YlIiIiSSJ0R0YlRikqKEYoRilJJ2xhbWJkYUdGJSEiIkkiVEdGJUYp

where NiMvKiYlInNHIiIiJSJjRyEiIiUkdGF1Rw== = t is called the proper time of the moving point, i.e. the time of a clock travelling along with the fixed point (observer) in the moving system. If the point's movement is accelerated, then NiMvJSJzRyomJSJjRyIiIi0lJEludEc2JComRidGJy0lJ2xhbWJkYUc2IyUiVEchIiJGL0Yn because NiMlJ2xhbWJkYUc= is no longer a constant. And the relativistic velocity, expressed through coordinates in the resting system : NiMtJSJ3RzYjJSJURw== = NiMqJi0lJ2xhbWJkYUc2IyUiVEciIiItJSVkaWZmRzYkLSUiWEdGJkYnRig= ors:='s':w(T):= lambda(T)*v(T);

NiM+LUkid0c2IjYjSSJUR0YmKiYtSSdsYW1iZGFHRiZGJyIiIi1JInZHRiZGJ0Ys

Hence we find the relativistic acceleration a :Diff(lambda*v,T)=a;

NiMvLUklRGlmZkc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiQqJkknbGFtYmRhR0YpIiIiSSJ2R0YpRi1JIlRHRilJImFHRik=

If the acceleration is constant, this results ineq1 := int(lhs(%),T)=int(rhs(%),T)+v(0);

NiM+SSRlcTFHNiIvKiZJJ2xhbWJkYUdGJSIiIkkidkdGJUYpLCYqJkkiYUdGJUYpSSJUR0YlRilGKS1GKjYjIiIhRik=

and withlambda:=1/(1-v^2/c^2)^(1/2);

NiM+SSdsYW1iZGFHNiIqJCwmIiIiRigqJkkidkdGJSIiI0kiY0dGJSEiIyEiIiNGLkYr

an easy transformation renderseq2 := simplify( (lhs(eq1)/c)^2 + 1 ) = (rhs(eq1)/c)^2 + 1;

NiM+SSRlcTJHNiIvKiZJImNHRiUiIiMsJiokRihGKSIiIiokSSJ2R0YlRikhIiJGLywmKiYsJiomSSJhR0YlRixJIlRHRiVGLEYsLUYuNiMiIiFGLEYpRighIiNGLEYsRiw=

The lefthand side of eq2 we recognize as NiMqJC0lJ2xhbWJkYUc2IyUiVEciIiM= , hence the relativistic velocity w(T) of, e.g. the moving origin as noticed from the resting systemeq:=Diff(X(T),T) = rhs(eq1)/sqrt(rhs(eq2));

NiM+SSNlcUc2Ii8tSSVEaWZmRzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHRiU2JC1JIlhHRiU2I0kiVEdGJUYwKiYsJiomSSJhR0YlIiIiRjBGNUY1LUkidkdGJTYjIiIhRjVGNSwmKiZGMiIiI0kiY0dGJSEiI0Y1RjVGNSMhIiJGPA==

int(rhs(%),T)+_C1;

NiMsJiooLCoqJkkiYUc2IiIiI0kiVEdGKEYpIiIiKihGJ0YrRipGKy1JInZHRig2IyIiIUYrRikqJEYtRilGKyokSSJjR0YoRilGK0YrRichIiIqJkYlRitGMyEiIyNGNEYpRitJJF9DMUdGKEYr

X:= unapply(%,T);

NiM+SSJYRzYiZio2I0kiVEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqKCwqKiZJImFHRiUiIiM5JEYxIiIiKihGMEYzRjJGMy1JInZHRiU2IyIiIUYzRjEqJEY1RjFGMyokSSJjR0YlRjFGM0YzRjAhIiIqJkYuRjNGOyEiIyNGPEYxRjNJJF9DMUdGJUYzRiVGJUYl

From the definition of the proper time NiMlJHRhdUc= of the moving observer, NiMvLSUiZEc2IyUkdGF1RyomLUYlNiMlIlRHIiIiJSdsYW1iZGFHISIi = NiMtJSJkRzYjJSJ0Rw== (see above), integration of NiMqJiIiIkYkJSdsYW1iZGFHISIi from above1/sqrt(rhs(eq2));

NiMqJCwmKiYsJiomSSJhRzYiIiIiSSJUR0YpRipGKi1JInZHRik2IyIiIUYqIiIjSSJjR0YpISIjRipGKkYqIyEiIkYw

results inInt(%,T) + _C;tau = value(%);

NiMsJi1JJEludEc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiQqJCwmKiYsJiomSSJhR0YpIiIiSSJUR0YpRjFGMS1JInZHRik2IyIiIUYxIiIjSSJjR0YpISIjRjFGMUYxIyEiIkY3RjJGMUkjX0NHRilGMQ==

NiMvSSR0YXVHNiIsJiomLUkjbG5HNiRJKnByb3RlY3RlZEdGK0koX3N5c2xpYkdGJTYjLCYqJiwmKihJImFHRiUiIiItSSJ2R0YlNiMiIiFGM0kiY0dGJSEiI0YzKihGMiIiI0Y4RjlJIlRHRiVGM0YzRjMqJkYyRjtGOEY5IyEiIkY7RjMqJCwoKihGMkY7RjhGOUY8RjtGMyoqRjJGM0Y0RjNGOEY5RjxGM0Y7KiYsJiokRjRGO0YzKiRGOEY7RjNGM0Y4RjlGMyNGM0Y7RjNGM0Y9Rj5GM0kjX0NHRiVGMw==

which can be written as tau:=T->c/abs(a)*ln(v(0)*a/(c*abs(a))+abs(a)*T/c+sqrt(1+(a*T+v(0))^2/(c^2)))+_C;

NiM+SSR0YXVHNiJmKjYjSSJUR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJiooSSJjR0YlIiIiLUkkYWJzR0kqcHJvdGVjdGVkR0YyNiNJImFHRiUhIiItSSNsbkc2JEYySShfc3lzbGliR0YlNiMsKCoqLUkidkdGJTYjIiIhRi9GNEYvRi5GNUYwRjVGLyooRjBGLzkkRi9GLkY1Ri8tSSVzcXJ0R0Y4NiMsJkYvRi8qJiwmKiZGNEYvRkJGL0YvRj1GLyIiI0YuISIjRi9GL0YvRi9JI19DR0YlRi9GJUYlRiU=

Now the transition from the coordinates (NiQlIlRHJSJYRw==) in the resting system to (NiQlInRHJSJ4Rw==) in the uniformly accelerated system, whereby the accelerated system moves along the NiMlIlhH-axis of the resting (inertial) system, starting from NiMvLSUidkc2IyIiIUYn , and NiMvJSJYRyUieEc= = 0 at NiMvJSJURyUidEc= = 0 :N.B.5 : The vector of the constant acceleration a has only one component a , in the direction of the NiMlIlhH - axis, so NiMvLSUkYWJzRzYjJSJhR0Yn .

defform(c=const,a=const,X=0,T=0,x=0,t=0,tau=0):

subs({v(0)=0,abs(a)=a},X(T)-X(0));

NiMsJiooLCYqJkkiYUc2IiIiI0kiVEdGKEYpIiIiKiRJImNHRihGKUYrRitGJyEiIiomRiVGK0YtISIjI0YuRilGKyomRi1GKUYnRi5GLg==

Therefor the law of motion, for an oberver fixed in the origin of the accelerated system, can be written asx = X - %; d( % );

