   #copyright

Introduction to special relativity

2007 Schools Wikipedia Selection. Related subjects: General Physics

The beginning

   Everything started with the Michelson-Morley experiment to determine
   the absolute speed of the Earth through the theorised "luminiferous
   aether." The experiment was a failure but it did show something: the
   speed of light is independent of the speed of the observer. This means
   that two persons that follow the same light ray at different speeds
   will measure the same relative speed for the ray.

   There were several articles before Einstein's trying to explain how
   this could happen. The most important was written by Lorentz, who
   postulated that light travels at a speed of c through the ether, and
   that when other objects move through the ether, they are compressed in
   the direction of motion, and their time slows down. He still had in
   mind absolute movement, and was thinking that a moving observer would
   measure the correct speed of light by a compensation of errors. If her
   reference frame is moving along with the light, then the speed of light
   relative to her is slower, but the contraction of the length of her
   instruments and the slowdown of her time are exactly enough for her to
   measure the speed of light as c.

   Einstein explained the same effect in a simpler way using the
   relativity principle. He imagined a ray of light bouncing between two
   mirrors. For an observer static with the mirrors this takes a given
   time. For an observer moving with respect to them this takes longer
   (the speed of light is the same and the space is more). From here he
   calculated the equation for transforming time.

   Later, using the previous equation, he discovered that simultaneity was
   not possible for all inertial systems. If we synchronize several clocks
   in a frame, they will look desynchronized from a frame at different
   speed.

   Finally, he used the previous results to obtain the same equation as
   Lorentz had for space contraction.

   After he did this, a lot of people, including himself, tried to find
   any inconsistencies in the system. A lot of supposed inconsistencies
   (or paradoxes) were proposed. He took the job of looking for an
   explanation for all of them one by one, until another scientist, named
   Hermann Minkowski, showed that the whole system of the special
   relativity is self-consistent and therefore no real paradox can appear.

Minkowski Space

   Also, the modern approach to the theory depends upon the concept of a
   four-dimensional universe that was first proposed by Minkowski in 1908,
   and further developed as a result of the contributions of Emmy Noether.
   This approach uses the concept of invariance to explore the types of
   coordinate systems that are required to provide a full physical
   description of the location and extent of things.

   The modern theory of special relativity begins with the concept of "
   length". In everyday experience, it seems that the length of objects
   remains the same no matter how they are rotated or moved from place to
   place. We think that the simple length of a thing is " invariant".
   However, as is shown in the illustrations below, what we are actually
   suggesting is that length seems to be invariant in a three-dimensional
   coordinate system.

   image:Coord1b.GIF

   The length of a thing in a two-dimensional coordinate system is given
   by Pythagoras' theorem:

          h^2 = x^2 + y^2 \,

   This two-dimensional length is not invariant if the thing is tilted out
   of the two-dimensional plane. In everyday life, a three-dimensional
   coordinate system seems to describe the length fully. The length is
   given by the three-dimensional version of Pythagoras's theorem:

          k^2 = x^2 + y^2 + z^2 \,

   The derivation of this formula is shown in the illustration below.

   image:Coord2.GIF

   It seems that, provided all the directions in which a thing can be
   tilted or arranged are represented within a coordinate system, the
   coordinate system can fully represent the length of a thing. However,
   it is clear that things may also be changed over a period of time. We
   must think of time as another dimension through which we move forward.
   This is shown in the following diagram:

   image:Coord3.GIF

   The path taken by a thing in both space and time is known as the
   space-time interval.

   Minkowski realised in 1908 that if things could be rearranged in time,
   then the universe might be four-dimensional. He boldly suggested that
   Einstein's recently published theory of special relativity was a
   consequence of this four-dimensional universe. He proposed that the
   space-time interval might be related to space and time by Pythagoras'
   theorem in four dimensions:

          s^2 = x^2 + y^2 + z^2 + (ict)^2 \,

   Where i is the imaginary unit (defined as one of the solutions of
   x^2 = −1), c is the speed of light (a constant), and t is the time
   interval spanned by the space-time interval, s. In this equation, the
   "second", usually a unit of time, becomes just another unit of length.
   In the same way as centimetres and inches are both units of length
   related by centimetres = conversion factor × inches, metres and seconds
   are related by metres = conversion factor × seconds. The conversion
   factor, c, has a value of approximately 300,000,000 meters per second.
   Because i^2 is equal to −1, the space-time interval can be represented
   by:

