There is Hadith in Bukhari and Muslim regarding the forbidden time of prayers and it emphasizes not to pray during sunrise or sunset because the SUN is traveling in between the two hones of Satan. According to science every second the sun is rising and setting somewhere in the globe.
The question is - according to the above Hadith and science one cannot pray any time on this earth. Please shed some light on this.

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

The forbidden times of prayers are as follows:

-         From Fajr to sunrise.

-         Noontime.

-         From Asr until sunset

The evidence about this is the Hadith reported by Bukhari and Muslim: “There is no prayer after Fajr prayer until sunrise, and there is no prayer after Asr prayer until sunset.”

Imam Muslim reported the Hadith narrated by Uqba Bin Amir who said: “The Prophet (Sallallahu Alaihi wa Sallam) forbade us to pray or to bury our dead during 3 particular times: when the sun rises until it goes high in the horizon, when the sun is at its meridian until its goes past the meridian and the shadow of objects starts to appear, and when the sun sets until it is totally down.”

As regards your statement that modern science affirmed that every second the Sun rises on one part of the world and sets on another, this scientific reality is known.  Allah Says (interpretation of meaning): {….He makes the night to go in the day and makes the day to go in the night…..}[39:5].

The Earth is round and it is continuously revolving around the Sun.  Therefore the Sun would be rising and setting continuously on the different parts of the world.  There is no contradiction between this and the forbiddance of praying at sunrise and sunset.  It is forbidden for anyone to pray at those times according to the place where he is located.  The five daily prayers are obligatory on each individual person according to the place where he is.  When the starting time of the prayer begins, it is obligatory on him to pray.  This of course differs from one place to another.  For instance, in some places it is time for Zuhr prayer, while in some other places, it is time for Isha prayer.

Your saying that it is not permissible for a person, who knows this reality to pray, is rejected.  Dhuha (forenoon) for instance is not connected to any of the five daily prayers.  According to this scientific reality, despite the fact that the Sun rises every second on some parts of the world, Dhuha still exists along the whole day on the globe. Does this mean that whoever knows this reality it is not permissible for him to pray the five daily prayers for the whole day.

In conclusion, the Hadith forbids the prayer during these times; this is in relation to the person's location, or according to sunrise or sunset in relation to him, like the five daily prayers.

Allah knows best.