NiMvSSJ4RzYiLChJIlhHRiUiIiIqKCwmKiZJImFHRiUiIiNJIlRHRiVGLUYoKiRJImNHRiVGLUYoRihGLCEiIiomRipGKEYwISIjI0YxRi1GMSomRjBGLUYsRjFGKA==NiMvLUkiZEc2IjYjSSJ4R0YmLCYtRiU2I0kiWEdGJiIiIiouLCYqJkkiYUdGJiIiI0kiVEdGJkYyRi0qJEkiY0dGJkYyRi1GLUYxRi0qJkYvRi1GNSEiIyMhIiRGMkY1RjdGM0YtLUYlNiNGM0YtISIi

and transition to time NiMlInRH renderssubs({T=t,d(T)=d(t)},%);

NiMvLUkiZEc2IjYjSSJ4R0YmLCYtRiU2I0kiWEdGJiIiIiouLCYqJkkiYUdGJiIiI0kidEdGJkYyRi0qJEkiY0dGJkYyRi1GLUYxRi0qJkYvRi1GNSEiIyMhIiRGMkY1RjdGM0YtLUYlNiNGM0YtISIi

s1 := solve(%,d(X));

NiM+SSNzMUc2IiooLCgqKC1JImRHRiU2I0kieEdGJSIiIiomLCYqJkkiYUdGJSIiI0kidEdGJUYyRi0qJEkiY0dGJUYyRi1GLUY1ISIjIyIiJEYyRjVGMkYtKihGMUY4RjNGOC1GKjYjRjNGLUYtKipGMUYtRjNGLUY6Ri1GNUYyRi1GLUYuIyEiJEYyRjVGNg==

This is the expression for the space differential NiMtJSJkRzYjJSJYRw== of the space-time invariant NiMqJC0lImRHNiMlIlNHIiIj , expressed in the coordinates of the accelerated system .If in the inertial system NiMvKiQtJSJkRzYjJSJTRyIiIywqKiYlImNHRiktRiY2IyUiVEdGKSIiIiokLUYmNiMlIlhHRikhIiIqJC1GJjYjJSJZR0YpRjUqJC1GJjYjJSJaR0YpRjU= , then this interval in the moving system is equal to:expand(s1^2): d(s)^2 = c^2*d(t)^2 - %% - d(y)^2 - d(z)^2: collect(%,{d(x),d(x)^2,d(t)^2,d(y)^2,d(z)^2});

NiMvKiQtSSJkRzYiNiNJInNHRiciIiMsLC1GJjYjSSJ4R0YnISIiKiQtRiY2I0kieUdGJ0YqRi8qJC1GJjYjSSJ6R0YnRipGLyomSSJjR0YnRiotRiY2I0kidEdGJ0YqIiIiKiosJiomSSJhR0YnIiIkRjxGQkY9KihGQUY9RjxGPUY5RipGPUY9KiYsJiomRkFGKkY8RipGPSokRjlGKkY9Rj1GOSEiIyMhIiRGKkY5RkhGOkY9Ri8=

simplify(-1/(c^2+a^2*t^2)*c^2-1/(c^2+a^2*t^2)*a^2*t^2);simplify(c^2-1/(c^2+a^2*t^2)*c^2*a^2*t^2);

NiMhIiI=NiMqJkkiY0c2IiIiJSwmKiZJImFHRiUiIiNJInRHRiVGKiIiIiokRiRGKkYsISIi

so that . d(s)^2=c^2*d(t)^2/(1+a^2*t^2/(c^2))-2*a*t*d(t)*d(x)/sqrt(1+a^2*t^2/(c^2))-d(x)^2-d(y)^2-d(z)^2;

NiMvKiQtSSJkRzYiNiNJInNHRiciIiMsLCooSSJjR0YnRiotRiY2I0kidEdGJ0YqLCYqKEkiYUdGJ0YqRjBGKkYtISIjIiIiRjVGNSEiIkY1KixGM0Y1RjBGNUYuRjUtRiY2I0kieEdGJ0Y1RjEjRjZGKkY0KiRGOEYqRjYqJC1GJjYjSSJ5R0YnRipGNiokLUYmNiNJInpHRidGKkY2

This metric, obviously, is not form-invariant in the initial (inertial) Galilean metric, as a result of the difference between the physical processes in the accelerated and the inertial coordinate system.The metric tensor in accelerated coordinates has the following components, attached to NiMqJiUiY0ciIiIqJC0lImRHNiMlInRHIiIjRiU=, NiMqJi0lImRHNiMlInhHIiIiLUYlNiMlInRHRig= , NiMqJC0lImRHNiMlInhHIiIj, NiMqJC0lImRHNiMlInlHIiIj, NiMqJC0lImRHNiMlInpHIiIj , respectively :coord:=[t, x, y, z]: g_compts:=array(symmetric,sparse,1..4,1..4):g_compts[1,1]:=1/(1+a^2*t^2/(c^2)): g_compts[1,2]:=-a*t/c/sqrt(1+a^2*t^2/(c^2)): g_compts[2,2]:=-1: g_compts[3,3]:=-1: g_compts[4,4]:=-1:We calculate the covariant metric tensor g1 and the contravariant metric tensor g1_inv :g1 := create([-1,-1], eval(g_compts));g1_inv := invert( g1, 'detg' ):

NiM+SSNnMUc2Ii1JJlRBQkxFR0kqcHJvdGVjdGVkR0YoNiM3JC9JK2luZGV4X2NoYXJHRiU3JCEiIkYuL0knY29tcHRzR0YlLUknbWF0cml4R0YlNiM3JjcmKiQsJiooSSJhR0YlIiIjSSJ0R0YlRjpJImNHRiUhIiMiIiJGPkY+Ri4sJCoqRjlGPkY7Rj5GPEYuRjcjRi5GOkYuIiIhRkI3JkY/Ri5GQkZCNyZGQkZCRi5GQjcmRkJGQkZCRi4=

The non-zero Christoffel symbols of the second kind, NiMpJiUmR2FtbWFHNiQlImpHJSJrRyUiaUc= = {i,jk} for this metric areD1g := d1metric( g1, coord ): Cf1_1 := Christoffel1 ( D1g ): Cf2_1 := Christoffel2( g1_inv, Cf1_1 ):displayGR(Christoffel2,%);

NiI=NiNJS1RoZX5DaHJpc3RvZmZlbH5TeW1ib2xzfm9mfnRoZX5TZWNvbmR+S2luZEc2Ig==NiNJNm5vbi16ZXJvfmNvbXBvbmVudHN+Okc2Ig==NiMvSSh+fGZyMiwxMXxockc2IiooSSJhR0YlIiIiSSJjR0YlISIiKiYsJiomRiciIiNJInRHRiVGLkYoKiRGKUYuRihGKEYpISIjIyEiJEYu