          s^2 = x^2 + y^2 + z^2 - (ct)^2 \,

   In modern formulations this formula for intervals is known as a metric
   tensor on the Minkowski space with the signature (−+++) or equivalently
   (+−−−). It is of particular importance that this form where one term
   has a sign different from the rest is in direct contrast to the
   above-mentioned example of euclidean space where all terms have the
   same sign, a signature of (++++). It is this difference in metric
   signature that causes the principal difference between geometry in a
   four-dimensional Euclidean space and geometry in the four-dimensional
   Minkowski space. The space-time interval is still given by:

          s^2 = x^2 + y^2 + z^2 - (ct)^2 \,

   or alternately

          s^2 = (ct)^2 - x^2 - y^2 - z^2 \,

   Space-time intervals are difficult to imagine; they extend between one
   place and time and another place and time, so the velocity of the thing
   that travels along the interval is already determined for a given
   observer. But attempts at visualisation can be made by graphing the
   displacement of a particle in one-dimensional motion (constrained to a
   straight line) against time. This can be imagined as a world line when
   the two other space dimensions are suppressed.

   If the universe is four-dimensional, then the space-time interval will
   be invariant, rather than spatial length. Whoever measures a particular
   space-time interval will get the same value, no matter how fast they
   are travelling. The invariance of the space-time interval has some
   dramatic consequences.

   The first consequence is the prediction that if a thing is travelling
   at a velocity of c metres per second, then all observers, no matter how
   fast they are travelling, will measure the same velocity for the thing.
   The velocity c will be a universal constant. This is explained below.

   When an object is travelling at c, the space time interval is zero:

          The space-time interval is s^2 = x^2 + y^2 + z^2 - (ct)^2 \,

          The distance travelled by an object moving at velocity v for t
          seconds is:

          x = vt \,

          So: s^2 = (vt)^2 - (ct)^2 \,

          But when the velocity v equals c:

          s^2 = (ct)^2 - (ct)^2 \,

          And hence the space time interval s^2 = 0 \,

   A space-time interval of zero only occurs when the velocity is c. This
   particular value of zero for the interval is unique in the sense that
   it is the only value of the space-time interval for which v can have
   one and only one value. This shows that not only is the velocity of
   light constant, it is the only velocity constant. Therefore, by
   assuming the particular form of the Minowski metric and postulating the
   invariance of space-time interval, we have an alternate approach to
   Einstein's special relativity where Einstein takes it as a postulate
   that the speed of light is constant. When observers observe something
   with a space-time interval of zero, they all observe it to have a
   velocity of c, no matter how fast they are moving themselves.

   The universal constant, c, is known for historical reasons as the
   "speed of light". In the first decade or two after the formulation of
   Minkowski's approach many physicists, although supporting special
   relativity, expected that light might not travel at exactly c, but
   might travel at very nearly c. There are now few physicists who believe
   that light (or any other EM wave) does not propagate at c.

   The second consequence of the invariance of the space-time interval is
   that clocks will appear to go slower on objects that are moving
   relative to you. Suppose there are two people, Bill and John, on
   separate planets that are moving away from each other. John draws a
   graph of Bill's motion through space and time. This is shown in the
   illustration below:

   image:Coord15.GIF

Clocks delays and rods contractions

   Being on planets, both Bill and John think they are stationary, and
   just moving through time. John spots that Bill is moving through what
   John calls space, as well as time, when Bill thinks he is moving
   through time alone. Bill would also draw the same conclusion about
   John's motion. To John, it is as if Bill's time axis is leaning over in
   the direction of travel and to Bill, it is as if John's time axis leans
   over.

   John calculates the length of Bill's space-time interval as:

          s^2 = (vt)^2 - (ct)^2 \,

   whereas Bill doesn't think he has traveled in space, so writes:

          s^2 = (0)^2 - (cT)^2 \,

   The space-time interval, s^2, is invariant. It has the same value for
   all observers, no matter who measures it or how they are moving in a
   straight line. Bill's s^2 equals John's s^2 so:

          (0)^2 - (cT)^2 = (vt)^2 - (ct)^2 \,

   and

          -(cT)^2 = (vt)^2 - (ct)^2 \,

   hence

          t = \frac{T}{\sqrt{1 - \frac{v^2}{c^2}}} \, .

