Space time :Mathematics of Universe

We experience and observe three dimensions of space generally like left-right, top-down, back-forward in our everyday life that we called those as spatial dimensions that are on ordinary space. It is not quite complete in case of universal space. We need a fourth value to specify the occurance of an event specially in universe. We consider that fourth one is time which doesn’t behave like other dimensions and extends beyond length, width and height and called as fourth dimension that is not a spatial one rather called temporal dimension. The four dimensional space diagram was first used by H. Minkowski in the year 1908 often we called world space or Minkowski space.

Showing Minkowski Space (relativity.com)

"The views of space and time which i wish to lay before you have sprung from the soil of experimental physics, and therein lies there strength. They are radical. Henceforth space by itself, and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."- Hermann Minkowski, 1908

If we consider two events, we can determine the quantity Δs² between the two events and we find that it is invariant in any inertial frame. The quantity Δs is known as the space-time interval between two events.

Considering space and time together and taking into account that the speed of light must be constant in any frame. We find the line element to satisfy such conditions is

ds² = c²dt² - dx² - dy² - dz²

There are three possibilities for invariant quantity Δs².                                                                               .

•    Timelike Interval

The relation is  Δs²<0:Δx²<c²Δt², two events can be casually connected and the interval is said to be timelike.

•     Spacelike Interval

The relation is Δs²>0:Δx²>c²Δt² and no signal can travel fast enough to connect the two events. The events are not casually connected said to have a spacelike separation.

•     Lightlike Interval

The relation is Δs²=0:Δx²=c²Δt² and the two events can be connected only by a light signal. The events are said to have a lightlike separation.                                        

We normally think that space and time behave totally independent. But Einstein put forward a theory that infact linked these together and that called space-time. A mathematical model was given which is a totally four dimensional structure that is a combination of three dimensional space and one dimensional time. Massive objects curved space time. This means that distances between two points and also times between two events are wrapped around massive objects and this has a strange consequences like time ticking differently on different planets and massless photons being bent by massive objects because of that space-time curvature.

The equation that describe all this –

${\displaystyle R_{\mu \nu }-{\tfrac {1}{2}}Rg_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }.}$

The left side of the equation tells about the curvature of space-time and right side of equation is a tensor relates the mass energy formula. The total expression reveals that the energy-momentum tensor is proportional to how the curvature (on left side expression) bends.

R_μν is the Ricci tensor defines the bending of space-time axis, g_μν is a metric tensor used to observe and measure the length of object in that geometry. 1/2 R is the correction factor introduced.

Positive and negative curvature:

Positive curvature in space time is caused by the gravity of mass and energy. Spheres we can say have positive curvature while saddles have negative curvature shown in figure below.

 

Showing positive and negative curvature (commons.wikimedia.org)

The theory of space-time explains the object moving with speed of light with the motion of massive object which is a relativistic effect predicted by Einstein which was one of the part of his theory named theory of relativity.       

Working of space-time:

A sheet of rubber best way that describes about the space-time curvature as a overview but it’s not actually if we discuss in analogy, the sheet of rubber is a two dimensional while space-time is four dimensional. Massive objects create curvature that is the cause to bend and objects move along the bending curvature. Gravity; a curved field described by an area of space under the influence of force in the space time fabric that is actually created by the presence of mass of matter and the motion of gravity occurs due to the twists and turns of curvature of space-time.

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The mission Gravity probe B (GP-B) was a mission to measure the shape of space-time vertex around the Earth and the mission used a telescope and four spherical gyroscopes and found according to the prediction given by Einstein.

"A human being is a part of the whole called by us universe; a part limited in time and space. He experiences himself, his thoughts and his feelings as something separate from the rest - a kind of optical delusion of consciousness."- Albert Einstein