Can someone please explain concept of spacetime in simple language? What is it and how it is important in the universe?
Wherever I have tried searching this concept, I have come across most complicated explanations.
A simple example will be appreciated.
The intuitive and traditional idea of space and time is that objects live in an infinite three-dimensional box, space, and that their motion in space happen in time in such a way that at each definite moment in time all objects have a position, and we can compare those positions because time flows the same for all objects.
Physicists discovered that there is no such box, and there is no such flow of time. This traditional space/time framework somewhat holds but only relatively to an object; it is not the same for all objects.
So there is no universal spatial background, and no universal time flow.
Spacetime is then the notion we use to still have a background after all. By forming a space (in the mathematical sense) combining traditional space and traditional time in an intricate way allowing space to rotate into time and the other way round, we can still get by with the idea that there is some smooth universal scene where everything happens.
The price to pay to see spacetime as a background is that this scene is completely static, sometimes called the block universe. But since all space and all time are intrinsically part of it, it actually cannot be conceived from an external point of view, and indeed Einstein's equations are strictly local and relational: they describe how the distribution of energy defines its own playground and how time and space can be seen in the way we are used to only instant by instant for specific observers, whose mutual perspectives are always shifting and transforming. In that view spacetime is far from static, it is more like a sort of fluid.
An experimentalist's answer:
In particle physics the data show that one is not dealing with a three dimensional space that has as a parameter time, but that space and time are united in a specific mathematical way in what is called a "four vector space".
In classical mechanics a vector in space is a three dimensional row or column with values from the field of real numbers that follow eucledian tranformation properties under translations and rotation.
Momentum for example $(p_x,p_y,p_z)$ are the three vector components whose length is $p$ and is given by
$$p=\sqrt{p_x^2+p_y^2+p_z^2}$$
Which is invariant under rotations and translation of the reference system.
Relativistic mechanics which hold for high energies and momenta, and which are necessary in order to make sense of the plethora of particle data, requires four vectors,
The momentum vector above is modified and called a four momentum ( $c=1$ in this):
$$(p_x,p_y,p_z,E)$$ .
In this the "length" of the four momentum becomes the $\sqrt{E^2-p_x^2-p_y^2-p_z^2}$ and is the mass of the particle under consideration, an invariant in all Lorenz transformation frames.
So in analogy to calling three dimensional space , space obeying euclidean rules, we call the four dimensional space of this special construction, which clarifies the transformation rules in particle interactions, space time to denote that one is dealing with the special fourvector quantities.
Spacetime is, like the name suggest, space and time together.
But there's more to it!
3D space isn't just horizontal 2D plane and height together. You can also rotate stuff in it, so for some 3D object, you don't have uniquely specified what is its height - this can change as you rotate it. Similar thing happens with the spacetime.
In 3D space, when you rotate stuff, you don't do much to it, you only change coordinate system. Distances stay the same. Square of the distance $s$ is given by \begin{equation} s^2=\Delta x^2+\Delta y^2+\Delta z^2 \end{equation}
In the spacetime, you can obviously rotate stuff, but you can also do the "rotations" which involve time. They are called "Lorentz transformations" (they are not quite rotations, see gif). So, once you have spacetime, space and time can get mixed up. What remains the same is the following quantity: \begin{equation} s^2=-c^2\Delta t^2+\Delta x^2+\Delta y^2+\Delta z^2 \end{equation} where $c$ is the speed of light.
One might ask "So what? This looks like just a fancy math/set of definitions/whatever...", but there are some interesting consequences. Relativity of simultaneity is one of them. In simple words, it means that (contrary to some philosophies of time) there is no uniquely defined "now" (except "here and now"), just like there is no uniquely defined "this much to the right" in space. Consequence of relativity of simultaneity is that faster than light (FTL) travel could in principle be used for time travel. However, so far we don't know of any laws of physics that would allow for FTL transfer of information, but that's another story.
Space and time is one single entity; we name it spacetime.
Figuratively speaking, events in spacetime is just like locations on earth; where northward and eastward are just like space and time.
You may ask why space and time have different units? As sometimes northward and eastward sometimes are written in different units (e.g. meters and mile), the same happens in spacetime, and the conversion factor between the meter and the second is light speed.
One of the tenets of relativity is to place both space and time on an equal footing. In special relativity, you will learn that not only like in Galilean relativity can one disagree on say velocities, but there is no absolute time - it is in fact also relative, motivating keeping time as having the same status as a spatial coordinate, rather than a parameter.
So, flat Minkowski spacetime is simply $\mathbb{R}^{1,n-1}$ with one timelike coordinate and $n-1$ spacelike coordinates, for an $n$-dimensional space. It has a special property however. If you think of distances in Euclidean space, we have that,
$$ds^2 = dx^2 + dy^2 + dz^2$$
as you are familiar with, from learning the Pythagorean theorem. However, in spacetime, we have to measure distances slightly differently, according to,
$$ds^2 = dt^2 - dx^2 - dy^2 - dz^2$$
or alternatively,
$$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$$
rather than taking the naive approach from Euclidean geometry. There are some profound implications from the fact that spacetime has signature $(1,n-1)$ as opposed to $(n,0)$, but this would be beyond the scope of your question.
As for why it is important, the concept of spacetime is used in the theory of gravitation known as general relativity and modern particle physics uses a framework called quantum field theory which is based on spacetime as well. However, recent developments suggest that this formulation may actually have drawbacks, but once again as for why is beyond the scope here.
Spacetime is a system in which neither the three yardsticks of space, nor the clock of time are invariant with respect to all observers and observed objects.
The appearance of yardsticks and clocks varies depending on the relative velocities of observers and observed objects, or on their acceleration, or on the position of an observer or an observed object within a gravitational field.
The four vectors (space and time) of spacetime are unified as events in each observer's reference frame, rather than as locations and times within a universal reference frame.
Whenever you draw a curve of a trajectory over time you are using spacetime as a concept. It is just a space with n+1 dimensions, where n is the number of space dimensions.
When we draw such diagrams on a piece of paper we usually omit one or two of the space dimensions (e.g. recording only the x position in space; y and z may be less important). That leaves room for using one axis for the time values of events.
That is, a 2-dimensional piece of paper can represent spacetime with n=1, in which you draw (x, t) curves. Similarly, a 3D diagram on paper or on a computer can represent subsets of spacetime with two explicit space dimensions.
Mathematically, this is just a Cartesian space of tuples of n+1 numbers. In these tuples you write the values of space coordinates, plus the time position, of events, and the set of these events forms a curve or surface in spacetime.
You can think of spacetime as a map. If you want to meet someone in this universe you have to agree an exact location and a time on the map. If you show up at correct location but at the wrong time then you are in the wrong place on the spacetime map.
