# Particles are a curve in space-time

What does geometry actually mean? |

Geometry is, to put it somewhat casually, the study of "**Intervals**We know the geometry of the from mathematics lessons **Drawing plane** (also **Euclidean plane** called): We know what the distance between two points is (hence we have a concept of the **length** a route and the **Arc length** a curve), and we know some elementary facts that occur in geometric figures such as triangles, squares or circles: for example the theorem about the sum of angles in a triangle, the Pythagorean theorem or the relationship u = 2pr between the circumference and radius of a circle. (In case you're wondering what **angle** have to do with lengths: The concept of angle is derived from the concept of length. In a triangle with given side lengths, all angles are uniquely determined. This - or the well-known story about the arc length of the circle sector - can be used to *define*what is an angle).

In order to better study such relationships, we use **Coordinate systems**. If, for example, two points A and B are given in the Euclidean plane, and a (Cartesian) coordinate system is selected, then two coordinates are assigned to each point. Are (x_{1}, y_{1}) the coordinates of A and (x_{2}, y_{2}) the coordinates of B, so play the **Coordinate differences** Dx = x_{2} - x_{1} and Dy = y_{2} - y_{1} an important role. Here is a diagram as a reminder:

If the distance between points A and B (i.e. the length of the segment AB) is denoted by Dl, then the equation applies

Dl^{2} = Dx^{ 2} + Dy^{ 2} . | (1) |

This follows from the Pythagorean theorem and is used every day when solving math problems. The distance Dlist is the (positive) square root of this expression.

The rule (1) is extremely important - it also bears the name **Metric** (or** Euclidean metric**): You can see it as the basis for the entire geometry of the Euclidean plane and use it to formally define the concept of distance (and also the concepts of the length of a line and the arc length of a curve) very precisely. She has a fantastic quality that we**Invariance under coordinate transformations**"call: If we decide, a *other*To use a twisted (Cartesian) coordinate system compared to the above, the coordinates are *the same* Points now through *other* Numbers given. We denote them with (x '_{1}, y '_{1}) for A and (x '_{2}, y '_{2}) for B. This is illustrated in the next diagram:

The method of calculating the square of the distance is, however *same* we in (1), because the Pythagorean theorem also applies in this situation: Dl^{2} = Dx '^{ 2} + Dy '^{ 2}. In whichever (Cartesian) coordinate system we form the "sum of the squares of the coordinate differences", the result is always the same:

Dx^{ 2} + Dy^{ 2 } = Dx '^{ 2} + Dy '^{ 2} . | (2) |

That is the mathematical justification for (1) as *Geometric quantity independent of the coordinate system*, to be interpreted as a square of distance.

But also that **three-dimensional** (**euclidean**) **room** we give a notion of distance by simply adding a coordinate. The metric that expresses spatial distances through coordinate differences is through

Dl^{2} = Dx^{ 2} + Dy^{ 2} + Double room^{ 2} . | (3) |

given, and the three-dimensional version of (2) is now called Dx^{ 2 }+ Dy^{ 2 }+ Double room^{ 2 } = Dx '^{ 2 }+ Dy '^{ 2 }+ Dz '^{ 2} , whereby the primed coordinates refer to any spatially rotated (Cartesian) coordinate system. We cannot imagine generalizations to higher-dimensional (Euclidean) spaces, but from the point of view of the basic equations they work according to the same pattern.

Equation (2) and its three-dimensional sister (3) express the freedom to represent the location of points *any* (Cartesian) coordinate systems to be used. This is particularly important where we have the **Applying geometry to physics**: Since there is no specific direction in space, there cannot be an excellent coordinate system either, and we must be free to orientate the coordinate axes arbitrarily (as long as they are normal to each other).

Equipped with regulations (1) and (3), we are able to work through geometrical facts *Calculations* analyze. Vector calculation, which is a powerful method for problem solving, can also be justified from them. The mathematical laws that follow from these regulations (and that of their higher-dimensional relatives) are collectively referred to as "**Euclidean geometry**" summarized.

