How Lorentz transformation is connected to reality and physics?

Question

I don't understand whats the meaning of Lorentz transformation in Special relativity.

In physics, the Lorentz transformations (or transformation) are coordinate transformations between two coordinate frames that move at constant velocity relative to each other

Here is what wiki says. Ok so it's about geometry right? Transformation of coordinates, so its some kind of projection. Ok I know what projections are, they are math staff that allows you to imagine (think) of something from different point of view. Cool I know what is orthographic projection, its coordinate transformation....

And then I read Consequences derived from the Lorentz transformation All that staff about time dilation, length contraction, barn-pole paradox.

Wow, wow, wow, hold on here are physics trying to say projections some how effect reality (how world undergo changes)? If I draw something on a paper in proper scale using orthographic projection, then measure it and it will tell me some bizarre thing, like Earth is flat. Does this mean the Earth is really flat?

Answer 1

Suppose you move a distance $x$ East then a distance $y$ North. How far have you moved?

Well, we all learned this at school, and the answer is that on a flat plane the answer is given by Pythagoras' theorem:

$$ s^2 = x^2 + y^2 $$

And this extends to three dimensions, so if we move a distance $z$ up we get:

$$ s^2 = x^2 + y^2 + z^2 $$

This quantity $s$ is just the magnitude of the vector $(x, y, z)$. But now suppose I'm using a different coordinate system to you. For example my $x$, $y$ and $z$ axes may be displaced or rotated relative to yours. In that case when I write down the vector for your movement in my coordinates $(x', y', z')$ it will look different to your representation - my $x'$ won't equal your $x$ and likewise for $y$ and $z$. But when I calculate the magnitude of the vector $(x', y', z')$ in my coordinates I'm going to get the same length $s$ that you get when you calculate the length in your coordinates. This has to be the case, because the vector hasn't changed. Changing the coordinates doesn't change the length of the vector. So, and this is the key bit:

the magnitude $s$ is an invariant of the geometry and all observers will calculate the same value for it

This might seem obvious, but it isn't. The equation for $s$ is called the metric and it defines the geometry of the surface. We generally write it in differential form. If the metric is:

$$ ds^2 = dx^2 + dy^2 + dz^2 $$

then this defines the flat Euclidean 3D space. Compare this to the metric for the surface of a sphere (polar coordinates this time):

$$ ds^2 = r^2d\theta^2 + r^2 \sin^2\theta d\phi^2 $$

This metric defines the geometry of a spherical surface.

By now you're probably wondering what this is all about. Well in special relativity we deal with a flat 4D spacetime $(t, x, y, z)$. So what is the metric for this geometry. The obvious guess would be to simply extend Pythagoras' theorem yo 4D:

$$ ds^2 = dt^2 + dx^2 + dy^2 + dz^2 $$

but this would be wrong. Experiment tells us that this isn't how the universe works. What we find is that the metric is actually:

$$ ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 $$

This is called the Minkowski metric. Note two things:

1. the $dt$ term has a minus sign

2. we multiply $dt$ by a constant with the dimensions of a velocity, $c$, to convert it to a length

The minus sign for $dt$ is the key difference, and all the weird stuff like time dilation and length contraction originate from this minus sign. For example see How do I derive the Lorentz contraction from the invariant interval?

I must emphasise that we didn't choose the minus sign for the $dt$ term in the metric. That's the way the geometry of the universe is and we were forced to use a minus sign to correctly describe physics.

Anyhow, you started out asking about the Lorentz transformations, and the Lorentz transformations are simply the linear transformations that keep the value of $ds$ constant. So I don't think the best way to understand it is to start with the Lorentz transformations. Given that the geometry of 4D spacetime is described by the Minkowski metric the Lorentz transformations follow automatically.

Answer 2

You are right that generally, coordinates have no physical meaning; they're akin to different projections on a map. In general relativity, you're allowed to use arbitrary coordinate systems. However, the physical predictions don't depend on the coordinate system because there's a geometric object, the metric, which stays the same; different coordinate systems will agree on lengths measured. To give an everyday example, you can still figure out that the Earth is curved given a world map, no matter what the projection is, as long as it has a distance scale at every point.

The situation in special relativity is different: we restrict to coordinate systems with a definite physical meaning. The coordinates $(t, x, y, z)$ are constructed by imagining a grid of rulers and clocks I've set up across space, so that the clocks are synchronized and the ruler are identical and perpendicular. Somebody else's coordinates $(t', x', y', z')$ are constructed the same way: the rulers have to be the orthogonal and the clocks have to be synchronized, but in their frame. Differences between $\Delta t$ and $\Delta t'$ really do correspond directly to differences in a physically measured time.

To link this to the first paragraph, the allowed transformations in special relativity are exactly those which keep the form of the metric simple; they are called Lorentz transformations. The 'general' in general relativity refers to expanding the allowed transformations to general coordinate transformations. Lorentz transformations are very special; they can be parametrized by just six numbers, while a general coordinate transformation needs infinitely many.

Answer 3

Imagine, you have a Rubik's cube and look at it from the direction of the white side. What you see is the white side and all other sides are not visible. Then you slowly rotate the cube and all of a sudden you see that the white side gets smaller while the blue side gets visible and it gets larger as you keep rotating the cube.

What has happened? You applied a coordinate transformation (a rotation) to you system and the projections (white side oriented in xy plane, blue side oriented in yz plane) changed! This is exactly what happens in special relativity if you apply a Lorentz transformation, the only difference being that Lorentz transformations are pseudorotations (hyperbolic rotations) rather than just rotations.

Answer 4

To use your terminology, indeed the Lorentz transformation allows to see things from different points of view, in that case from observers with different relative speeds.

It appears that space and time are dependent on that sort of point of view. The thing that is being projected is spacetime, a new entity encompassing as an integrated whole what we usually see as qualitatively different things, space and time.

What is "relative" in special relativity is the articulation between space and time. There is no absolute, canonical, ever-valid way of splitting spacetime into space and time. But all observers see such a split, so we cannot do without a reference to a specific point of view.

The Lorentz transformation allows us to understand how all observers can be right while they are not seing the same thing. It makes it possible to describe physics from any reference frame in a seamless manner.