If you are curious of scientific progress, I’m sure that you have at least heard of the Lorentz Transformation. If you have not heard about it let alone see, today must be your lucky day. Here it is:
These equations look very trivial, but let its simplicity not delude you. It leads to some of the most profound truths ever realized by man like Time Dilation.
In this article, I will try to explain the meaning of these equations.
Imagine you are in a region of space such that there is no force acting upon you(perhaps at the edge of the cosmos) so that you are not accelerating.
A reference frame which is not undergoing acceleration is termed Inertial Reference Frame.
You have your accurate meter sticks and a very accurate wristwatch. Being someone who likes quantitative descriptions you define x, y and z orthogonal directions with yourself placed at the origin.
It so happens that your friend is also with you except that you perceive him to be moving. You measure his velocity to be a constant v using your meter sticks and wristwatch of course. By sheer coincidence he also has his accurate meter sticks and a very accurate wristwatch. By another stroke of luck he has defined his x, y and z orthogonal directions in the same directions you chose with him being at the origin of his coordinate axis. Let us call it x’, y’ and z’ to avoid confusion.
Fortunately he is headed towards you along the x-axis which you had previously defined. When he reaches you(both origins are at the same space point), both of you reset your wristwatches to read a time 00:00 to avoid confusions about time in the future.
In this diagram you are P while your friend is P’.
Some time after you had exchanged greetings at the origin and your friend kept moving in your +x direction , an event B takes place somewhere, say a bulb suddenly lights up. Being someone who tracks events with their location and time of occurrence, you add an entry in your log book as:
B happened at (x, y, z, t)
Here x, y, z and t stand for the values corresponding to event B.
Like you, your friend also adds an entry in his log book as:
B happened at (x’, y’, z’, t’)
Here x’, y’, z’ and t’ stand for the values corresponding to event B.
Intuition tells you that the following must be true:
The above relationships are termed as Galilean Transformations.
but is it really true?
In reality what you would observe is that the correct relationships are given by:
These relationships are termed as Lorentz Transformation.
It seems so counter intuitive from your everyday experience that t’ and t are not equal!
Let us explore why that might be the case.
Speed of light, c is close to 299 792 458 m/s.
In our everyday lives, we don’t experience speeds comparable to c. We usually deal with v << c. In the Lorentz Transformation, if we use a v<<c, these equations reduce to Galilean Transformation, which was our intuition of course!
v << c implies v/c~ 0
When dealing with multiple events where we are interested in the time interval and space intervals between these events, we can modify the Lorentz Transformation to :
Not surprisingly this holds,
This is because your friend perceives you as moving with velocity -v.
Since both of you are inertial frames, there really is nothing special about either one of you!
In short, there is no preferred reference frame in the universe.
Additional Notes:
- If event A is a cause of event B, then there does not exist a reference frame P where B will occur before A unless P moves faster than the speed of light. Thus the saying, faster than speed of light travel will break causality!
- The equations can be easily modified for v being in any direction not necessarily x axis. But, rotating the spatial axis so that v is along x is quite trivial. So Lorentz Transformation is usually defined in this way.
- Under Lorentz Transformation, speed of light will always be measured to be c in any frame of reference.
- If the origins don’t coincide at any point of time or if time is not synchronized when the origins coincide, a suitable offset can be added to the Lorentz Transformation equations.
- If you are interested to read about Time Dilation and Length Contraction, I have explained it in this article:
With that, it’s a wrap. Thanks for reading.
Credits
All the Images containing Math Expressions were created with the help of Mathcha Online.
