Deuteronomy 21:4

and the sages of that city shall bring the heifer down into an uncultivated field, where the ground hath not been tilled by work, nor sowed; and let them there behead the heifer from behind her with an axe (or knife, dolch) in the midst of the field.

uncultivated rough.

there they shall break the neck [literally: they shall neck] i.e., they will remove the heifer’s neck.

AND THE ELDERS OF THAT CITY SHALL BRING DOWN THE HEIFER UNTO A VALLEY ‘EITHAN’ — “which is ‘hard,’ one that has never been tilled.” This is Rashi’s language. And our Rabbis have said that the field becomes forbidden from ever being tilled or sown. If so, [the words which is neither tilled nor sown] are admonitions that he who sows or cultivates [a rough valley in which a heifer’s neck has been broken] transgresses this negative commandment. The meaning of the verse is thus that the elders should take forth the heifer to [what will forever become] “a rough valley” because it will never again be tilled or sown. The meaning of the expression And the priests the sons of Levi shall come near is in order to recite Forgive, O Eternal, Thy people Israel, whom Thou hast redeemed etc. And the purport of the verses is as follows: And the priests the sons of Levi shall come near; and all the elders of that city, who are nearest unto the slain man, shall wash their hands; and they shall speak and say: ‘Our hands have not shed this blood.’ The washing of hands is to be done only by some of the [elders] mentioned; so also their speaking [‘Our hands have not shed etc.’] is done by some of them, similar to what is stated, And thou shalt eat before the Eternal thy G-d etc. the tithe of thy corn, of thy wine, and of thine oil, and the firstlings of thy herd etc. [meaning “he who is fit to eat,” for it cannot mean that all are to eat, for the Second Tithe may be eaten by all ritually clean Israelites, while the firstlings are eaten only by the priests; here, too, the washing of hands and speaking are done only by some, but not all, of the elders].
Now on the matter of breaking the heifer’s neck Rabbi Abraham ibn Ezra explained that G-d commanded it to be done for the city which is nearest unto the slain man because had they not committed a similar sin, it would not have happened that a man be killed near them, and G-d’s thoughts are infinitely profound to us. But the Rabbi [Moshe ben Maimon] said in the Moreh Nebuchim that the reason for the breaking of the heifer’s neck is to discover the murderer and to cleanse [the guilt for the innocent] blood. In most cases the murderer comes from the place which is round about him that is slain, and when the elders go out and engage in the measuring of the cities, and the elders of the nearest city then bear witness before the Creator that they were not …

אל נחל איתן, where people do not normally walk.

וערפו, a discreet death, not one drawing attention to itself. The animal does not experience pangs of death, not knowing of its impending fate. Naturally, people knew even less about this impending procedure. This procedure parallels what the murderer had done when he ambushed an unsuspecting victim in a location which was isolated so that the court never found out who the murderer was. Had the court known of the identity of the murderer they would have dealt with him instead of this heifer having to be killed.