It should be noted, that the correct dimensional expression for NiMqJCYlJkdhbW1hRzYkIiIiRiciIiM= = NiMlKH58ZnIyLDExfGhyRw== (in MAPLE notation) must be NiMqJiUiYUciIiIqJiUiY0ciIiMtJSVzcXJ0RzYjKiQsJkYlRiUqKEYkRiglInRHRigqJEYnRighIiJGJSIiJEYlRjE= , because NiMtJSVkaWZmRzYkJiUiZ0c2JCUiaUclImpHJiUieEc2IyIiIg== inserts an extra NiMqJiIiIkYkJSJjRyEiIg== from NiMvJiUieEc2IyIiIiomJSJjR0YnJSJ0R0Yn .We can see the geometrical sense of the relativity of acceleration . The acceleration connects with the Christoffel symbols :NiMqJiYlI2R1RzYjJSJpRyIiIiUjZHNHISIi = NiMvKiglImNHIiIiKSYlJkdhbW1hRzYkRiZGJiUiaUdGJiYlImdHRiohIiIqJiYlImFHNiNGK0YmKiZGJUYmLSUlc3FydEc2IywmRiZGJiomJSJ2RyIiIyokRiVGOkYuRi5GJkYu , NiMmJSJ1RzYjJSJpRw== being the spatial part of the 4-velocity ( i = 2,3,4 see N.B.3 above) .The Christoffel symbols obey the inhomogeneous transformational law and, therefore, the acceleration is relative, i. e. it can be " turned off " by means of a coordinate transformation ! And it should be noted, that our system does not concern any gravitational processes, because the space-time system is flat :D2g := d2metric( D1g, coord ): RMN := Riemann( g1_inv, D2g, Cf1_1 ): displayGR(Riemann,%);

NiI=NiNJM1RoZX5SaWVtYW5uflRlbnNvckc2Ig==NiNJNm5vbi16ZXJvfmNvbXBvbmVudHN+Okc2Ig==NiNJJU5vbmVHNiI=NiI=NiNJPWNoYXJhY3Rlcn46flstMSx+LTEsfi0xLH4tMV1HNiI=

Therfor the meaning of the equivalence principle has to be restricted to a heuristic device (see the first part in Introduction to Relativistic Astrophysics and Cosmology Through Maple , published by V.L. Kalashnikov).

Now let us choose, as the time coordinate, the proper time of the accelerated system and make some transformations (remember : NiMvLSUidkc2IyIiIUYn , NiMvLSUkYWJzRzYjJSJhR0Yn )subs({v(0)=0,abs(a)=a,_C=0},tau(T)) = c*arcsinh(a*T/c)/a; tau = c*arcsinh(a*T/c)/a;

NiMvKihJImNHNiIiIiJJImFHRiYhIiItSSNsbkc2JEkqcHJvdGVjdGVkR0YtSShfc3lzbGliR0YmNiMsJiooRihGJ0kiVEdGJkYnRiVGKUYnKiQsJiooRigiIiNGJSEiI0YyRjZGJ0YnRicjRidGNkYnRicqKEYlRictSShhcmNzaW5oR0YsNiNGMUYnRihGKQ==NiMvSSR0YXVHNiIqKEkiY0dGJSIiIi1JKGFyY3NpbmhHNiRJKnByb3RlY3RlZEdGLEkoX3N5c2xpYkdGJTYjKihJImFHRiVGKEkiVEdGJUYoRichIiJGKEYwRjI=

tau:='tau';

NiM+SSR0YXVHNiJGJA==

sinh(a*tau/c) = sinh( arcsinh(a*T/c));

NiMvLUklc2luaEc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiMqKEkiYUdGKSIiIkkkdGF1R0YpRi1JImNHRikhIiIqKEYsRi1JIlRHRilGLUYvRjA=

d(lhs(%)) = d(rhs(%)); s2 := solve(%%,T); # this is T s3 := solve(%%,d(T)); # this is d(T)

NiMvKiotSSVjb3NoRzYkSSpwcm90ZWN0ZWRHRihJKF9zeXNsaWJHNiI2IyooSSJhR0YqIiIiSSR0YXVHRipGLkkiY0dGKiEiIkYuRi1GLkYwRjEtSSJkR0YqNiNGL0YuKihGLUYuRjBGMS1GMzYjSSJUR0YqRi4=NiM+SSNzMkc2IiooLUklc2luaEc2JEkqcHJvdGVjdGVkR0YqSShfc3lzbGliR0YlNiMqKEkiYUdGJSIiIkkkdGF1R0YlRi9JImNHRiUhIiJGL0YxRi9GLkYyNiM+SSNzM0c2IiomLUklY29zaEc2JEkqcHJvdGVjdGVkR0YqSShfc3lzbGliR0YlNiMqKEkiYUdGJSIiIkkkdGF1R0YlRi9JImNHRiUhIiJGLy1JImRHRiU2I0YwRi8=

We can now express NiMtJSJkRzYjJSJYRw== through the coordinates in the accelerated system by taking the law of motion for the point of origin of the accelerated coordinate system from above:X-c^2*(sqrt(1+a^2*T^2/(c^2)))/a;

NiMsJkkiWEc2IiIiIiooSSJjR0YlIiIjLCYqKEkiYUdGJUYpRighIiNJIlRHRiVGKUYmRiZGJiNGJkYpRiwhIiJGMA==

d(x) = d(subs(T=s2,%));

NiMvLUkiZEc2IjYjSSJ4R0YmLCYtRiU2I0kiWEdGJiIiIiosSSJjR0YmRi0sJiokLUklc2luaEc2JEkqcHJvdGVjdGVkR0Y1SShfc3lzbGliR0YmNiMqKEkiYUdGJkYtSSR0YXVHRiZGLUYvISIiIiIjRi1GLUYtI0Y7RjxGMkYtLUklY29zaEdGNEY3Ri0tRiU2I0Y6Ri1GOw==

s4:=solve(%,d(X));

NiM+SSNzNEc2IiomLCYqJi1JImRHRiU2I0kieEdGJSIiIiwmKiQtSSVzaW5oRzYkSSpwcm90ZWN0ZWRHRjNJKF9zeXNsaWJHRiU2IyooSSJhR0YlRi1JJHRhdUdGJUYtSSJjR0YlISIiIiIjRi1GLUYtI0YtRjtGLSoqRjlGLUYwRi0tSSVjb3NoR0YyRjVGLS1GKjYjRjhGLUYtRi1GLiNGOkY7

Thus the space-time interval in the accelerated coordinates becomes:d(s)^2 = c^2*s3^2 - s4^2 - d(y)^2 - d(z)^2:simplify(%,trig,radical): collect(%,{d(tau)^2,d(x)^2,d(x),d(y)^2,d(z)^2});

NiMvKiQtSSJkRzYiNiNJInNHRiciIiMsLComSSJjR0YnRiotRiY2I0kkdGF1R0YnRioiIiIqJC1GJjYjSSJ4R0YnRiohIiIqLEYzRjEtSSVjc2duRzYkSSpwcm90ZWN0ZWRHRjtJKF9zeXNsaWJHRic2Iy1JJWNvc2hHRjo2IyooSSJhR0YnRjFGMEYxRi1GNkYxRi1GMS1JJXNpbmhHRjpGQEYxRi5GMSEiIyokLUYmNiNJInlHRidGKkY2KiQtRiY2I0kiekdGJ0YqRjY=

or NiMvKiQtJSJkRzYjJSJzRyIiIywsKiYlImNHRiktRiY2IyUkdGF1R0YpIiIiKiQtRiY2IyUieEdGKSEiIiosRilGMEYsRjBGMkYwLSUlc2luaEc2IyooJSJhR0YwRi9GMEYsRjVGMEYtRjBGNSokLUYmNiMlInlHRilGNSokLUYmNiMlInpHRilGNQ== The metric tensors here have attached to NiMqJiUiY0ciIiIqJCktJSJkRzYjJSR0YXVHIiIjRiVGJQ==, NiMqJi0lImRHNiMlInhHIiIiLUYlNiMlJHRhdUdGKA== , NiMqJC0lImRHNiMlInhHIiIj, NiMqJC0lImRHNiMlInlHIiIj, NiMqJC0lImRHNiMlInpHIiIj , respectively :coord := [tau, x, y, z]:g_compts := array(symmetric,sparse,1..4,1..4): g_compts[1,1] := 1: g_compts[1,2] := -sinh(a*tau/c): g_compts[2,2] := -1: g_compts[3,3] := -1: g_compts[4,4] := -1: g2 := create([-1,-1], eval(g_compts)); # covariant metric tensor g2_inv := invert( g2, 'detg' ): # contravariant metric tensor

