Difference between revisions of "Time dilation"

Latest revision as of 23:19, 30 January 2021

In special relativity, the time dilation that will be experienced can be expressed with the formula: $\dfrac{t_1}{\sqrt{1-v^2/c^2}}=t_2$

where $t_1$ is the "proper" time experienced by the moving object.

$v$ is the relative velocity the ovject is moving to the observer.

$c$ is the speed of light; it can be simply expressed as 1, but then the velocity will have to be given in terms of $c$ .

$t_2$ is the time the events appear to occur, with the time dilation added.

This formula can be derived using a simple example. If there was a light clock that consisted of two mirrors with a light beam bouncing between them, the distance between the mirrors would be $ct_1$ . If, the clock was inside a moving vehicle, than the distance the clock travels would equal $vt_2$ . Finally, to an observer of the light clock, the light beam, instead of bouncing up and down, would move diagonally. The distance it appears to travel is $ct_2$ . This forms a triangle, and using the Pythagorean theorem, it would obtain:

$c^2(t_2)^2=c^2(t_1)^2+v^2(t_2)^2$

Subtracting by $v^2(t_2)^2$ yields

$c^2(t_2)^2-v^2(t_2)^2=c^2(t_1)^2$

Dividing by $c^2$ yields

$(t_2)^2[1-(v^2/c^2)]=(t_1)^2$

Dividing by $[1-(v^2/c^2)]$ yields

$(t_2)^2=\dfrac{(t_1)^2}{1-v^2/c^2}$

Finding the root of both sides finally yields

$\dfrac{t_1}{\sqrt{1-v^2/c^2}}=t_2$

@@ Line 1: / Line 1: @@
 In special relativity, the time dilation that will be experienced can be expressed with the formula:
-<math>\dfrac{t_1}{\sqrt{1-c^2/v^2}}</math>
+<math>\dfrac{t_1}{\sqrt{1-v^2/c^2}}=t_2</math>
+where
+<math>t_1</math> is the "proper" time experienced by the moving object.
+<math>v</math> is the relative velocity the ovject is moving to the observer.
+<math>c</math> is the speed of light; it can be simply expressed as 1, but then the velocity will have to be given in terms of <math>c</math>.
+<math>t_2</math> is the time the events appear to occur, with the time dilation added.
+This formula can be derived using a simple example. If there was a light clock that consisted of two mirrors with a light beam bouncing between them, the distance between the mirrors would be <math>ct_1</math>. If, the clock was inside a moving vehicle, than the distance the clock travels would equal <math>vt_2</math>. Finally, to an observer of the light clock, the light beam, instead of bouncing up and down, would move diagonally. The distance it appears to travel is <math>ct_2</math>. This forms a triangle, and using the Pythagorean theorem, it would obtain:
+<math>c^2(t_2)^2=c^2(t_1)^2+v^2(t_2)^2</math>
+Subtracting by <math>v^2(t_2)^2</math> yields
+<math>c^2(t_2)^2-v^2(t_2)^2=c^2(t_1)^2</math>
+Dividing by <math>c^2</math> yields
+<math>(t_2)^2[1-(v^2/c^2)]=(t_1)^2</math>
+Dividing by <math>[1-(v^2/c^2)]</math> yields
+<math>(t_2)^2=\dfrac{(t_1)^2}{1-v^2/c^2}</math>
+Finding the root of both sides finally yields
+<math>\dfrac{t_1}{\sqrt{1-v^2/c^2}}=t_2</math>

Page

Toolbox

Search

Difference between revisions of "Time dilation"

Latest revision as of 23:19, 30 January 2021