(1) and (3) show that in mathematics we are free to use a concept of distance within a set *define*. But we can also give other sets a concept of distance, and one of these sets is our real topic: space-time.

The spacetime metric - a mathematical observation |

In the geometry of the (Euclidean) **level** or the three-dimensional (Euclidean) **Space** distances always mean **Lengths**. The special theory of relativity suggests that **Spacetime** give a structure that can also be called "distance concept", and which is based on **Lengths and times** relates.

First we leave out the two spatial coordinates y and z (i.e. we consider the two-dimensional model of space-time) and fix an inertial system. If two events A and B are given, space-time coordinates are assigned to them. Are (t_{1}, x_{1}) the coordinates of A and (t_{2}, x_{2}) the coordinates of B, so play the **Coordinate differences** Dt = t_{2} - t_{1} and Dx = x_{2} - x_{1} an important role. Here is the corresponding spacetime diagram:

If, in the case of Euclidean geometry, we have the freedom to use any (Cartesian) coordinate system, in special relativity we have the freedom to use a **any inertial system** to use. About the relationship between the space-time coordinates *of an event* in *two different inertial systems* we talked about the Lorentz transformation in the section. Let us contrast:

- In Eulidic geometry the relationship between two coordinate systems (if they have the same origin) is through one
**rotation**given. - In the special theory of relativity, the relationship between two inertial systems (if their clocks are set to zero at the moment when the spatial coordinate origins coincide) by one
**Lorentz transformation**given ..

Lorentz transformations therefore play a role analogous to rotations of the coordinate system. Like those, they allow one to switch between different frames of reference when describing a situation.

annotationFor those who want to know exactly: In the full (four-dimensional) theory, Lorentz transformations can also take place in other than the x-direction, and the two inertial systems involved can be spatially rotated with respect to one another, but we are usually content with this in this section the two-dimensional model of space-time (which is only characterized by a time coordinate t and a space coordinate x and therefore allows visualizations in the form of space-time diagrams). It contains everything that is essential for us.Furthermore, the two inertial systems can still be offset from one another (i.e. their spatial coordinate origins do not have to meet one another at all). In this case one speaks of

Poincaré transformations, analogous to the fact that Cartesian coordinate systems of Euclidean space do not have to have the same origin.

If we next to the original also one *other* consider moving inertial system, get *same* Events *other* Space-time coordinates, say (t '_{1}, x '_{1}) for A and (t '_{2}, x '_{2}) for B. A remarkable relationship now applies between the coordinate differences in the first and those in the second inertial system:

c^{2}German^{ 2}-Dx^{ 2 } = c^{2}Dt '^{ 2}-Dx '^{ 2} . | (4) |

With the help of the formulas

| (5a) (5b) |

and

| (6) |

this can easily be recalculated for the Lorentz transformation in the x direction. Basically, we already encountered relation (4) as a secondary result in the section on Bond's k-calculus. (For those who want to read up there and compare formulas: we only looked at the coordinates of a single event. The other one sat, so to speak, at the origin of the spacetime diagram).

Equation (4) says that the expression c^{2}German^{ 2 }- Dx^{ 2 }in *any one* Inertial system can be calculated - and the same thing comes out every time! It stands to reason to use an expression with this property **Lorentz invariant** (the two-dimensional theory). What does that mean? It means, in a modern language, that we have one here *Geometric quantity independent of the inertial system* have found. It only depends on the two events A and B, but not on the specific reference system that we have chosen for the calculation!

Since the form of the expression c^{2 }German^{ 2 }- Dx^{ 2 } legitimately to the square of the distance Dx^{ 2 }+ Dy^{ 2 } reminiscent of Euclidean geometry - see (1) - and since (4) is a similar (invariance) statement to (2), we give this quantity its name **Spacetime metric** (or simply **Metric**) and designate them with Ds^{2 }:

Ds^{2 }^{ } = c^{2}German^{ 2}-Dx^{ 2} . | (7) |

If the two omitted coordinates y and z are added again, the result is also

Ds^{2 }^{ } = c^{2}German^{ 2}-Dx^{ 2}-Dy^{ 2}-Dz^{ 2} | (8) |

the expression for the metric of the full, four-dimensional space-time. It also has the same value in every inertial system, so it represents one **Lorentz invariant** (the four-dimensional theory).

