general relativity

Examples of general relativity in the following topics:

• The Relativistic Universe

  • Special relativity indicates that humans live in a four-dimensional space-time where the 'distance' $s$ between points in space-time can be regarded as:
  • In 1916, Einstein found the importance of these space-times in his theory of general relativity.
  • General relativity, or the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1916.
  • General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or space-time.

• Simultaneity

  • The relativity of simultaneity is the concept that simultaneity is not absolute, but depends on the observer's reference frame.
  • The question of whether the events are simultaneous is relative: in some reference frames the two accidents may happen at the same time, in other frames (in a different state of motion relative to the events) the crash in London may occur first, and still in other frames, the New York crash may occur first.
  • If the two events are causally connected ("event A causes event B"), then the relativity of simultaneity preserves the causal order (i.e.
  • If we imagine one reference frame assigns precisely the same time to two events that are at different points in space, a reference frame that is moving relative to the first will generally assign different times to the two events.
  • Formulate conclusions of the theory of special relativity, noting the assumptions that were made in deriving it

• Four-Dimensional Space-Time

  • Let us examine two observers who are moving relative to one another at a constant velocity.
  • For general events we can define the quantity:
  • (Using the principles of relativity, you can prove this for general separations, not just light rays).
  • This phenomenon is known as the relativity of simultaneity and may be counterintuitive.
  • Finally, let's discuss an important result of special relativity -- that the energy $E$ of an object moving with speed $v$ is:

• Relativistic Energy and Mass

  • In special relativity, as the object approaches the speed of light, the object's energy and momentum increase without bound.
  • In special relativity, an object that has a mass cannot travel at the speed of light.
  • In order for these laws to hold in all reference frames, special relativity must be applied.
  • When the relative velocity is zero, is simply equal to 1, and the relativistic mass is reduced to the rest mass.
  • While Newton's second law remains valid in the form the derived form is not valid because in is generally not a constant.

• Reference Frames and Displacement

  • In order to describe an object's motion, you need to specify its position relative to a convenient reference frame.
  • More precisely, you need to specify its position relative to a convenient reference frame.
  • Mathematically, the position of an object is generally represented by the variable x.
  • Displacement is the change in position of an object relative to its reference frame.
  • His location relative to the airplane is given by x.

• Gauge Pressure and Atmospheric Pressure

  • Depending on the altitude relative to sea level, the actual atmospheric pressure will be less at higher altitudes and more at lower altitudes as the weight of air molecules in the immediate atmosphere changes, thus changing the effective atmospheric pressure.
  • Atmospheric pressure is a measure of absolute pressure and can be affected by the temperature and air composition of the atmosphere but can generally be accurately approximated to be around standard atmospheric pressure of 101,325 Pa.
  • In this equation p0 is the pressure at sea level (101,325 Pa), g is the acceleration due to gravity, M is the mass of a single molecule of air, R is the universal gas constant, T0 is the standard temperature at sea level, and h is the height relative to sea level.
  • Gauge pressure is a relative pressure measurement which measures pressure relative to atmospheric pressure and is defined as the absolute pressure minus the atmospheric pressure.
  • Pressure instruments connected to the system will indicate pressures relative to the current atmospheric pressure.

• General Case

  • The Doppler effect is the apparent change in frequency of a wave when the observer and the source of the wave move relative to each other.
  • The change in sound perception can be explained through relativity.
  • In classical physics, where the speeds of source and the receiver relative to the medium are lower than the velocity of waves in the medium, the relationship between observed frequency (f) and emitted frequency (fo) is given by:$f=\frac{(c+{v_r})}{(c+{v_s})}*f_0$where c - velocity of the sound waves in the medium, vr- velocity of the observer or reciever, vs - velocity of the sound source, and f0 - original frequency of the sound waves.
  • where $\Delta v$ is the velocity of the receiver relative to the source: it is positive when the source and the receiver are moving towards each other.

• Mass

  • Four of the seven base units in the SI system are defined relative to the kilogram, so the stability of this measurement is crucial for accurate and consistent measurements.
  • At its 2011 meeting, the General Conference on Weights and Measures (CGPM) agreed that the kilogram should be redefined in terms of the Planck constant.
  • A graph of the relative change in mass of selected kilogram prototypes.

• Examples

  • In other words any non-periodic function defined on a finite interval can be used to generate a periodic function just by cloning the function over and over again.
  • Figure~\ref{sawtooth} shows the periodic extension of the function $f(x) = x$ relative to the interval $[0,1]$.
  • What would the periodic extension of $f(x) = x$ look like relative to the interval $[-.5,.5]$?

• Maxwell's Predictions and Hertz' Confirmation

  • Maxwell's prediction of the electromagnetic force was confirmed by Hertz who generated and detected electromagnetic waves.
  • Magnetic fields are generated by moving charges or by changing electric fields .
  • Since changing electric fields create relatively weak magnetic fields, they could not be easily detected at the time of Maxwell's hypothesis.
  • The apparatus used by Hertz in 1887 to generate and detect electromagnetic waves.
  • An RLC circuit connected to the first loop caused sparks across a gap in the wire loop and generated electromagnetic waves.