Concept Summary

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Topic | Discussion | Learning Aids
Reference frame | 28.1 EVENTS AND INERTIAL REFERENCE FRAMES An event is a physical “happening” that occurs at a certain place and time. To record the event an observer uses a reference frame that consists of a coordinate system and a clock. Different observers may use different reference frames. |
Inertial reference frame | The theory of special relativity deals with inertial reference frames. An inertial reference frame is one in which Newton’s law of inertia is valid. Accelerating reference frames are not inertial reference frames. |
Relativity postulate Speed of light postulate | 28.2 THE POSTULATES OF SPECIAL RELATIVITY The theory of special relativity is based on two postulates. The relativity postulate states that the laws of physics are the same in every inertial reference frame. The speed of light postulate says that the speed of light in a vacuum, measured in any inertial reference frame, always has the same value of c, no matter how fast the source of the light and the observer are moving relative to each other. |
Proper time interval | 28.3 THE RELATIVITY OF TIME: TIME DILATION The proper time interval between two events is the time interval measured by an observer who is at rest relative to the events and views them occurring at the same place. An observer who is in motion with respect to the events and |
Dilated time interval | who views them as occurring at different places measures a dilated time interval The dilated time interval is greater than the proper time interval, according to the time-dilation equation: |
Time-dilation equation |

| (28.1)

| In this expression, v is the relative speed between the observer who measures and the observer who measures |
Proper length | 28.4 THE RELATIVITY OF LENGTH: LENGTH CONTRACTION The proper length L0 between two points is the length measured by an observer who is at rest relative to the points. An observer moving with a relative speed v parallel to the line between the two points does not measure the proper length. |
Contracted length | Instead, such an observer measures a contracted length L given by the length-contraction formula: |
Length-contraction formula |

| (28.2)

| Length contraction occurs only along the direction of the motion. Those dimensions that are perpendicular to the motion are not shortened. |
| The observer who measures the proper length may not be the observer who measures the proper time interval. |

Use Self-Assessment Test 28.1 to evaluate your understanding of Sections 28.1, 28.2, 28.3 and 28.4 .

| 28.5 RELATIVISTIC MOMENTUM An object of mass m, moving with speed v, has a relativistic momentum whose magnitude p is given by |
Relativistic momentum |

| (28.3)

| 28.6 THE EQUIVALENCE OF MASS AND ENERGY Energy and mass are equivalent. The total energy E of an object of mass m, moving at speed v, is |
Total energy |

| (28.4)

| The rest energy E0 is the total energy of an object at rest (v = 0 m/s): |
Rest energy |

| (28.5)

| An object’s total energy is the sum of its rest energy and its kinetic energy KE, or . Therefore, the kinetic energy is |
Kinetic energy |

| (28.6)

| The relativistic total energy and momentum are related according to |
Relation between total energy and momentum |

| (28.7)

The ultimate speed | Objects with mass cannot attain the speed of light, which is the ultimate speed for such objects. |
| 28.7 THE RELATIVISTIC ADDITION OF VELOCITIES According to special relativity, the velocity addition formula specifies how the relative velocities of moving objects are related. For objects that move along the same straight line, this formula is |
Velocity-addition formula |

| (28.8)

| where vAB is the velocity of object A relative to object B, vAC is the velocity of object A relative to object C, and vCB is the velocity of object C relative to object B. The velocities can have positive or negative values, depending on whether they are directed along the positive or negative direction. Furthermore, switching the order of the subscripts changes the sign of the velocity, so that, for example, |
| Use Self-Assessment Test 28.2 to evaluate your understanding of Sections 28.5, 28.6 and 28.7 . |

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