Space Time Explained: Physics & E=mc^2

[nst.john]

I hear the term 'space time' thrown around a lot, and I have a basic, and I mean BASIC, understanding of the concept, can someone explain to me space time, its importance to the study of physics and how it relates to E=mc^2.

 

[WannabeNewton]

A space-time is a 4-dimensional Lorentzian manifold i.e. a set of events that are given a differentiable structure and a Lorentzian metric tensor. See here for more: http://en.wikipedia.org/wiki/Space-time#Mathematics_of_spacetimes

 

[DrewD]

If you are just learning about special relativity, you don't need to worry about manifolds and other mathematical objects that are important to the study of the subject at some level, but impede your understanding of the basics. At a very basic level the term is used since time and space should not be considered separately and time behaves similarly to the normal cartesian coordinates (the signs are opposite, but you'll see this eventually). Normally you would consider a point around you to be describable by three coordinates like $(x,y,z)$. In relativity, events are the points and they require a fourth coordinate and may be written as $(t,x,y,z)$. It should be clear why this is called space-time. This is a sufficient understanding IMO to learn the basics of special relativity. Early on the connection between relativity and geometry was recognized and the connection has been fruitful since.

Getting a feeling for the geometry is a good idea, but basic ideas like $E=mc^2$ don't require much geometric thought. This can be derived from the Lorentz transformations (which are derivable from the postulates of special relativity) and some "simple" thought experiments. A deep understanding of metrics and manifolds is unnecessary.

 

[robphy]

Before stepping off into special relativity and four-dimensional spaces...
it might be good to first recognize that the (x-position)-vs-time graph in PHY 101 is a spacetime diagram.
The "points" on that diagram are called [as mentioned earlier] events. (Think of fingersnaps... not birthday parties.)

The distinction between Newtonian physics and relativistic physics occurs in the causal relationships between events on that diagram. (What events can influence other events...etc.) Eventually, those relationships can be codified and quantified by certain non-Euclidean geometries.

 

[WannabeNewton]

Probably the best way to get introduced to this stuff IMO (regardless of your math experience) is to check out Geroch's book "Relativity from A to B". He will talk about the differences between the causal structure of Galilean vs. Lorentzian space-times. robphy recommended the book to me some time ago and I loved it myself.

 

[atyy]

Before stepping off into special relativity and four-dimensional spaces...
it might be good to first recognize that the (x-position)-vs-time graph in PHY 101 is a spacetime diagram. The "points" on that diagram are called [as mentioned earlier] events. (Think of fingersnaps... not birthday parties.)

The distinction between Newtonian physics and relativistic physics occurs in the causal relationships between events on that diagram. (What events can influence other events...etc.) Eventually, those relationships can be codified and quantified by certain non-Euclidean geometries.

Probably the best way to get introduced to this stuff IMO (regardless of your math experience) is to check out Geroch's book "Relativity from A to B". He will talk about the differences between the causal structure of Galilean vs. Lorentzian space-times. robphy recommended the book to me some time ago and I loved it myself.

Here's a site that gives a brief introduction to spacetime as determining causal structure:
Space-time http://www.einstein-online.info/elementary/specialRT/spacetime
E=mc² http://www.einstein-online.info/elementary/specialRT/emc

 

[Phy_Man]

If you are just learning about special relativity, you don't need to worry about manifolds and other mathematical objects that are important to the study of the subject at some level, but impede your understanding of the basics.

Excellent point.

At a very basic level the term is used since time and space should not be considered separately and time behaves similarly to the normal cartesian coordinates (the signs are opposite, but you'll see this eventually).

That’s only true to a certain extent. While it’s true that in some sense space and time are treated on the same footing mathematically but not physically. I like the way Einstein stated it in his article A Brief Outline of the Development of the Theory of Relativity, Nature, February 17, 1921. On page783 Einstein writes

From this it follows that, in respect to its role in the equations of physics, though not with regard to its physical significance, time is equivalent to space co-ordinates (apart from the relations of reality). From this point of view, physics is, as it were, a Euclidean geometry of four-dimensions, or more correctly, a statics in a four-dimensional Euclidean continuum.

When Einstein uses the term Euclidean above it’s being used to describe the geometry of spacetime in special relativity, i.e. that it’s flat.

Tolman had a clever way to explain the nature of the difference between space and time. From Relativity, Thermodynamics and Cosmology by Richard C. Tolman, page 29

In using this language it is important to guard against the fallacy of assuming that all directions in the hyper-space are equivalent, and of assuming that the extension in time is of the same nature as extension in space merely because it may be convenient to think of them as plotted along perpendicular axes. A similar fallacy would be to assume that pressure and volume are the same kind of quantity because they are plotted at right angles in the diagram on a pv card. That there must be a difference between spatial and temporal axes in our hyper-space is made evident, by contrasting the physical possibility of rotating a metre stick from an orientation where it measures distance in the x-direction to one where it measures distances in the y-direction., with the impossibility of rotating it into a direction where it would measure time intervals – in other words the impossibility of rotating a clock into a rod.