NiM+SSNnMkc2Ii1JJlRBQkxFR0kqcHJvdGVjdGVkR0YoNiM3JC9JK2luZGV4X2NoYXJHRiU3JCEiIkYuL0knY29tcHRzR0YlLUknbWF0cml4R0YlNiM3JjcmIiIiLCQtSSVzaW5oRzYkRihJKF9zeXNsaWJHRiU2IyooSSJhR0YlRjZJJHRhdUdGJUY2SSJjR0YlRi5GLiIiIUZBNyZGN0YuRkFGQTcmRkFGQUYuRkE3JkZBRkFGQUYu

The Christoffel symbols areD1g := d1metric( g2, coord ): Cf1_2 := Christoffel1 ( D1g ): Cf2_2 := Christoffel2( g2_inv, Cf1_2 ): displayGR(Christoffel2,%);

NiI=NiNJS1RoZX5DaHJpc3RvZmZlbH5TeW1ib2xzfm9mfnRoZX5TZWNvbmR+S2luZEc2Ig==NiNJNm5vbi16ZXJvfmNvbXBvbmVudHN+Okc2Ig==NiMvSSh+fGZyMSwxMXxockc2IioqLUklc2luaEc2JEkqcHJvdGVjdGVkR0YqSShfc3lzbGliR0YlNiMqKEkiYUdGJSIiIkkkdGF1R0YlRi9JImNHRiUhIiJGLy1JJWNvc2hHRilGLEYyRi5GL0YxRjI=NiMvSSh+fGZyMiwxMXxockc2IiooLUklY29zaEc2JEkqcHJvdGVjdGVkR0YqSShfc3lzbGliR0YlNiMqKEkiYUdGJSIiIkkkdGF1R0YlRi9JImNHRiUhIiJGMkYuRi9GMUYy

in a flat space-time:D2g := d2metric( D1g, coord ): RMN := Riemann( g2_inv, D2g, Cf1_2 ): displayGR(Riemann,%);

NiI=NiNJM1RoZX5SaWVtYW5uflRlbnNvckc2Ig==NiNJNm5vbi16ZXJvfmNvbXBvbmVudHN+Okc2Ig==NiNJJU5vbmVHNiI=NiI=NiNJPWNoYXJhY3Rlcn46flstMSx+LTEsfi0xLH4tMV1HNiI=

The expressions obtained above are sufficient for the consideration of the so-called "Clock Paradox".As it is known, this paradox concerns the difference between the clock reading (NiMlIlRH) of the resting and (NiMvJSJ0RyUkdGF1Rw==) of the moving observer, due to the slowing-down of the time in the theory of special relativity : NiMvJSJURyomJSdsYW1iZGFHIiIiJSJ0R0Yn . The inertial motion is relative, and we cannot decide whose clock "lags". The decision will result from the comparison of the clocks after the return of the "traveler".But this return breaks the inertial character of the motion. And there is an opinion, that accelerations demand the use of the theory of general relativity for the description of this situation. That this is not necessary, will be shown in the following.

Let a traveller move back and forth, in both directions first accelerated during NiQlI1QxRyUjdDFH, then with constant velocity during NiQlI1QyRyUjdDJH, then decelerated during NiMlI1QxRw== , NiMlI3QxRw==. The way back and forth obviously takes NiMvJSJURywmKiYiIiUiIiIlI1QxR0YoRigqJiIiI0YoJSNUMkdGKEYo and NiMvJSR0YXVHLCYqJiIiJSIiIiUjdDFHRihGKComIiIjRiglI3QyR0YoRig=in the resting (always inertial) and the moving (traveler's) coordinate systems, respectively.

Let us consider the travel from the point of view of the observer in the inertial system. From the definition of the proper time NiMlJHRhdUc= (see above), we get on the traveller's clock :tau1 := c*arcsinh(a*T1/c)/a;# accelerated motion

NiM+SSV0YXUxRzYiKihJImNHRiUiIiItSShhcmNzaW5oRzYkSSpwcm90ZWN0ZWRHRixJKF9zeXNsaWJHRiU2IyooSSJhR0YlRihJI1QxR0YlRihGJyEiIkYoRjBGMg==

w:=T1->a*T1/sqrt(1+a^2*T1^2/c^2):# this is the constant velocity (after T1) of the inertial motion tau2 := simplify( T2*sqrt(1-w(T1)^2/c^2) );# the interval of inertial motion

NiM+SSV0YXUyRzYiKiZJI1QyR0YlIiIiKiZJImNHRiUiIiMsJiokRipGK0YoKiZJImFHRiVGK0kjVDFHRiVGK0YoISIiI0YoRis=

tau_proper := 4*tau1 + 2*tau2;# taking into account both, forward and reverse motion

NiM+SSt0YXVfcHJvcGVyRzYiLCYqKEkiY0dGJSIiIi1JKGFyY3NpbmhHNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJTYjKihJImFHRiVGKUkjVDFHRiVGKUYoISIiRilGMUYzIiIlKiZJI1QyR0YlRikqJkYoIiIjLCYqJEYoRjhGKSomRjFGOEYyRjhGKUYzI0YpRjhGOA==

tau_diff := tau_proper - 4*T1 - 2*T2;# time difference between the clocks of moving and resting observer

NiM+SSl0YXVfZGlmZkc2IiwqKihJImNHRiUiIiItSShhcmNzaW5oRzYkSSpwcm90ZWN0ZWRHRi1JKF9zeXNsaWJHRiU2IyooSSJhR0YlRilJI1QxR0YlRilGKCEiIkYpRjFGMyIiJSomSSNUMkdGJUYpKiZGKCIiIywmKiRGKEY4RikqJkYxRjhGMkY4RilGMyNGKUY4RjhGMiEiJUY2ISIj

This is the time lag of the moving clock as noticed by the observer in the resting system, a negative value. E.g. :plot3d(subs({c=1,T1=10*T2},tau_diff),a=0..1,T2=0..10,axes=boxed, title=`time difference`);

