I Time as a Dimension: Exploring Galilean and Lorentz Transformation

[Pushoam]

In Galilean transformation,it is said that time is a physical quantity while space is dimension. What do we mean by this i.e. time is a physical quantity while space is dimension ?
Isn't space a physical quantity in Galilean transformation?
Here, too, we plot graph of position(on y-axis) as a function of time (on x-axis).In Lorentz Transformation, it is said that both space and time are dimension.
I don't understand that what difference does it bring to our understanding of the concept "time"when we say time is dimension and when we say time is not dimension?
How does Lorentz Transformation lead us to the conclusion that time is dimension and to the concept of space - time?

Here, too, we plot graph of position(on x-axis) and time (on y-axis).

Do the two plots differ? If yes, in what way?

How is space- time different from space and time?

Does one need to understand these concepts well for solving numerical problems?

 

[Orodruin]

In Galilean transformation,it is said that time is a physical quantity while space is dimension.

Where did you read this?

 

[Pushoam]

I don't remember the source. I just have this impression. If it is wrong , please tell me so.

 

[Orodruin]

It is impossible to answer your question without further qualification to what you mean by "a physical quantity". Typically, I would take it to mean roughly "something that can be measured using some measurement standard". This does not discount neither space nor time. I would call both dimensions both classically and relativistically.

 

[pervect]

I'm not sure if this will answer the OP's question, but I thought it might be helpful. Before special relativity, time was regarded as being independent and separate from space.

Special relativity (henceforth SR) introduced the idea that space-time was somehow unified into a whole. The question naturally arises - what does this mean? What's the difference between space-time, and space plus time?

To attempt to answer this question, let's look at space. To make life simple, we'll look at a two dimensional space, a plane. Suppose we have this plane, why do we consider it as being a single two-dimensional space, rather than two separate one-dimensional, independent entities?

The basic reason for this is the existence of rotational symmetries. We usually view both north-south displacements (or vectors, if you prefer) and east-west displacements (or vectors) as being part of the same thing, space. Mathematically, we say that a plane is a two dimensional vector space.

The basic reason that we regard north-south and east-west as being the same is the existence of rotational symmetry. If we rotate our plane (or we make a map of the plane, and rotate the axis on the map), distances don't change at all, the rotated plane or the rotated map of the plane are equivalent. If we rotate our map 90 degrees, in fact north becomes east, and east becomes south.

The reason we regard space-time as being unified, is that a similar symmetry exists in special relativity, which is called the boost symmetry. If we have two observers, moving relative to each other, we can regard either observer as being stationary at will. The laws of physics don't change when we change our state of motion in SR, the speed of light (for instance) is constant for all observers regardless of their speed or state of motion.

For a more formal (but not too formal) treatment, look up "The Parable of the Surveyor" in Taylor & Wheeler's "Space-time physics". The first few chapters of this first edition of this classic text, including "The Parable of the Surveyor", can be found online on the author's website, <<here>>.

 

[Ibix]

With Galilean transforms it's easy to see time as a parameter, something fundamentally different from a direction. Rather like watching a DVD - there's a qualitative difference between directions in the screen and the DVD timer.

The Lorentz transforms make that model harder to believe because they are easiest to interpret as describing a four dimensional manifold with one direction being slightly different from the others. There's a meaningful notion of (hyperbolic) rotation of the time axis towards one of the spacelike axes, which is totally alien to the Galilean transforms.

Edit: pervect beat me to it, I see.