The importance of the spacetime metric |

Now that we have found such a beautiful structure, the assumption is obvious that it also has a physical meaning. Indeed it is. The sign of Ds^{2} tells us whether the two events A and B are related to each other **causal contact** can stand and what meaning Ds^{2} Has. We distinguish three cases in which A and B can be related to each other:

- Is
**D.****s**, so it is possible to get from one event to another with a speed v^{2 }> 0timely to each other. In this case there is an inertial system I 'in which they *at the same place*occur. (To find it graphically in the two-dimensional theory with coordinates (t, x), its t'-axis only has to be selected parallel to the segment AB). The earlier can influence the later event.

Ds^{2}is (except for a factor c^{2}) the square of the**Proper time**that passes for a clock that moves in a straight line uniformly from the earlier to the later event. (*Proof*s: In the inertial system I 'both take place at the same place, therefore Dx' = 0, from which Ds^{2}= c^{2}Dt '^{ 2}follows. But Dt 'is precisely the time that passes for the said clock).

- Is
**D.****s**, so it is not possible to get from one event to another with a velocity v £ c. One then says that the two events lie^{2 }< 0**space-like**to each other. In this case there is an inertial system I 'in which they*at the same time*occur. (To find it graphically in the two-dimensional theory with coordinates (t, x), its x'-axis only has to be selected parallel to the segment AB). Neither of the two events can influence the other.

Ds^{2}is the negative of the**spatial distance**between the two events, measured in the inertial system in which they take place simultaneously. He is sometimes too "**Own length**" called. (*proof*: In the inertial system I 'both events take place simultaneously, therefore Dt' = 0, from which Ds^{2}= - Dx '^{ 2 }- Dy '^{ 2 }- Dz '^{ 2 }follows).

The unfortunate spelling of Ds^{2}as*square*do not bump - it has proven itself as an analogy to Dl^{2}naturalized in (1), although it is actually wrong, because Ds^{2}can also be negative.

- Is
**D.****s**, a light signal (running in a straight line) can connect the two events. One then says that the two events lie^{2 }= 0**light-like**to each other. The earlier event can influence the later one with the help of a light signal.

In this case, the equation resulting from (8) has c^{2}German^{ 2}-Dx^{ 2}-Dy^{ 2}-Dz^{ 2 }= 0 an interesting meaning: If one of the two events is recorded, it represents a sphere that expands (or contracts) at the speed of light - the front of a spherical wave.

That’s the **the physical meaning of the space-time metric explained**. It represents the best that we consider to be **"Distance" between two events in spacetime** can define - depending on the causal situation of the events, it describes one time (proper) time, another time a spatial distance, each with precise (and meaningful) measurement rules. We present the three different meanings that the size Ds^{2} can assume, for two-dimensional spacetime in a diagram:

If the two events lie temporally (red) or space-like (blue) to one another, then Ds^{2} (except for the factor c^{2}) the proper time or the proper length. The third case (green) is recognized by the fact that Ds^{2} = 0, i.e. c^{2 }German^{ 2 }= Dx^{ 2} is - then the two events lie on a photon world line. (The fact that for such - light-like lying to one another - events Ds^{2} = 0, sometimes suggests that "there is no time for a photon").

We want to say a word about the causal relationships in space-time: If the event A is recorded, then the set of all events lying like light in A forms the **Cone of light** of A (whose interior is in the **Past light cone**- all events that can influence A - and the **Future light cone**- all events that can be influenced by A - decays). The amount outside the cone of light is also called that **presence** from A (all events that occur simultaneously with A in a suitable inertial frame):

The totality of all cones of light thus determines which events can be causally connected to one another (one also speaks of the** Causal structure of spacetime**).