-%'PLOT3DG6&-%%GRIDG6%;$""!F*$"""F*;F)$"#5F*7;7;I*undefinedGI*protectedGF3F2F2F2F2F2F2F2F2F2F2F2F2F2F2F2F2F2F2F2F2F2F2F2F27;F)$!(g!*[*!")$!)(*y!G(F7$!*()=]I#F7$!*g.O/&F7$!*eV)***)F7$!+a/j79F7$!*/nC.#!"($!*&z*zu#FD$!*!>*ya$FD$!*p'z@WFD$!*_.1O&FD$!*WYkN'FD$!*c!f-uFD$!*Z&G$\)FD$!*Y=Oi*FD$!+,#Q*y5FD$!+Kfp)>"FD$!+IGK@8FD$!+b"[lW"FD$!+_g8u:FD$!+3,)Qq"FD$!+l))fN=FD$!+6@8p>FD$!+3yL/@FD7;F)$!)\RSOF7$!*"=!=_#F7$!*F_J1(F7$!+v*)*RP"F7$!+U$)*3@#F7$!+=KAyJF7$!+LFkYUF7$!+.5p%R&F7$!+^Th1mF7$!+f-oqyF7$!*K%*z<*FD$!+.*o@0"FD$!+Vrj*="FD$!+pRxH8FD$!+^8Bs9FD$!+;Ts;;FD$!+,f,j<FD$!+c(34">FD$!+a`Bg?FD$!+UI&3@#FD$!+l#REO#FD$!+))z[:DFD$!+$)oIpEFD$!+t],CGFD7;F)$!)jR$o(F7$!*<o(3ZF7$!+I1j#="F7$!+![:)=@F7$!+=G(y?$F7$!+C%4WS%F7$!+f.gzcF7$!+e$fW,(F7$!+c$pfR)F7$!+rc([")*F7$!+n)Qk7"FD$!+q"RRF"FD$!+JRfB9FD$!+WG4v:FD$!+Vw=G<FD$!+9nn#)=FD$!+jTRQ?FD$!+"*=?&>#FD$!+VP)HN#FD$!+96k6DFD$!+%f*3rEFD$!+wjDJGFD$!+O#y?*HFD$!+?**\`JFD7;F)$!*"4!4E"F7$!*([**poF7$!+5;6*e"F7$!++bM(p#F7$!+H,MNRF7$!+7X%3E&F7$!+X)p)[mF7$!+$e?O3)F7$!+#yVXb*F7$!+@lU06FD$!+%*Rud7FD$!+Nv+79FD$!+-8"zc"FD$!+uf@D<FD$!+O3t$)=FD$!+4:IV?FD$!+#H,Q?#FD$!+R\7lBFD$!+%3%=FDFD$!+FQ!**o#FD$!+G,A`GFD$!+;y2<IFD$!+>!H9=$FD$!+U>BYLFD7;F)$!*ro**z"F7$!*O$fV))F7$!+"pBZ#>F7$!+,@F[JF7$!+U!z@[%F7$!+)RD*))eF7$!+tJ4ZtF7$!+t@TV))F7$!+'e7p."FD$!+Jl"=>"FD$!+!o:'[8FD$!+oY)p]"FD$!+I]nm;FD$!+1G\F=FD$!+;PG*)>FD$!+iI#>:#FD$!+B'3`J#FD$!+wbNzCFD$!+eF*Rk#FD$!+%*)f"4GFD$!+![0[(HFD$!+l^)39$FD$!+&[gtI$FD$!+py>uMFD7;F)$!*4%QaBF7$!+SxSf5F7$!+9Z?-AF7$!+w'Hs]$F7$!+NyV2\F7$!+\eppjF7$!+?UYvyF7$!+vNQ8%*F7$!+X4g(4"FD$!+d0#eD"FD$!+)=GcT"FD$!+f*\nd"FD$!+uq(*Q<FD$!+_n9->FD$!+Z'Hh1#FD$!+%H@3B#FD$!+?h8'R#FD$!+]J+iDFD$!+rGOGFFD$!+!*[;&*GFD$!+lhOiIFD$!+P'H*HKFD$!+zJ#yR$FD$!+a(=gc$FD7;F)$!*MCN!HF7$!+%\EL@"F7$!+PW6MCF7$!+%>a$*z$F7$!+#pBzC&F7$!+%*yR]nF7$!+cW^!H)F7$!+*)pPe)*F7$!+$o_Z9"FD$!+/?N08FD$!+0qJn9FD$!+-:TI;FD$!+<cX%z"FD$!+F#4$f>FD$!+e5'[7#FD$!+%G@5H#FD$!+`jrdCFD$!+$G&)[i#FD$!+^pZ#z#FD$!+y![/'HFD$!+2<wGJFD$!+DgQ(H$FD$!+_MHmMFD$!+E*faj$FD7;F)$!*W(*\V$F7$!+\Fn[8F7$!+cAUIEF7$!+*G5=/%F7$!+0E8FbF7$!+tw.gqF7$!+m)zgi)F7$!+b2l@5FD$!+qCc#="FD$!+9>&\M"FD$!+3*Q&3:FD$!+qf6t;FD$!+#>D&Q=FD$!+BOk/?FD$!+oOPr@FD$!+nljQBFD$!+[yO1DFD$!+kS^uEFD$!+@/.VGFD$!+.!z=,$FD$!+Mu-"=$FD$!+]yW]LFD$!+(4;,_$FD$!+-6,!p$FD7;F)$!*6-@%RF7$!+xk8o9F7$!+]kl)z#F7$!+,RYYUF7$!+uaigdF7$!+I'RtJ(F7$!+!)>j.*)F7$!+Sm;^5FD$!+I_j87FD$!+k(>uP"FD$!+k.FU:FD$!++@+3<FD$!+#HuW(=FD$!+VudT?FD$!+gYA4AFD$!+NeMxBFD$!+*e$)ea#FD$!+j.z9FFD$!++j-%)GFD$!+owb`IFD$!+pcNBKFD$!+8bR$R$FD$!+=dljNFD$!+Vv6MPFD7;F)$!*o'z@WF7$!+^g8u:F7$!++FYWHF7$!+$31<U%F7$!+bE3ffF7$!+NL#\`(F7$!+GSYP"*F7$!+J:'f2"FD$!+)yx'R7FD$!+Z*zXS"FD$!+#eU/d"FD$!+M*)4P<FD$!+&=AW!>FD$!+bOJs?FD$!+Z]pSAFD$!+.K]4CFD$!+LkoyDFD$!+->?[FFD$!+9P,=HFD$!+#\"4)3$FD$!+-$4%eKFD$!+C[%*GMFD$!+9(y'*f$FD$!+(4%fqPFD7;F)$!*$fFt[F7$!+Qiso;F7$!+**f5sIF7$!+AXhtXF7$!+s72IhF7$!+$H3;s(F7$!+b=ZP$*F7$!+1#>r4"FD$!+qv&=E"FD$!+v]nF9FD$!+CkO%f"FD$!+-)z<w"FD$!+"y*zH>FD$!+hlL)4#FD$!+.*=tE#FD$!+Y$*oOCFD$!+p3S1EFD$!+5XTwFFD$!+=wpYHFD$!+mDA<JFD$!+od'zG$FD$!+Bp!*eMFD$!+U%G+j$FD$!+!)\J,QFD7;F)$!*rQqH&F7$!+Q[h`<F7$!+Dz%[=$F7$!+$y"p1ZF7$!+$y-"ziF7$!+"z\P)yF7$!+iPt5&*F7$!+Z1T:6FD$!+v:+"G"FD$!+XCeZ9FD$!+=['\h"FD$!+w$4Iy"FD$!+;-h^>FD$!+c\o?@FD$!+_#o,H#FD$!+,u+gCFD$!+!Gf,j#FD$!+c")e+GFD$!+!4k7(HFD$!+Y<;UJFD$!+(\fKJ$FD$!+J(QX[$FD$!+>L)fl$FD$!+!>zv#QFD7;F)$!