Not caring is an empty, dead feeling. Not being concerned is a very alive vibrant feeling that can come by keeping the awareness in the chest with no other thought, fully present. One is full and the other is empty.
For this to be a full feeling, training and practice of repeating a word mantra (a mantra is any word that has an emotional impact or value to you) to get your mind focused and being able to sense within your body is necessary to be able to consistently do this on demand or have as a lasting or frequent state.
The slightest touch of negative emotions turns it from full to empty and miserable.
Negative emotions empty your heart from any connection to the fullness and awareness of not being concerned and turn it into not caring.
The experience of not being concerned happens without effort spontaneously on its own, when you have quieted your mind from being concerned. It is important to be clear about this; the effort is in focusing your mind, there is no effort in not being concerned.
When you connect through full attention and awareness in the heart then the peace and fullness develops. The level you attain is proportionate to the strength of your mind not to drift and allow negative emotions or false beliefs to enter – refer to the article; ‘Negative Emotions Are False and How to End Them’
To get the state of not caring, you have to put in effort to ‘not care’ that is active mental effort to not care. It may happen automatically by negative thoughts, but you can put in effort to make yourself not care about something that you would normally care about.
Not being concerned happens on its own spontaneously if you have your mind in your heart, purely focused there without any other thoughts or worries. It happens when you realize the emptiness of life and your own existence.
Of course, this now sounds terrible, who would want to think that their life is pointless? That existence that you are so occupied with is your ego, a fragment, it is not you and it is not free from the limited life destined to end in a short time. Not being concerned can only happen when you are no longer fixated on this life as your entire existence, that you know or at least accept that there is more than this and your conscious thoughts are merely a fragment of your reality. Your reality is not concerned about this life or anything that happens in it because it knows that it is far more.
You can take time to contemplate the possibility of other levels of existence. You can call that God, angels, spirits, anything that can exist beyond the material realm. It is something that your 5 senses cannot grasp, yet you know for certain exists.
If there is such a thing, then it must be probable that you are also of such a nature and more than limited to just this body.
When you can feel that to be true, if you do the exercise of objective observation of any situation, or simply when you are not doing anything, to contemplate the very moment and nothing more, no thoughts of a past event or future plan, that moment alone, you will experience not being concerned.
Only when you experience it, will you understand.