   So, if John sees Bill measure a time interval of 1 second (T = 1)
   between two ticks of a clock that is at rest in Bill's frame (modeled
   by the condition (X = 0), John will find that his own clock measures
   between these same ticks an interval t, called coordinate time, which
   is greater than one second. It is said that clocks in motion slow down,
   relative to those on observers at rest. This is known as "relativistic
   time dilation of a moving clock". The time that is measured in the rest
   frame of the clock (in Bill's frame) is called the proper time of the
   clock.

   John will also observe measuring rods at rest on Bill's planet to be
   shorter than his own measuring rods, in the direction of motion. This
   is a prediction known as "relativistic length contraction of a moving
   rod". If the length of a rod at rest on Bill's planet is X, then we
   call this quantity the proper length of the rod. The length x of that
   same rod as measured on John's planet, is called coordinate length, and
   given by

          x = X \sqrt{1 - \frac{v^2}{c^2}} \, .

Clocks desynchronization

   The last consequence is that clocks will appear to be out of phase with
   each other along the length of a moving object. This means that if one
   observer sets up a line of clocks that are all synchronised so they all
   read the same time, then another observer who is moving along the line
   at high speed will see the clocks all reading different times. This
   means that observers who are moving relative to each other see
   different events as simultaneous. This effect is known as "Relativistic
   Phase" or the "Relativity of Simultaneity". Relativistic phase is often
   overlooked by students of special relativity, but if it is understood,
   then phenomena such as the twin paradox are easier to understand.

   The net effect of the four-dimensional universe is that observers who
   are in motion relative to you seem to have time coordinates that lean
   over in the direction of motion, and consider things to be simultaneous
   that are not simultaneous for you. Spatial lengths in the direction of
   travel are shortened, because they tip upwards and downwards, relative
   to the time axis in the direction of travel, akin to a rotation out of
   three-dimensional space.

   image:Coord6.GIF

   Great care is needed when interpreting space-time diagrams. Diagrams
   present data in two dimensions, and cannot show faithfully how, for
   instance, a zero length space-time interval appears.

   image:Coord8.GIF

Common misconceptions

   It is a common misconception that special relativity only applies to
   objects that are moving quickly. This is entirely untrue. In the main
   page, it is shown that the kinetic energy of an object at all speeds is
   a relativistic quantity. Kinetic energy is relativistic because,
   although relativistic changes in mass are tiny in ordinary units, these
   result in large changes in energy in ordinary units, due to E = mc^2
   where c^2 is about 90,000,000,000,000,000 m^2/s^2. Newtonian physics
   describes the interplay between kinetic and potential energy, without
   explaining the origin of kinetic energy or inertia; it just assumes
   these things, whereas special relativity explains them at a deeper
   level.

   It should be pointed out however, that Minkowski was partly motivated
   in his proposal of a four-dimensional universe by Franz Taurinus'
   discovery that Euclidean geometry, with the exception of the fifth
   postulate, applies on the surface of a sphere, with a radius measured
   in imaginary units (i), and cannot be assumed as a fundamental geometry
   in any universe (see Non-Euclidean geometry and Walter(1999)).

Caveats and Warnings

   The discussion given above has been confined to what is known as "flat
   space-time". The general, differential form of the space-time interval
   is given in the article Special relativity. The modern description of
   the universe uses the term (3+1)D rather than 4D to show how time is
   not like the spatial dimensions. This corresponds to the difference of
   the 4D Euclidean metric and the (3+1)D Minowski metric mentioned
   earlier.

   The time dilation equation

          t = T / \sqrt{1 - v^2/c^2} .

   is valid for a clock that is at rest in Bill's frame, measuring proper
   time T. This equation is only valid when the two tick events of the
   clock happen at the same place in Bill's frame; in other words, for two
   events satisfying X = 0.

   The length contraction equation

          x = X \sqrt{1 - v^2/c^2} .

   is valid for a rod that is at rest in Bill's frame, with proper length
   X. If the rod is to be measured in John's frame, then John must make
   sure to measure both end points of the rod at the same time in his own
   frame, so he must use two events satisfying t = 0.