*W+Vp&F7$!+))4@I=F7$!+in@&G$F7$!+oCMC[F7$!+x&)G5kF7$!+L![g-)F7$!+FLXi'*F7$!+=,SJ6FD$!+sHr(H"FD$!+V,%\Y"FD$!+2v!Hj"FD$!+`j[,=FD$!+b)y0(>FD$!+C*3,9#FD$!+5i,5BFD$!+U?D![#FD$!+-mx]EFD$!+"*ob@GFD$!+m_c#*HFD$!+N$yP;$FD$!+Gh<NLFD$!+g9u1NFD$!+U%f%yOFD$!+#3<.&QFD7;F)$!*ZKm1'F7$!+(4x'**=F7$!+Z*)>vLF7$!+#\)=H\F7$!+;+wElF7$!+.v0_")F7$!+Oha'z*F7$!+U1^X6FD$!+UEW78FD$!+RSA!["FD$!+7Ip[;FD$!+i*Hx"=FD$!++SC()>FD$!+RW;d@FD$!+;_VFBFD$!+A6,)\#FD$!+q^&)oEFD$!+6n$*RGFD$!+r+B6IFD$!+@Nr#=$FD$!+=&oVN$FD$!++"zh_$FD$!+Q98)p$FD$!+oM@qQFD7;F)$!*jXdT'F7$!+L^(H'>F7$!+$GvjX$F7$!+DAGB]F7$!+<d%4j'F7$!+z&=XE)F7$!+p#=g"**F7$!+cf1e6FD$!+'zMbK"FD$!+)p'z$\"FD$!+G0qi;FD$!+,Y8K=FD$!+[S,-?FD$!+oGFs@FD$!+?(eGM#FD$!+(RHP^#FD$!+o.&[o#FD$!+BI>cGFD$!+#RLx-$FD$!+&>^%*>$FD$!+>"H8P$FD$!+lANVNFD$!+Ex]:PFD$!+RUy()QFD7;F)$!*vjLu'F7$!+W^!4-#F7$!+P)=+`$F7$!+pPD3^F7$!+m&fZs'F7$!+X)zbO)F7$!+7=K-5FD$!+%G=$p6FD$!+KqDP8FD$!+,&Rf]"FD$!+DRAv;FD$!+]b+X=FD$!+%y/_,#FD$!+K(fd=#FD$!+58icBFD$!+S*\x_#FD$!+VK6*p#FD$!+#R%oqGFD$!+74WUIFD$!+'yjV@$FD$!+%yOkQ$FD$!+OfkeNFD$!+P"z4t$FD$!+JeU.RFD7;F)$!*a_60(F7$!+_j8u?F7$!+oh;(f$F7$!+,[T&=&F7$!+V7t4oF7$!+/5&pX)F7$!+_3+75FD$!+BsYz6FD$!+C2#yM"FD$!+sV(o^"FD$!+!o%\'o"FD$!+@Iec=FD$!+gW1F?FD$!+G2)y>#FD$!+>c)*oBFD$!+w=MSDFD$!+=!>>r#FD$!+%z"p$)GFD$!+Z!Rc0$FD$!+zGuFKFD$!+4!))**R$FD$!+"HhBd$FD$!+I8&[u$FD$!+*=[u"RFD7;F)$!*Q#oStF7$!+]>BB@F7$!+;)p'eOF7$!+&>LeD&F7$!+>))4()oF7$!+$\5+a)F7$!+A()y?5FD$!+=Hn)="FD$!+#=&Rd8FD$!+M)yn_"FD$!+cxp'p"FD$!+q(eq'=FD$!+)3"zP?FD$!+l'R)3AFD$!+`3;!Q#FD$!+g%><b#FD$!+%p'[BFFD$!+0(Qa*GFD$!+3bbnIFD$!+)>?)RKFD$!+U$=AT$FD$!+Avt%e$FD$!+xpOdPFD$!+At4IRFD7;F)$!+P8T8w!"*$!+#zo'o@F7$!+([P_r$F7$!+WvQ?`F7$!+yc(y&pF7$!+WF)eh)F7$!+)H2)G5FD$!+Ta1(>"FD$!+@x6m8FD$!+#p'zN:FD$!+KG)fq"FD$!+Yyew=FD$!+cYaZ?FD$!+06!)=AFD$!+OeJ!R#FD$!+0b0iDFD$!+KG*Rt#FD$!+=_51HFD$!+JPPyIFD$!+_By]KFD$!+4uJBMFD$!+Mr'ff$FD$!+M8soPFD$!+;6dTRFD7;F)$!+`-oqyFecn$!+TI&3@#F7$!+o;YnPF7$!+^w!)z`F7$!+L(**G-(F7$!+kY\&o)F7$!+Go:O5FD$!+,;v/7FD$!+^45u8FD$!+Yd/W:FD$!+6CZ9<FD$!+ZqH&)=FD$!+<fXc?FD$!+m%**yA#FD$!+(R)e*R#FD$!+I5\rDFD$!+Z9eVFFD$!+y"Qe"HFD$!+NKC)3$FD$!+*R"ygKFD$!+!oRMV$FD$!++p?1OFD$!+sL2zPFD$!+]1._RFD7;F)$!+!z9P6)Fecn$!+JE8]AF7$!+tA%e"QF7$!+.]qMaF7$!+)=qG3(F7$!+0wh\()F7$!+p0#H/"FD$!+v*>=@"FD$!+W"Q9Q"FD$!+xMi^:FD$!+#ynAs"FD$!+T6H$*=FD$!+SGjk?FD$!+zcCOAFD$!+GA43CFD$!+7B9!e#FD$!+X7P_FFD$!+y&eZ#HFD$!+zrG(4$FD$!+^D%*pKFD$!+DBrUMFD$!+^ee:OFD$!+2Rb)y$FD$!+X%3;'RFD7;F)$!+&>JOM)Fecn$!+gs!oG#F7$!+ZT!3'QF7$!+Dgf&[&F7$!+s`PQrF7$!+.!**)3))F7$!+"Go"\5FD$!+tYM=7FD$!+bs?)Q"FD$!+$G6'e:FD$!+hMXH<FD$!+*[d1!>FD$!+3a;s?FD$!+b@$RC#FD$!+**>#fT#FD$!+Nh5)e#FD$!+(*4YgFFD$!+jq'H$HFD$!+zzg0JFD$!+3*p$yKFD$!+S5C^MFD$!+67@COFD$!+;;F(z$FD$!+uXTqRFD7;F)$!+aRWh&)Fecn$!+=k8@BF7$!+S2r-RF7$!+I+#H`&F7$!+&49**=(F7$!+Zl)Q'))F7$!+x)e\0"FD$!+3#)QC7FD$!+xRZ%R"FD$!+,x2l:FD$!+f05O<FD$!+d&pu!>FD$!+\#H"z?FD$!+Gl.^AFD$!+7s:BCFD$!+[PY&f#FD$!+"fLzw#FD$!+p![0%HFD$!+O:H8JFD$!+e2:'G$FD$!+PW6fMFD$!+^G<KOFD$!+wvJ0QFD$!+n7ayRFD7;F)$!+#>u!o()Fecn$!+"*eM`BF7$!+'*[(=%RF7$!+JK0xbF7$!+BA"zB(F7$!+wo/:*)F7$!+yCMg5FD$!++P+I7FD$!+ySH+9FD$!+t33r:FD$!+l$pAu"FD$!+%f*y8>FD$!+=%)e&3#FD$!+BXidAFD$!+\^')HCFD$!+()QG-EFD$!+L"f[x#FD$!+!*HdZHFD$!+m/T?JFD$!+u)eLH$FD$!+*Q2kY$FD$!+#pY&ROFD$!+8(oF"QFD$!+5k1')RFD-%&TITLEG6#Q0time~difference6"-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%Q"aF]foQ#T2F]foQ!F]fo