So why "geometry"? |

The formal similarity of equations (7) or (8) with (1) or(3) suggests that we can "operate geometry" on the basis of the spacetime metric as well as on the basis of the Euclidean metric. This is actually the case:

In analogy to Euclidean geometry, the space-time metric (7) or (8) establishes the so-called **Minkowski geometry** (or **Lorentz geometry**, because of the minus sign in the metric, too **pseudo-Euclidean geometry** called). The minus signs in the metric mean that a lot looks different here than in the usual drawing plane or in three-dimensional space. The key point is common to both geometries: Certain expressions do not depend on the way of description (the choice of the reference system in which they are handled). You provide independent - just "**geometric**This view of space-time has many consequences, and it has changed physics profoundly over the last hundred years. We want to name just a few aspects to make this clear:

- The
**Relativity Principle**(see the section Postulates) demands that the physical laws in all inertial systems should have the same form. From the geometrical point of view this means that the basic equations of every physical theory should be expressed by geometrical objects of space-time, so that their independence from the way of description is ensured from the outset. This has now become a central "design" principle in modern physics. Geometric perspectives are not only applied to space-time today, but also to many other quantities that occur in physics (e.g. the "quantity of all electromagnetic field configurations").

- The formal similarities to Euclidean geometry often make it possible to use our geometrical notions. So (as we noted above) the
**Lorentz transformations**as analogies of**Rotations of coordinate systems**be understood. Or, to give another example: the - physically and mathematically a bit cumbersome - relativistic addition of speed can be used as a spatiotemporal variant of the**Theorem of the sum of angles in a triangle**be understood (as noted at the end of the relevant section)!

- We mentioned in the sections on the Lorentz contraction and the Lorentz transformation that
**spatial distances of events**, the*not at the same time*take place, cannot simply be read off in a spacetime diagram. If a moving inertial system is shown in a space-time diagram, the physical units of its ("inclined") axes do not match those of the plane of the drawing. (This is why we spoke of the "unit problem" in the two previous sections).

That may seem irritating at first. The matter can be explained very easily with the help of the term metric: It is simply the expression (7) for the metric, which (with the exception of a minus sign) represents the square of the physically measured distance! The plane is used to draw spacetime diagrams, but its Euclidean geometry does not match the geometry of spacetime. If one tries to find the analog of the "unit circle" in order to have a convenient representation of the units on all conceivable axes of moving inertial systems at hand, one comes across one**hyperbole**or in the full version on**Hyperboloid**, which is now in the form of the so-called "**Mass bowl**"has become quite indispensable in particle physics. With the help of the formula (7) you can easily convince yourself that all events affecting the points on the red hyperbolacorrespond to the spatial distance 1 from the origin O (i.e. from the event t = x = 0) (if it is measured in the inertial frame in which O and the event in question occur simultaneously). The x 'axis of a moving inertial system is shown as an example. This solves the "unit problem" of the x'-axes. (In an analogous way, the hyperbola is t

^{2}- x^{2}/ c^{2}= 1 for the**Proper times**, i.e. responsible for the units on the time axes - draw them!)

- In the section on the twin paradox and the geodesics of spacetime, a criterion for force-free movement was formulated ("The world line of a force-free movement is one
**Geodesic**"), which corresponds entirely to the spirit of the geometrical perspective. It can be transferred with profit to the general theory of relativity, in which the concepts of straightness and uniformity are no longer available, but a spatiotemporal concept of distance (the so-called**Riemannian metric**, one**curved**Version of the Minkowski metric) and hence the concept of geodesics.

- Geometric methods can be used to efficiently solve specific problems. The mathematics that allows us to do this was in part developed before the theory of relativity, but it was only with it that it conquered physics. For example, we can define objects that correspond to the vectors known from math lessons (the so-called
**Four-vectors**) to make our lives easier. The so-called**Tensor calculus**(and in general the area of**Differential geometry**) falls into the same category.

The geometric view of space-time only gains its full meaning in the general theory of relativity, but it has its roots in the special theory of relativity.

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