   It can be tempting to combine the above equations for time dilation and
   length contraction to some kind of 'invariant spacetime area' xt = XT
   or some kind of 'velocity that transforms like' x/t = X/T (1-v^2/c^2),
   but taking into account the above, and combining them with the Lorentz
   transformation, the equations immediately reduce to the trivial case
   where all the quantities are zero: x = t = X = T = 0, in which case the
   'area equation' reduces to 0 = 0 and the quantities x/t and X/T are not
   defined. In a sense the combined equations would only be valid for a
   rod with zero length that is also a clock that does not tick.

   It should also be made clear that the length contraction result only
   applies to rods aligned in the direction of motion. At right angles to
   the direction of motion, there is no contraction.

   It might also be noted that special relativity, in the form given
   above, is only a stepping stone to general relativity. In special
   relativity, space-time is absolute, and appears as a background against
   which events occur; whereas in general relativity, space-time is
   construed to be a dynamic product of the gravitational field, and the
   Minkowski metric no longer applies in general. Instead, every situation
   demands its own metric which are solutions to the famous Einstein field
   equations.

Measurement Units, and Simplifications to the Lorentz Equations

   The equations above do not specify what the units of measurement are
   for times and distances. The following discussion, however, will only
   focus on the section on the Lorentz equations. Suppose we choose some
   arbitrary time unit, and some arbitrary distance unit. Then the
   equations above are still valid, provided the constant c\, is the speed
   of light measured with respect to these units. For example, if the unit
   of time is the second, and the unit of distance is the meter, then the
   value of c\, is exactly 299,792,458 meters per second.

          Side note: The 299,792,458 figure is exact, for the following
          reason. In the past, the second was defined as a subdivision of
          a day (the time between successive solar noons). The meter was
          defined independently of time, as a fraction of the distance
          from the equator to the north pole, and was standardized as the
          length of ruler kept in Paris. Yet with the advent of atomic
          clocks, and the discovery of the invariance of the speed of
          light, the definition of the second and meter were altered. The
          second now is defined with respect to the vibration-periods of
          certain atoms in certain energy states. The meter now is defined
          as 1/299,792,458 th of the distance that a beam of light travels
          in one second. So, rather than estimate the speed of light as a
          function of the length of a physical object, which is imprecise,
          we now use the absolute constant, the speed of light, and the
          definition of the second (presumably based on a highly invariant
          physical entity) to define what we mean by a "meter."

   Now, suppose rather than use meters to measure distance, we let our
   unit of measurement be the distance light travels in one second, which
   is called a "light-second." In terms of meters, one light second is
   exactly defined as 299,792,458 meters. One beneficial result of using
   these units of measurement is that the value of c\, is now 1 in the
   equations above ( since light travels exactly ONE light-second, every
   second ). So all the c\, 's can be removed. The only thing to remember
   is that now all distances should be in terms of light-seconds, not
   meters. Note, also, that there is nothing special about using seconds
   as the unit of time, as long as the unit of distance is the distance
   traveled by a beam of light in one unit of time. For example, if time
   is measured in years, then distance should be measured in light-years.

   The simplified equations are:

          t' = \gamma \left(t - v x \right)
          x' = \gamma (x - v t)\,
          y' = y\,
          z' = z\,

   where \gamma = \frac{1}{\sqrt{1 - v^2}}

   In their Spacetime Physics Taylor and Wheeler present another way to
   look at and think about the variables and equations when they are
   written this way (i.e. with "units where c = 1"). The variable c\, is
   kept as the normal speed of light, but the v-variable is now defined as

          v = v_{conv}/c\,

   where v_{conv}\, is the conventional velocity that appears in the
   original non-simplified equations. Then, depending on the problem or
   situation at hand one can choose to
     * either measure and express time in units of conventional distance
       (e.g. meters), and define the t-variable as

                t = c\;t_{conv}\,

     * or measure and express distances in units of conventional time
       (e.g. seconds or years), and define the spatial variables x, y, z
       as

                x = x_{conv}/c \qquad y = y_{conv}/c \qquad z = z_{conv}/c
                \,

   Similar definitions can then be used for the other variables like
   energy:

          E = E_{conv}/c^2\,

   etc... This way the speed of light effectively disappears from all the
   equations.

   Retrieved from "
   http://en.wikipedia.org/wiki/Introduction_to_special_relativity"
   This reference article is mainly selected from the English Wikipedia
   with only minor checks and changes (see www.wikipedia.org for details
   of authors and sources) and is available under the GNU Free
   Documentation License. See also our Disclaimer.