Now let us consider this situation from the point of view of the traveller. His clock rests in the origin of his coordinate system. And now the first clock moves in a space-time with the metric g1 on a geodesic line. The law of this motion is given by the following equations:Cf2_1:=subs({tau=t},Christoffel2( g1_inv, Cf1_1 )):subs({v[0]=0,abs(a)=a,_C=0},geodesic_eqns( coord, s, Cf2_1 ));

NiM8Ji8tSSVkaWZmR0kqcHJvdGVjdGVkR0YnNiQtSSJ5RzYiNiNJInNHRistSSIkR0YnNiRGLSIiIyIiIS8sJi1GJjYkLUkieEdGK0YsRi4iIiIqKkkiYUdGK0Y5SSJjR0YrISIiKiYsJiomRjtGMUkidEdGK0YxRjkqJEY8RjFGOUY5RjwhIiMjISIkRjEtRiY2JC1JJHRhdUdGK0YsRi1GMUY5RjIvLUYmNiRGSEYuRjIvLUYmNiQtSSJ6R0YrRixGLkYy

So NiMvLCYtJSVkaWZmRzYkJiUidUc2IyUieEclInNHIiIiKiolImFHRi0mRik2I0YtRi1GMEYtKiYlImNHIiIjLSUlc3FydEc2IyokLCZGLUYtKihGL0Y0JSJ0R0Y0KiRGM0Y0ISIiRi0iIiRGLUY9Ri0iIiE= . Here the additional c in the denominator results again from NiMvJiUieEc2IyIiIiomJSJjR0YnJSJ0R0Yn and NiMmJSJ1RzYjJSJpRw== , i = 1,2,3,4 is the 4-velocity. As NiMvJiUidUc2IyUieEcqKiZGJTYjIiIiRislImNHISIiJSNkeEdGKy0lImRHNiMlInRHRi0= and NiMvKiYlImRHIiIiJSNkc0chIiIqKiYlInVHNiNGJkYmJSJjR0YoRiVGJi1GJTYjJSJ0R0Yo , we have for the velocity of motion :diff(u(t),t) + a/(1+a^2*t^2/c^2)^(3/2) = 0; dsolve({%,u(0)=0},u(t)): simplify(%,radical,symbolic);

NiMvLCYtSSVkaWZmR0kqcHJvdGVjdGVkR0YnNiQtSSJ1RzYiNiNJInRHRitGLSIiIiomSSJhR0YrRi4sJiooRjAiIiNGLUYzSSJjR0YrISIjRi5GLkYuIyEiJEYzRi4iIiE=

NiMvLUkidUc2IjYjSSJ0R0YmLCQqKkYoIiIiSSJjR0YmRissJiomSSJhR0YmIiIjRihGMEYrKiRGLEYwRisjISIiRjBGL0YrRjM=

normal(%,'expanded');

NiMvLUkidUc2IjYjSSJ0R0YmLCQqKkYoIiIiSSJjR0YmRissJiomSSJhR0YmIiIjRihGMEYrKiRGLEYwRisjISIiRjBGL0YrRjM=

and that is, of course the relativistic velocity -w(t) :-subs(T1=t,w(T1));

NiMsJCooSSJhRzYiIiIiSSJ0R0YmRicsJiooRiUiIiNGKEYrSSJjR0YmISIjRidGJ0YnIyEiIkYrRi8=

From the point of view of the observer, who is resting in the moving system, the resting system is moving in the negative x-direction ! So we getdiff(x(t),t)=%; dsolve({%,x(0)=0},x(t)): sol := simplify(%);

NiMvLUklZGlmZkdJKnByb3RlY3RlZEdGJjYkLUkieEc2IjYjSSJ0R0YqRiwsJCooSSJhR0YqIiIiRixGMCwmKihGLyIiI0YsRjNJImNHRiohIiNGMEYwRjAjISIiRjNGNw==

NiM+SSRzb2xHNiIvLUkieEdGJTYjSSJ0R0YlKigsKComSSJhR0YlIiIjRipGLyEiIiokSSJjR0YlRi9GMComRjJGLyomLCZGLSIiIkYxRjZGNkYyISIjI0Y2Ri9GNkY2Ri5GMEY0I0YwRi8=

The proper time of the first (inertial) observer from the metric here is NiMvLSUiZEc2IyUkdGF1RyomJSNkc0ciIiIlImNHISIi = NiMqJiUjZHRHIiIiLSUlc3FydEc2IywoJiUiZ0c2JEYlRiVGJSoqIiIjRiUmRis2JEYlRi5GJSUjZHhHRiUqJiUiY0dGJUYkRiUhIiJGJSomKiYlI2RzR0YlRiRGNEYuKiRGM0YuRjRGNEYld(t)*sqrt(get_compts(g1)[1,1]+2*get_compts(g1)[1,2]*diff(rhs(sol),t)/c-diff(rhs(sol),t)^2/c^2):d(tau)=simplify(%);

NiMvLUkiZEc2IjYjSSR0YXVHRiYtRiU2I0kidEdGJg==

Hence NiMvJSV0YXUxRyUjdDFH and the proper time of the second clock, resting relative to the accelerated system, NiMvLSUiZEc2IyUjVDFHKiYlI2R0RyIiIi0lJXNxcnRHNiMmJSJnRzYkRipGKkYq :Int(sqrt(get_compts(g1)[1,1]),t=0..t1); value(%): simplify(%,radical,symbolic);

NiMtSSRJbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkKiQqJCwmKihJImFHRigiIiNJInRHRihGL0kiY0dGKCEiIyIiIkYzRjMhIiIjRjNGLy9GMDsiIiFJI3QxR0Yo

NiMsJCoqSSJjRzYiIiIiLCYtSSNsbkc2JEkqcHJvdGVjdGVkR0YsSShfc3lzbGliR0YmNiMqJEYlIiIjRictRio2IyomLCYqJkkiYUdGJkYnSSN0MUdGJkYnRicqJiwmKiZGNkYwRjdGMEYnRi9GJyNGJ0YwLUklY3NnbkdGKzYjRjZGJ0YnRidGPEYnISIjRidGPEYnRjYhIiIjRkBGMA==

2*ln(a)-2*ln(c)+2*ln(t1*a+(c^2+t1^2*a^2)^(1/2))-2*ln(a);

NiMsJi1JI2xuRzYkSSpwcm90ZWN0ZWRHRidJKF9zeXNsaWJHNiI2I0kiY0dGKSEiIy1GJTYjLCYqJkkiYUdGKSIiIkkjdDFHRilGMkYyKiQsJiomRjEiIiNGM0Y3RjIqJEYrRjdGMiNGMkY3RjJGNw==

and because of c/a*ln(a*t1/c+sqrt(1+a^2*t1^2/(c^2)))=c*arcsinh(a*t1/c)/a;

NiMvKihJImNHNiIiIiJJImFHRiYhIiItSSNsbkc2JEkqcHJvdGVjdGVkR0YtSShfc3lzbGliR0YmNiMsJiooRihGJ0kjdDFHRiZGJ0YlRilGJyokLCZGJ0YnKihGKCIiI0YyRjZGJSEiI0YnI0YnRjZGJ0YnKihGJUYnLUkoYXJjc2luaEdGLDYjRjFGJ0YoRik=

we get T1:=rhs(%);

NiM+SSNUMUc2IiooSSJjR0YlIiIiLUkoYXJjc2luaEc2JEkqcHJvdGVjdGVkR0YsSShfc3lzbGliR0YlNiMqKEkiYUdGJUYoSSN0MUdGJUYoRichIiJGKEYwRjI=

After acceleration we have the section of inertial motion with the law of motion x = -w(t1)*t;

NiMvSSJ4RzYiLCQqKkkiYUdGJSIiIkkjdDFHRiVGKSwmRilGKSooRigiIiNGKkYtSSJjR0YlISIjRikjISIiRi1JInRHRiVGKUYx

But here one cannot use the Lorentz-transformations, because they correspond to the Galilean (diagonal) metric, and the transformation of g1 into such a metric, obviously is discontinuous !

Taking the following non-diagonal metric d(s)^2 = c^2*d(t)^2*(1-v^2/(c^2))-2*v*d(x)*d(t)-d(x)^2-d(y)^2-d(z)^2;

NiMvKiQtSSJkRzYiNiNJInNHRiciIiMsLCooSSJjR0YnRiotRiY2I0kidEdGJ0YqLCYiIiJGMiomSSJ2R0YnRipGLSEiIyEiIkYyRjIqKEY0RjItRiY2I0kieEdGJ0YyRi5GMkY1KiRGOEYqRjYqJC1GJjYjSSJ5R0YnRipGNiokLUYmNiNJInpHRidGKkY2

coord := [t, x, y, z]: g_compts := array(symmetric,sparse,1..4,1..4): g_compts[1,1] := 1-v^2/c^2: g_compts[1,2] := -v/c:g_compts[2,2] := -1: g_compts[3,3] := -1:g_compts[4,4] := -1:g3 := create([-1,-1], eval(g_compts)); g3_inv := invert( g3, 'detg' ):

NiM+SSNnM0c2Ii1JJlRBQkxFR0kqcHJvdGVjdGVkR0YoNiM3JC9JK2luZGV4X2NoYXJHRiU3JCEiIkYuL0knY29tcHRzR0YlLUknbWF0cml4RzYkRihJKF9zeXNsaWJHRiU2IzcmNyYsJiIiIkY5KiZJInZHRiUiIiNJImNHRiUhIiNGLiwkKiZGO0Y5Rj1GLkYuIiIhRkE3JkY/Ri5GQUZBNyZGQUZBRi5GQTcmRkFGQUZBRi4=

which is inertial, because:D1g := d1metric( g3, coord ): Cf1_3 := Christoffel1 ( D1g ): Cf2_3 := Christoffel2( g3_inv, Cf1_3 ): displayGR(Christoffel2,%);

NiI=NiNJS1RoZX5DaHJpc3RvZmZlbH5TeW1ib2xzfm9mfnRoZX5TZWNvbmR+S2luZEc2Ig==NiNJNm5vbi16ZXJvfmNvbXBvbmVudHN+Okc2Ig==NiNJJU5vbmVHNiI=

one sees that this metric, for t = t1 transforms continuously into g1.So in this metric the geodesic equations for the motion of the first clock are :geodesic_eqns( coord, s, Cf2_3 );

NiM8Ji8tSSVkaWZmR0kqcHJvdGVjdGVkR0YnNiQtSSJ6RzYiNiNJInNHRistSSIkR0YnNiRGLSIiIyIiIS8tRiY2JC1JInRHRitGLEYuRjIvLUYmNiQtSSJ4R0YrRixGLkYyLy1GJjYkLUkieUdGK0YsRi5GMg==

and because NiMvLS0lIkRHNiMlInhHNiMiIiEsJCUidkchIiI=d(t)*sqrt(get_compts(g3)[1,1]+2*get_compts(g3)[1,2]*(-v)/c-(-v)^2/c^2):d(tau) = simplify(%);

NiMvLUkiZEc2IjYjSSR0YXVHRiYtRiU2I0kidEdGJg==

This means NiMvJSV0YXUyRyUjdDJH , and the proper time of the second clock NiMvJSRkVDJHKiYlI2R0RyIiIi0lJXNxcnRHNiMmJSJnRzYkRidGJ0Yn :t2 := 't2': Int(sqrt(get_compts(g3)[1,1]),t=0..t2); value(%): T2:=simplify(subs(v=a*t1/sqrt(1+a^2*t1^2/c^2),%),radical,symbolic);

NiMtSSRJbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkKiQsJiIiIkYsKiZJInZHRigiIiNJImNHRighIiMhIiIjRixGLy9JInRHRig7IiIhSSN0MkdGKA==NiM+SSNUMkc2IiooSSJjR0YlIiIiLCYqJkkiYUdGJSIiI0kjdDFHRiVGLEYoKiRGJ0YsRigjISIiRixJI3QyR0YlRig=

Then the time difference between the second and the first clock equals:t_diff := 4*T1 + 2*T2 - 4*t1 - 2*t2;

NiM+SSd0X2RpZmZHNiIsKiooSSJjR0YlIiIiLUkoYXJjc2luaEc2JEkqcHJvdGVjdGVkR0YtSShfc3lzbGliR0YlNiMqKEkiYUdGJUYpSSN0MUdGJUYpRighIiJGKUYxRjMiIiUqKEYoRiksJiomRjEiIiNGMkY4RikqJEYoRjhGKSNGM0Y4SSN0MkdGJUYpRjhGMiEiJUY7ISIj

This expression, compared to the one obtained above for tau_diff shows that the time difference between the two clocks is an invariant ! Hence there is no paradox , q.e.d.

In order to emphasize the above fact, one can consider the limit for NiRmKjYjJSN0MUc3IjYkJSlvcGVyYXRvckclJmFycm93RzYiIiIhRipGKkYqZio2IyUiYUdGJkYnRiolKWluZmluaXR5R0YqRipGKg== so that v is the finite value (which means instant acceleration) :v^2 = x/(1+x/c^2); # x = NiMqJiUiYUciIiMlI3QxR0Yl sol := solve(%,x);

NiMvKiRJInZHNiIiIiMqJkkieEdGJiIiIiwmRipGKiomRilGKkkiY0dGJiEiI0YqISIiNiM+SSRzb2xHNiIqKEkidkdGJSIiI0kiY0dGJUYoLCYqJEYpRigiIiIqJEYnRighIiJGLg==

and the time difference becomes4*c*ln(sqrt(sol)/c+sqrt(1+sol/(c^2)))/a+2*t2/sqrt(1+sol/(c^2))-2*t2: simplify(%,radical,symbolic):expand(%): simplify(limit(%,a= infinity));

NiMsJCooSSN0Mkc2IiIiIiwmKiQsJiokSSJjR0YmIiIjRicqJEkidkdGJkYtISIiI0YnRi1GMEYsRidGJ0YsRjAhIiM=

that is NiMqKCIiIyIiIiUjdDJHRiUsJiomRiVGJSUnbGFtYmRhRyEiIkYlRiVGKkYl : This expression corresponds to the one, which can be obtained from the theory of special relativity in the case of inertially moving clocks. But in our case, the observers are not the same and it is the expression obtained which is the invariant !

A purely geometrical interpretation of the acceleration as the non-zero Christoffel symbols in a flat pseudo-Euclidian space-time, without restriction to the Galilean (diagonal) metric, allows the description of the accelerated motion within the framework of the theory of special relativity. The opinion, that accelerations demand the use of the theory of general relativity for the description of the situation, is not correct.

Classic Worksheet, by means of MAPLE 9.01, by Dr. Friedrich Futschik, Eindhoven, The Netherlands, Mai 2005Errors in this worksheet should not be attributed to prof. dr. Kalashnikow.